Attractors of Hamilton nonlinear partial differential equations

TL;DR

This paper surveys the theory of attractors in nonlinear Hamiltonian PDEs, discussing stability, solitons, and numerical results, and proposes a new conjecture linking these attractors to fundamental quantum phenomena.

Contribution

It provides a comprehensive survey of attractor theory in Hamiltonian PDEs and introduces a novel conjecture connecting attractors to quantum mechanics interpretations.

Findings

Results on global attraction to stationary states and solitons

Numerical simulations illustrating attractor behavior

A new conjecture linking attractors to quantum phenomena

Abstract

We survey the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. These are results on global attraction to stationary states, to solitons and to stationary orbits, on adiabatic effective dynamics of solitons and their asymptotic stability. Results of numerical simulation are given. The obtained results allow us to formulate a new general conjecture on attractors of $G$ -invariant nonlinear Hamiltonian partial differential equations. This conjecture suggests a novel dynamical interpretation of basic quantum phenomena: Bohr's transitions between quantum stationary states, wave-particle duality and probabilistic interpretation.

Equations

$$\psi (x,t)\to S(x),t\to +\infty .$$

$$\ddot{\psi }(x,t)=\Delta \psi (x,t)+F(\psi (x,t)),x\in {\mathbb{R}}^n,$$

$${\int }_{|x|<R}[|\dot{\psi }(x,t){|}^2+|\nabla \psi (x,t){|}^2+|\psi (x,t){|}^2]dx\to 0,t\to \pm \infty$$

$$\psi (x,t)\sim e^{g_{\pm }t}{\psi }_{\pm }(x),t\to \pm \infty .$$

$$\psi (x,t)\to S_{\pm }(x),t\to \pm \infty$$

$$\psi (x,t)\sim {\psi }_{\pm }(x-v_{\pm }t),t\to \pm \infty ,$$

$$\psi (x,t)\sim \sum _k{\psi }_{\pm }(x-v_{\pm }^{k}t)+w_{\pm }(x,t),t\to \pm \infty ,$$

$$\psi (x,t)\sim {\psi }_{\pm }(x)e^{-i{\omega }_{\pm }t},t\to \pm \infty$$

$$i\dot{\psi }(x,t)=-\Delta \psi (x,t)+V(x)\psi (x,t),x\in {\mathbb{R}}^n$$

$$\psi (x,t)=\sum C_k{\psi }_k(x)e^{-i{\omega }_{k}t}+{\int }_0^{\infty }C(\omega )\psi (\omega ,x)e^{-i\omega t}d\omega ,$$

$$\ddot{\psi }(x,t)={\psi }^{\prime \prime }(x,t),x\in \mathbb{R}.$$

$$\dot{\psi }(t)=D_{\pi }\mathcal{H},\dot{\pi }(t)=-D_{\psi }\mathcal{H}$$

$$\mathcal{H}(\psi ,\pi )=\frac{1}{2}\int [|\pi (x){|}^2+|{\psi }^{\prime }(x){|}^2]dx,(\psi ,\pi )\in {\mathcal{E}}_c:=H_c^1(\mathbb{R})\oplus [L^2(\mathbb{R})\cap L^1(\mathbb{R})],$$

$$\Vert \psi {\Vert }_{H_c^1(\mathbb{R})}:=\Vert {\psi }^{\prime }{\Vert }_{L^2(\mathbb{R})}+|\psi (0)|.$$

$$\psi (x)\to C_{\pm },x\to \pm \infty .$$

$$\psi (x,t)\to S_{\pm }(x)=\frac{C_{+}+C_{-}}{2}\pm \frac{1}{2}{\int }_{-\infty }^{\infty }\pi (y)dy,t\to \pm \infty ,$$

$$\dot{\psi }(x,t)=\frac{{\psi }^{\prime }(x+t)-{\psi }^{\prime }(x-t)}{2}+\frac{\pi (x+t)+\pi (x-t)}{2}\to 0,t\to \pm \infty ,$$

$$\ddot{\psi }(x,t)={\psi }^{\prime \prime }(x,t)+\chi (x)F(\psi (x,t)),x\in \mathbb{R},$$

$$\chi \in C_0^{\infty }(\mathbb{R}),\chi (x)\ge 0,\chi (x)\not\equiv 0;F(\psi )=-\nabla U(\psi ),\psi \in {\mathbb{R}}^N;U(\psi )\in C^2({\mathbb{R}}^N).$$

$$\mathcal{H}(\psi ,\pi )=\frac{1}{2}\int [|\pi (x){|}^2+|{\psi }^{\prime }(x){|}^2+\chi (x)U(\psi (x,t))]dx,(\psi ,\pi )\in {\mathcal{E}}_c^N={\mathcal{E}}_c\otimes {\mathbb{R}}^N.$$

$$U(\psi )\to \infty ,|\psi |\to \infty .$$

$$\mathcal{H}(Y(t))=\mathrm{const},t\in \mathbb{R}.$$

$$\Vert (\psi ,\pi ){\Vert }_{{\mathcal{E}}_c^N,R}=\Vert {\psi }^{\prime }{\Vert }_R+|\psi (0)|+\Vert \pi {\Vert }_R,R=1,2,\dots ,$$

$$\mathrm{dist}[Y_1,Y_2]=\sum _1^{\infty }{2}^{-R}\frac{\Vert Y_1-Y_2{\Vert }_{{\mathcal{E}}_c^N,R}}{1+\Vert Y_1-Y_2{\Vert }_{{\mathcal{E}}_c^N,R}},Y_1,Y_2\in {\mathcal{E}}_c^N.$$

$${\psi }^{\prime \prime }(x)+\chi (x)F(\psi (x))=0,x\in \mathbb{R}.$$

$$Y(t)\stackrel{{\mathcal{E}}_F^N}{⟶}\mathcal{S},t\to \pm \infty$$

$$\mathrm{dist}[Y(t),\mathcal{S}]:=\underset{S\in \mathcal{S}}{\inf }\mathrm{dist}[Y(t),S]\to 0,t\to \pm \infty .$$

$$Y(t)\stackrel{{\mathcal{E}}_F^N}{⟶}{S}_{\pm }\in \mathcal{S},t\to \pm \infty .$$

$${\int }_0^{\infty }[|\dot{\psi }(-a,t){|}^2+|{\psi }^{\prime }(-a,t){|}^2+|\dot{\psi }(a,t){|}^2+|{\psi }^{\prime }(a,t){|}^2]dt<\infty .$$

$$\psi (\pm a,t)\sim C_{\pm },{\psi }^{\prime }(\pm a,t)\sim 0,t\to \infty .$$

Full text

The obtained results allow us to formulate a new general conjecture on attractors of $G$ -invariant nonlinear Hamiltonian partial differential equations.   This conjecture suggests a novel  dynamical interpretation of basic quantum phenomena: Bohr’s transitions between quantum stationary states, de Broglie’s wave-particle duality and Born’s probabilistic interpretation.

**Attractors of Hamilton nonlinear

**

partial differential equations

A.I. Komech 111 Supported partly by Austrian Science Fund (FWF) P28152-N35

Institute for Information Transmission Problems RAS

akomech@iitp.ru

E.A. Kopylova 222 Supported partly by grant of RFBRa 18-01-00524

Institute for Information Transmission Problems RAS

ek@iitp.ru

To the memory of Mark Vishik

Key words: Hamilton equations; nonlinear partial differential equations; wave equation; Maxwell equations; Klein – Gordon equation; principle of limiting amplitude; principle of limiting absorption; attractor; steady states; soliton; stationary orbits; adiabatic effective dynamics; symmetry group; Lee group;  Schrödinger equation; quantum transitions; wave-particle duality.

1. 1 Introduction

2. 2 Global attraction to stationary states

3. 2.1 1D nonlinear wave equations

4. 2.2 String coupled to nonlinear oscillators

5. 2.3 Wave-particle system

6. 2.3.1 Liénard-Wiechert asymptotics

7. 2.3.2 Free wave equation

8. 2.3.3 Scattering of energy to infinity

9. 2.3.4 Convolution representation and relaxation of acceleration and velocity

10. 2.3.5 A compact attracting set

11. 2.3.6 Global attraction

12. 2.4 Maxwell–Lorentz equations: radiation damping

13. 2.5 Wave equation with concentrated nonlinearities

14. 2.6 Remarks

15. 3 Global attraction to solitons

16. 3.1 Translation-invariant wave-particle system

17. 3.1.1 Canonical transformation and reduced system

18. 3.1.2 Orbital stability of solitons

19. 3.1.3 Strong Huygens principle and soliton asymptotics

20. 3.2 Invariance of symplectic structure

21. 3.3 Translation-invariant Maxwell-Lorentz system

22. 3.4 The case of weak interaction

23. 4 Adiabatic effective dynamics of solitons

24. 4.1 Wave-particle system with a slowly varying external potential

25. 4.2 Mass–Energy equivalence

26. 5 Global attraction to stationary orbits

27. 5.1 Method of omega-limit trajectories

28. 5.2 Spectral representation and limiting absorption principle

29. 5.3 Nonlinear analogue of Kato’s theorem

30. 5.4 Splitting onto dispersion and bound components

31. 5.5 Omega-compactness

32. 5.6 Reduction of spectrum of omega-limit trajectories to spectral gap

33. 5.7 Reduction of spectrum of omega-limit trajectories to a single point

34. 5.7.1 Equation for omega-limit trajectories and spectral inclusion

35. 5.7.2 Titchmarsh convolution theorem

36. 5.8 Remarks on dispersion radiation and nonlinear energy transfer

37. 6 Asymptotic stability of stationary orbits and solitons

38. 6.1 Asymptotic stability of stationary orbits. Orthogonal projection

39. 6.2 Asymptotic stability of solitons. Symplectic projection

40. 6.3 Generalizations and Applications

41. 6.4 Further generalizations

42. 6.5 Linear dispersion

43. 7 Numerical Simulation of Soliton Asymptotics

44. 7.1 Kinks of relativistic-invariant Ginzburg–Landau equations

45. 7.2 Numerical observation of soliton asymptotics

46. 7.3 Adiabatic effective dynamics of relativistic solitons

47. A Attractors and Quantum Mechanics

48. A.1 On dynamical interpretation of quantum jumps

49. A.2 Bohr’s postulates by perturbation theory

50. A.3 Conclusion

1 Introduction

This paper is a survey of the results on long time behaviour and attractors for nonlinear Hamilton partial differential equations that appeared since 1990.

Theory of attractors for nonlinear PDEs originated from the seminal paper of Landau [159] published in 1944, where he suggested the first mathematical interpretation of the onset of turbulence as the growth of the dimension of attractors of the Navier–Stokes equations when the Reynolds number increases.

The foundation for corresponding mathematical theory was laid in 1951 by Hopf who established for the first time the existence of global solutions to the 3D Navier–Stokes equations [74]. He introduced the ‘‘method of compactness’’ which is a nonlinear version of the Faedo-Galerkin approximations. This method relies on a priori estimates and Sobolev embedding theorems. It has strongly influenced the development of the theory of nonlinear PDEs, see [165].

Modern development of the theory of attractors for general dissipative systems, i.e. systems with friction (the Navier–Stokes equations, nonlinear parabolic equations, reaction-diffusion equations, wave equations with friction, etc.), as originated in the 1975–1985’s in the works of Foias, Hale, Henry, Temam, and others [55, 68, 73], was developed further in the works of Vishik, Babin, Chepyzhov, and others [8, 29]. A typical result of this theory in the absence of external excitation is global convergence to stationary states: for any finite energy solution to dissipative autonomous equation in a region $\Omega\subset\mathbb{R}^{n}$, there is a convergence

[TABLE]

Here $S(x)$ is a stationary solution with suitable boundary conditions, and this convergence holds as a rule in the $L^{2}(\Omega)$-metric. In particular, the relaxation to an equilibrium regime in chemical reactions is due to the energy dissipation.

A development of a similar theory for Hamiltonian PDEs seemed unmotivated and impossible in view of energy conservation and time reversal for these equations. However, as it turned out, such a theory is possible and its shape was suggested by a novel mathematical interpretation of fundamental postulates of quantum theory:

I. Transitions between quantum stationary orbits (Bohr 1913).

II. Wave-particle duality (de Broglie 1924).

III. Probabilistic interpretation (Born 1927).

Namely, postulate I can be interpreted as global attraction (1.8) of all quantum trajectories to an attractor formed by stationary orbits (see Appendix), and postulate II can be interpreted as decay into solitons (1.7). The probabilistic interpretation also can be justified by the asymptotics (1.7). More details can be found in [104].

Investigations of the 1990–2019’s suggest that such long time asymptotics of solutions are in fact typical for nonlinear Hamiltonian PDEs. These results are presented in this article. This theory differs significantly from the theory of attractors of dissipative systems where the attraction to stationary states is due to an energy dissipation caused by a friction. For Hamiltonian equations the friction and energy dissipation are absent, and the attraction is caused by radiation which irrevocably brings the energy to infinity.

The modern development of the theory of nonlinear Hamiltonian equations dates back to Jörgens [89], who has established the existence of global solutions for nonlinear wave equations of the form

[TABLE]

developing the Hopf method of compactness. The subsequent studies in this direction were well reflected by J.-L. Lions [165].

First results on the long time asymptotics of solutions to nonlinear Hamiltonian PDEs were obtained by Segal [190, 192], Morawetz and Strauss [176, 177, 201]. In these papers local energy decay is proved for solutions to equations (1.2) with defocusing type nonlinearities $F(\psi)=-m^{2}\psi-\kappa|\psi|^{p}\psi$, where $m^{2}\geq 0$, $\kappa>0$, and $p>1$. Namely, for sufficiently smooth and small initial states, one has

[TABLE]

for any finite $R>0$. Moreover, the corresponding nonlinear wave and scattering operators are constructed. In the works of Strauss [202, 203], the completeness of scattering is established for small solutions to more general equations. The decay (1.3) means that the energy escapes each bounded region for large times.

For convenience, characteristic properties of all finite energy solutions to an equation will be referred to as global, in order to distinguish them from the corresponding local properties for solutions with initial data sufficiently close to an attractor.

All the above-mentioned results on local energy decay (1.3) mean that the corresponding local attractor of small initial states consists of the zero point only. First results on global attraction for nonlinear Hamiltonian PDEs were obtained by one of the authors in the 1991–1995’s for 1D models [95, 97, 98], and were later extended to nD equations. Let us note that global attraction to a (proper) attractor is impossible for any finite-dimensional Hamiltonian system because of energy conservation.

Global attraction for Hamiltonian PDEs is derived from an analysis of the irreversible energy radiation to infinity, which plays the role of the dissipation. Such analysis requires subtle methods of harmonic analysis: the Wiener Tauberian theorem, the Titchmarsh convolution theorem, the theory of quasi-measures, the Paley-Wiener estimates, eigenfunction expansions for nonselfadjoint Hamiltonian operators based on M.G. Krein theory of $J$-selfadjoint operators, and others.

The results obtained so far indicate a certain dependence of long-time asymptotics of solutions on symmetry group of an equation: for example, it may be the trivial group $G=\{e\}$, or the unitary group $G=U(1)$, or the group of translations $G=\mathbb{R}^{n}$. Namely, the results suggest the conjecture that for ‘‘generic’’ nonlinear Hamilton autonomous PDEs with a Lie symmetry group $G$, any finite energy solution admits the asymptotics

[TABLE]

Here, $e^{g_{\pm}t}$ is a representation of one-parameter subgroup of the symmetry group $G$ which corresponds to the generators $g_{\pm}$ from the corresponding Lie algebra, while $\psi_{\pm}(x)$ are some ‘‘scattering states’’ depending on the considered trajectory $\psi(x,t)$. Both pairs $(g_{+},\psi_{+})$ and $(g_{-},\psi_{-})$ are solutions to the corresponding nonlinear eigenfunction problem.

In the case of the trivial symmetry group, the conjecture (1.4) means global attraction to the corresponding stationary states

[TABLE]

(see Fig. 1), where $S_{\pm}(x)$ depend on considered trajectory $\psi(x,t)$, and the convergence holds in local seminorms, i.e., in norms of type $L^{2}(|x|<R)$ with any $R>0$. The convergence (1.5) in global norms (i.e., corresponding to $R=\infty$) cannot hold due to the energy conservation.

In particular, the asymptotics (1.5) can be easily demonstrated for the d’Alembert equation, see (2.1)– (2.7). In this example the convergence (1.5) in global norms obviously fails due to the presence of travelling waves $f(x\pm t)$. Similarly, a solution to 3D wave equation with a unit propagation velocity is concentrated in spherical layers $|t|-R<|x|<|t|+R$ if initial data has a support in the ball $|x|\leq R$. Therefore, the solution converges to zero when $t\to\pm\infty$, although its energy remains constant. This convergence corresponds to the well-known strong Huygens principle. Thus, attraction to stationary states (1.5) is a generalization of the strong Huygens principle to non-linear equations. The difference is that for linear wave equation the limit is always zero, while for non-linear equations the limit can be any stationary solution.

Further, in the case of symmetry group of translations $G=\mathbb{R}^{n}$ asymptotics (1.4) means global attraction to solitons (traveling waves)

[TABLE]

for generic translation-invariant equation. In this case the convergence holds in local seminorms in the comoving frame of reference, that is, in $L^{2}(|x-v_{\pm}t|<R)$ for any $R>0$. The validity of such local asymptotics in comoving reference systems suggests that there may be several such solitons, which provide the refined asymptotics

[TABLE]

where $w_{\pm}$ are some dispersion waves, being solutions to corresponding free equation, and convergence holds now in some global norm. A trivial example gives the d’Alembert equation (2.1) with solutions $\psi(x,t)=f(x-t)+g(x+t)$.

Asymptotics with several solitons (1.7) were discovered first in 1965 by Kruskal and Zabusky in numerical simulations of the Korteweg–de Vries equation (KdV). Later on, global asymptotics of this type were proved for nonlinear integrable translation-invariant equations (KdV and others) by Ablowitz, Segur, Eckhaus, van Harten, and others using the method of inverse scattering problem [48].

Finally, for the unitary symmetry group $G=U(1)$, asymptotics (1.4) mean global attraction to ‘‘stationary orbits’’ (or ‘‘solitary waves’’)

[TABLE]

in the same local seminorms (see Fig. 3). These asymptotics were inspired by Bohr’s postulate on transitions between quantum stationary states (see Appendix for details). Our results confirm such asymptotics for generic $U(1)$-invariant nonlinear equations of type (5.4) and (5.16)–(5.18). More precisely, we have proved global attraction to the manifold of all stationary orbits, though the attraction to a particular stationary orbitы, with fixed $\omega_{\pm}$, is still open problem.

The existence of stationary orbits $\psi(x)e^{i\omega t}$ for a broad class of $U(1)$-invariant nonlinear wave equations (1.2) was extensively studied in the 1960–1980’s. The most general results were obtained by Strauss, Berestycki and P.-L. Lions [17, 18, 200]. Moreover, Esteban, Georgiev and Séré constructed stationary orbits for nonlinear relativistically-invariant Maxwell–Dirac equations (A.5). The orbital stability of stationary orbits has been studied by Grillakis, Shatah, Strauss and others [62, 63].

Let us emphasize that we conjecture asymptotics (1.8) for generic $U(1)$-invariant equations. This means that long time behavior of solutions may be quite different for $U(1)$-invariant equations of ‘‘positive codimension’’. In particular, for solutions to linear Schrödinger equation

[TABLE]

the asymptotics (1.8) generally fail. Namely, any finite energy solution admits the spectral representation

[TABLE]

where $\psi_{k}$ and $\psi(\omega,\cdot)$ are corresponding eigenfunctions of discrete and continuous spectrum, respectively. The last integral is a dispersion wave which decays to zero in the norms $L^{2}(|x|<R)$ with any $R>0$ (under appropriate conditions on the potential $V(x)$). Respectively, the attractor is the linear span of the eigenfunctions $\psi_{k}$. Thus, the long-time asymptotics does not reduce to a single term like (1.8), so the linear case is degenerate in this sense. Let us note that our results for equations (5.4) and (5.16)–(5.18) are established for strictly nonlinear case: see condition (5.12) below, which eliminates linear equations.

For more sophisticated symmetry groups $G=U(N)$, asymptotics (1.4) mean the attraction to $N$-frequency trajectories, which can be quasi-periodic. In particular, the symmetry groups $SU(2)$, $SU(3)$ and others were suggested in 1961 by Gell-Mann and Ne’eman for strong interaction of baryons [60, 179]. The suggestion relies on discovered parallelism between empirical data for the baryons, and the ‘‘Dynkin scheme’’ of Lie algebra $su(3)$ with $8$ generators (the famous ‘‘eightfold way’’). This theory resulted in the scheme of quarks and in the development of the quantum chromodynamics [3, 67], and in the prediction of a new baryon with prescribed values of its mass and decay products. This particle, the $\Omega^{-}$-hyperon, was promptly discovered experimentally [12].

This empirical correspondence between Lie algebra generators and elementary particles presumably gives an evidence in favor of the general conjecture (1.4) for equations with Lie symmetry groups.

Let us note that our conjecture (1.4) specifies the concept of ‘‘localized solution/coherent structures’’ from ‘‘Grande Conjecture’’ and ‘‘Petite Conjecture’’ of Soffer [194, p.460] in the context of $G$-invariant equations. The Grande Conjecture is proved in [128] for 1D wave equation coupled to nonlinear oscillator (2.20). Moreover, a suitable versions of the Grande Conjecture are also proved in [81, 82] for 3D wave, Klein–Gordon and Maxwell equations coupled to relativistic particle with sufficiently small charge (3.34); see Remark 3.12. Finally, for any matrix symmetry group $G$, (1.4) implies the Petite Conjecture since the localized solutions $e^{g_{\pm}t}\psi_{\pm}(x)$ are quasiperiodic then.

Below we dwell upon available results on the asymptotics (1.5)–(1.8). In Sections 2 and 3 we review results on global attraction  to stationary states and to solitons, respectively. Section 4.1 concerns adiabatic effective dynamics of solitons, and Section 4.2 concerns the mass-energy equivalence. In Section 5 we give a concise complete proof of the attraction to stationary orbits. Sections 6.1 and 6.2 concern asymptotic stability of stationary orbits and solitons, and Section 6.3 – various generalizations. In Section 7 we present results of numerical simulation of soliton asymptotics for relativistic-invariant equations. In Appendix we comment on the relations between general conjecture (1.4) and Bohr’s postulates in Quantum Mechanics.

In conclusion let us comment on previous related surveys in this area. The survey [100] presents the results only for 1D equations. The results on asymptotic stability of solitons were described in detail in [77] for linear equations coupled to a particle, and in [139] – for relativistic-invariant Ginzburg–Landau equations. In present article we give only a short statement of these results (Sections 2.1, 2.2 and 6.3). Finally, present survey gives much more information on our methods than [103]. Our main novelties are as follows:

i) Streamlined and simplified proofs of the results [134, 133, 132] on global attraction to stationary states and to solitons for systems of relativistic particle coupled to scalar wave equation and to the Maxwell equation. These results give the first rigorous justification of famous radiation damping in Classical Electrodynamics. We omit unessential technical details, but explain carefully our approach which relies on the Wiener Tauberian Theorem in Sections 2.3, 2.4 and 3.1.

ii) The complete proof of nonlinear analog of the Kato theorem on the absence of embedded eigenvalues (Section 5.3) which is a crucial point in the proof of global attraction to stationary orbits for $U(1)$-invariant equations in [101, 105, 106, 109, 107, 108, 110, 111, 140, 141, 145, 146, 32, 33].

iii) The informal arguments on the dispersion radiation and the nonlinear spreading of spectrum (Section 5.8) which mean the nonlinear energy transfer from lower to higher harmonics and lie behind our application of the Titchmarsh Convolution theorem.

iv) Recent results [140, 141, 145, 146] on global attraction for nonlinear wave, Klein-Gordon and Dirac equations with concentrated nonlinearities. We give a detailed survey of the methods and results in Section 2.5.

These methods and ideas are presented here for the first time in review literature.

Acknowledgments

The authors express a deep gratitude to H. Spohn and B. Vainberg for long-time collaboration on attractors of Hamiltonian PDEs, as well as to A. Shnirelman for many useful long-term discussions. We are also grateful to V. Imaykin and A. A. Komech for collaboration lasting many years.

2 Global attraction to stationary states

In this section we review the results on global attraction to stationary states (1.5) that were received in 1991-1999 for nonlinear Hamiltonian PDEs. The first results of this type were obtained for one-dimensional nonlinear wave equations [95, 97, 98, 99, 100]. Later on these results were extended to three-dimensional wave equations and Maxwell’s equations coupled to a charged relativistic particle [134, 133], and also to the three-dimensional wave equations with concentrated nonlinearity. In [47, 208] the attraction (1.5) was established for finite systems of oscillators coupled to an infinite-dimensional thermostat.

The global attraction (1.5) can be easily demonstrated on trivial (but instructive) example of the d’Alembert equation

[TABLE]

All derivatives here and below are understood in the sense of distributions. This equation is formally equivalent to the Hamilton system

[TABLE]

with Hamiltonian

[TABLE]

where $H^{1}_{c}({\mathbb{R}})$ is the space of continuous functions $\psi(x)$ with finite norm

[TABLE]

Let moreover,

[TABLE]

For such initial data $(\psi(x,0),\dot{\psi}(x,0))=(\psi(x),\pi(x))\in{\cal E}_{c}$ the d’Alembert formula gives

[TABLE]

where the convergence is uniform on every finite interval $|x|<R$. Moreover,

[TABLE]

where convergence holds in $L^{2}(-R,R)$ for each $R>0$. Thus, the attractor is the set of states $(\psi(x),\pi(x))=(C,0)$, where $C\in{\mathbb{R}}$ is any constant. Note that the limits (2.6) for positive and negative times may be different.

2.1 1D nonlinear wave equations

In [99], global attraction to stationary states has been proved for nonlinear wave equations of type

[TABLE]

where

[TABLE]

The equation (2.8) can be formally written as Hamilton’s system (2.2) with Hamiltonian

[TABLE]

We assume that the potential is confining, i.e.

[TABLE]

In this case, it is easy to prove that a finite energy solution $Y(t)=(\psi(t),\pi(t))\in C({\mathbb{R}},{\cal E}_{c}^{N})$ exists and is unique for any initial state $Y(0)\in{\cal E}_{c}^{N}$, and the energy is conserved:

[TABLE]

Definition 2.1**.**

i) ${\cal E}_{F}^{N}$ denote the space ${\cal E}_{c}^{N}$ endowed with the seminorms

[TABLE]

*where $\|\cdot\|_{R}$ denotes the norm in $L^{2}_{R}:=L^{2}(-R,R)$.

ii) The convergence in ${\cal E}_{F}^{N}$ is equivalent to the convergence in every seminorm (2.12).*

The space ${\cal E}_{F}^{N}$ is not complete, and the convergence in ${\cal E}_{F}^{N}$ is equivalent to the convergence in the metric

[TABLE]

The main result of [99] is the following theorem, which is illustrated by the Figure 1. Denote by ${\cal S}$ the set of stationary states $(\psi(x),0)\in{\cal E}_{c}^{N}$, where $\psi(x)$ is the solution of the stationary equation

[TABLE]

Theorem 2.2**.**

i)* Let conditions (2.9) and (2.10) hold. Then any finite energy solution $Y(t)=(\psi(t),\pi(t))\in C({\mathbb{R}},{\cal E}_{c}^{N})$ attracts to ${\cal S}$:*

[TABLE]

in the metric (2.13). It means that

[TABLE]

ii)* Suppose additionally that the function $F(\psi)$ is real-analytic for $\psi\in{\mathbb{R}}^{N}$. Then ${\cal S}$ is a discrete subset of ${\cal E}_{c}^{N}$, and for any finite energy solution $Y(t)=(\psi(t),\pi(t))\in C({\mathbb{R}},{\cal E}_{c}^{N})$*

[TABLE]

Sketch of the proof. It suffices to consider only the case $t\to\infty$. Our proof of (2.14) and (2.16) in [99] relies on new method of omega-limit trajectories which is a development of the method of omega-limit points used in [98] . Later on this method played an essential role in the theory of global attractors for $U(1)$-invariant PDEs [101, 105, 106, 109, 107, 108, 110, 111, 140, 145, 146, 32, 33].

First, we note that the finiteness of energy radiated from the segment $[-a,a]\supset{\rm supp{\hskip 1.42262pt}{\hskip 1.42262pt}}\chi$ implies the finiteness of the ‘‘dissipation integral’’ [99, (6.3)]:

[TABLE]

This means, roughly, that

[TABLE]

More precisely, the functions $\psi(\pm a,t)$ and $\psi^{\prime}(\pm a,t)$ are slowly varying for large times, so their shifts form omega-compact families. Namely, from an arbitrary sequence $s_{k}\to\infty$, one can choose a subsequence $s_{k^{\prime}}\to\infty$ for which

[TABLE]

where the constants $C_{\pm}$ depend on the subsequence, and the convergence holds in $C[0,T]$ for any $T>0$. It remains to prove that

[TABLE]

in $C(0,T;H^{1}[-a,a])$ for any $T>0$. In other words, each omega limit trajectory is a stationary state.

Roughly speaking we need to justify the correctness of the boundary value problem for a nonlinear differential equation (2.8) in the half-strip $-a\leq x\leq a$, $t>0$, with the Cauchy boundary conditions (2.17) on the sides $x=\pm a$. Then the convergence (2.18) of boundary values implies the convergence (2.19) of the solution inside the strip.

Our main idea is to use evident symmetry of wave equation with respect to interchange of variables $x$ and $t$ with a simultaneous change of the sign of the potential $U$. However, in this equation with the ‘‘time’’ $x$ the condition (2.10) makes new potential $-U$ unbounded from below! Consequently, this dynamics with $x$ as the time variable is not correct on the interval $|x|\leq a$. For example, in the case $U(\psi)=\psi^{4}$, the equation (2.8) for solutions of type $\psi(x,t)=\psi(x)$ reads $\psi^{\prime\prime}(x)-4\psi^{3}(x)=0$. Solutions of this ordinary equation with finite Cauchy’s initial data at $x=-a$ can become infinite at any point $x\in(-a,a)$. However, in our situation a local correctness is sufficient due to a priori bounds, which follow from energy conservation (2.11) by the conditions (2.9) and (2.10).

Remark 2.3**.**

i) The energy of the limit states $S_{\pm}$ may be less than the conserved energy of the corresponding solution. This limit jump of energy is similar to the well-known property of the norm for weak convergence of a sequence in the Hilbert space.

ii) The discreteness of the set ${\cal S}$ is essential for the asymptotics (2.16). For example, convergence (2.16) fails for the solution $\psi(x,t)=\sin[\log(|x-t|+2)]$ in the case when $F(\psi)=0$ for $|\psi|\leq 1$. **

2.2 String coupled to nonlinear oscillators

I. First results on global attraction to stationary states (2.14) and (2.16) were established in [95, 97, 128] for the case of point nonlinearity (‘‘Lamb system’’):

[TABLE]

This equation describes transversal oscillations of a string with vector displaysments $\psi(t)\in{\mathbb{R}}^{N}$ coupled to an oscillator attached at $x=0$, and acting on the string with a force $F(\psi(0,t))$ orthogonal to the string; $m>0$ is the mass of a particle attached to the string at the point $x=0$. For linear force function $F(\psi)=-k\psi$, such system was first considered by H. Lamb [157].

The conserved energy reads

[TABLE]

We denote $Z:=\{z\in\mathbb{R}^{N}:F(z)=0\}$. Obviously, every finite energy stationary solution of the equation (2.20) is a constant function $\psi_{z}(x)=z\in Z$. Let us denote by ${\cal S}$ the manifold of all finite energy stationary states,

[TABLE]

This set is discrete in ${\cal E}_{c}$, if $Z$ is discrete in $\mathbb{R}^{N}$. Now the proof of attractions (2.14) and (2.16) relies on the reduced equation for the oscillator

[TABLE]

where $\dot{w}_{\rm in}\in L^{2}(0,\infty)$. This equation follows from the d’Alembert representation for the solution $\psi(x,t)$ at $x>0$ and $x<0$.

In [128] a stronger asymptotics in the global norm of the Hilbert space ${\cal E}_{c}$ are obtained instead of asymptotics (2.16) in local seminorms. This is achieved by identifying the corresponding d’Alembert outgoing and incoming waves. In [129, 130] the asymptotic completeness of the corresponding nonlinear scattering operators has been proved.

II. In [98] we have extended the results [95, 97] on global attraction to stationary states to the case of a string with several oscillators:

[TABLE]

This equation reduces to a system of $M$ ordinary equations with delay. Its study required new approach relying on a special analysis of omega-limit points of trajectories.

Note that detailed proofs of all results [95, 97, 98, 99] are available in the survey [100].

2.3 Wave-particle system

In [134], the first result on global attraction to stationary states (1.5) is obtained  for three-dimensional real scalar wave field coupled to a relativistic particle. The scalar field satisfies 3D wave equation

[TABLE]

where $\rho\in C^{\infty}_{0}(\mathbb{R}^{3})$ is a fixed function representing the charge density of the particle, and $q(t)\in\mathbb{R}^{3}$ is the particle position. The particle motion obeys the Hamilton equation with relativistic kinetic energy $\sqrt{1+p^{2}}$:

[TABLE]

Here $-\nabla V(q)$ is external force corresponding to real potential $V(q)$, and the integral term is a self-force. Thus, wave function $\psi$ is generated by charged particle, and plays the role of a potential acting on the particle, along with the external potential $V(q)$.

The system (2.22)–(2.23) can formally be represented in Hamiltonian form

[TABLE]

with Hamiltonian (energy)

[TABLE]

By $\|\cdot\|$ we denote the norm in the Hilbert space $L^{2}:=L^{2}({\mathbb{R}}^{3})$, and $\|\cdot\|_{R}$ denotes the norm in $L^{2}(B_{R})$, where $B_{R}$ being the ball $|x|\leq R$. Let $\mathaccent 28695{H}^{1}:=\mathaccent 28695{H}^{1}({\mathbb{R}}^{3})$ be the completion of the space $C_{0}^{\infty}({\mathbb{R}}^{3})$ in the norm $\|\nabla\psi(x)\|$.

Definition 2.4**.**

i) ${\cal E}:=\mathaccent 28695{H}^{1}\oplus L^{2}\oplus\mathbb{R}^{3}\oplus\mathbb{R}^{3}$ is the Hilbert phase space of tetrads $(\psi,\pi,q,p)$ with finite norm

[TABLE]

ii) ${\cal E}_{\sigma}$ for $\sigma\in{\mathbb{R}}$ is the space of $Y=(\psi,\pi,q,p)\in{\cal E}$ with $\psi\in C^{2}({\mathbb{R}}^{3})$ and $\pi\in C^{1}({\mathbb{R}}^{3})$ satisfying the estimate

[TABLE]

iii) ${\cal E}_{F}$ is the space ${\cal E}$ with metric of type (2.13), where the corresponding seminorms are defined as

[TABLE]

Obviously, the energy (2.25) is a continuous functional on ${\cal E}$, and ${\cal E}_{\sigma}\subset{\cal E}$ for $\sigma>3/2$. The convergence in ${\cal E}_{F}$ is equivalent to the convergence in every seminorm (2.27). We assume the external potential be confining:

[TABLE]

In this case the Hamiltonian (2.25) is bounded below:

[TABLE]

where

[TABLE]

The following lemma is proved in [134, Lemma 2.1].

Lemma 2.5**.**

*Let  $V(q)\in C^{2}(\mathbb{R}^{3})$ satisfies the condition (2.30).  Then for any initial state $Y(0)\in{\cal E}$ there exists a unique  finite energy solution $Y(t)=(\psi(t),\pi(t),q(t),p(t))\in C(\mathbb{R},{\cal E})$, and

i) for every $t\in{\mathbb{R}}$ the map $W_{t}:Y_{0}\mapsto Y(t)$ is continuous both on ${\cal E}$ and on ${\cal E}_{F}$;

ii) the energy ${\cal H}(Y(t))$ is conserved, i.e.*

[TABLE]

iii) a priori estimates hold

[TABLE]

iv) if (2.28) holds, then also

[TABLE]

Remark 2.6**.**

In the case of point particle $\rho(x)=\delta(x)$, the system (2.22)–(2.23) is incorrect, since in this case any solution of the wave equation (2.22) is singular at the point $x=q(t)$, and, accordingly, the integral in (2.23) is not defined. Energy functional (2.25) in this  case is not bounded from below, because the integral in (2.29) diverges and is equal to $-\infty$.  Indeed, in the Fourier transform, this integral has the form

[TABLE]

where $\hat{\rho}(k)\equiv 1$. This is the famous ‘‘ultraviolet divergence.’’ Thus, the self-energy of point charge is infinite, that suggested Abraham to introduce the model of an ‘‘extended electron’’ with a continuous charge density $\rho(x)$.   **

Denote $Z=\{q\in{\mathbb{R}}^{3}:\nabla V(q)=0\}$. It is easy to verify that stationary states of the system (2.22)–(2.23) have the form $S_{q}=(\psi_{q},0,q,0)$, where $q\in Z$ and $\Delta\psi_{q}(x)=\rho(x-q)$. Therefore, $\psi_{q}(x)$ is the Coulomb potential

[TABLE]

Respectively, the set of all stationary states of this system is

[TABLE]

If the set $Z\subset{\mathbb{R}}^{N}$ is discrete, then the set ${\cal S}$ is also discrete in ${\cal E}$ and in ${\cal E}_{F}$. Finally, assume that the ‘‘form-factor’’ $\rho$ satisfies Wiener’s condition

[TABLE]

Remark 2.7**.**

The Wiener condition means a strong coupling of scalar wave field $\psi(x)$ to the particle. It is a suitable version of the ‘‘Fermi Golden Rule’’ for the system (2.22)–(2.23): the perturbation $\rho(x-q)$ is not orthogonal to all eigenfunctions of continuum spectrum of the Laplacian $\Delta$. **

For simplicity of the exposition we assume that

[TABLE]

The main result of [134] is as follows.

Theorem 2.8**.**

i)* Let the conditions (2.28) and (2.34) hold, and $\sigma>3/2$. Then for any initial state $Y(0)=(\psi_{0},\pi_{0},q_{0},p_{0})\in{\cal E}_{\sigma}$ the corresponding solution $Y(t)=(\psi(t),\pi(t),q(t),p(t))\in C({\mathbb{R}},{\cal E})$ to the system (2.22)–(2.23) attracts to the set of stationary states:*

[TABLE]

*where attraction holds in the metric (2.13) defined by the seminorms (2.27).

ii) Let, additionally, the set $Z$ be discrete in $\mathbb{R}^{3}$. Then*

[TABLE]

The key point in the proof of this is the relaxation of the acceleration

[TABLE]

This relaxation has long been known in classical electrodynamics as ‘‘radiation damping’’. Namely, the Liénard–Wiechert formulas for retarded potentials suggest that a particle with a non-zero acceleration radiates energy to infinity. The radiation cannot last forever, because the total energy of the solution is finite. These arguments result in the conclusion (2.38) that can be found in any textbook on classical electrodynamics.

However, rigorous proof is not so obvious and it was done for the first time in [134]. The proof relies on calculation of total energy amount radiated to infinity using the Liénard-Wiechert formulas. The central point is the representation of this amount in the form of a convolution and subsequent application of the Wiener Tauberian theorem.

Below we give a streamlined version of this proof for $t\to+\infty$.

Remark 2.9**.**

i) The condition (2.28) is not necessary for relaxation   (2.38). The relaxation also takes place under the condition (2.30) (see Remark 2.12).

ii) The Wiener condition   (2.34) also is not necessary for relaxation   (2.38). For example,   (2.38) obviously holds in the case when $V(x)\equiv 0$ and $\rho(x)\equiv 0$. More generally, such relaxation also holds when $V(x)\equiv 0$ and the norm $\|\rho\|$ is sufficiently small, see (3.34). **

2.3.1 Liénard-Wiechert asymptotics

Let us recall long range asymptotics of the Liénard-Wiechert potentials [134, 133]. Denote by $\psi_{r}(x,t)$ the retarded potential

[TABLE]

and set $\pi_{r}(x,t)=\dot{\psi}_{r}(x,t)$. Denote $T_{r}:=\overline{q}_{0}+R_{\rho}$.

Lemma 2.10**.**

The following asymptotics hold

[TABLE]

uniformly in $t\in[T_{r},T]$ for any $T>T_{r}$. Here $\omega(x)=x/|x|$, and $\overline{\pi}(\omega(x),t)$ is given in (2.42).

Proof.

The integrand of (2.39) vanishes for $|y|>T_{r}$. Then $|x-y|\leq t$ for $t-|x|>T_{r}$, and (2.39) implies

[TABLE]

because $\displaystyle n=\frac{x-y}{|x-y|}=\omega(x)+{\cal O}(|x|^{-1})$ for bounded $|y|$. Hence, it suffices to prove asymptotics (2.40) for $\pi_{r}$ only. We have

[TABLE]

Replacing $t$ by $|x|+t$ in definition of $\tau$, we obtain

[TABLE]

since

[TABLE]

Hence (2.41) implies (2.40) with

[TABLE]

∎

2.3.2 Free wave equation

Consider now the solution $\psi_{K}(x,t)$ of free wave equation with initial conditions

[TABLE]

The Kirchhoff formula gives

[TABLE]

Here $S_{t}(x)$ is the sphere $\{y:~{}|y-x|=t\}$. Denote $\pi_{K}(x,t)=\dot{\psi}_{K}(x,t)$.

Lemma 2.11**.**

Let $Y_{0}\in{\cal E}_{\sigma}$. Then for any $R>0$ and any $T_{2}>T_{1}\geq 0$

[TABLE]

Proof.

Formula (2.44) implies

[TABLE]

Here $S_{1}:=S_{1}(0)$. From (2.26) it follows that

[TABLE]

Therefore,

[TABLE]

The integral with $\nabla\pi_{K}(x,t)$ can be estimated similarly. ∎

2.3.3 Scattering of energy to infinity

Now we obtain a bound on the total energy radiated to infinity which we will represent as a ‘‘radiation integral’’. This integral has to be bounded a priori by (2.32). Indeed, the energy ${\cal H}_{R}(t)$ at time $t\in{\mathbb{R}}$ in the ball $B_{R}$ is defined by

[TABLE]

Consider the energy $I_{R}(T_{1},T_{2})$ radiated from the ball $B_{R}$ during the time interval $[T_{1},T_{2}]$ with $T_{2}>T_{1}>0$:

[TABLE]

This energy is bounded a priori, because by (2.32) the energy ${\cal H}_{R}(T_{1})$ is bounded from above, while ${\cal H}_{R}(T_{2})$ is bounded from below. Thus,

[TABLE]

where $I$ does not depend on $T_{1}$, $T_{2}$ and $R$. Further, one has

[TABLE]

Hence, (2.46) implies

[TABLE]

The solution admits the splitting $\pi=\pi_{r}+\pi_{K}$, $\psi=\psi_{r}+\psi_{K}$, and hence,

[TABLE]

Lemmas 2.10 and 2.11 together with the Cauchy-Schwarz inequality imply

[TABLE]

where $I_{1}<\infty$ does not depend on $T$ and $R$. Taking the limit $R\to\infty$ and then $T\to\infty$ we obtain the finiteness of the energy radiated to infinity:

[TABLE]

2.3.4 Convolution representation and relaxation of acceleration and velocity

Applying a partial integration in (2.42), we obtain

[TABLE]

The function $\overline{\pi}(\omega,t)$ is globally Lipschitz continuous in $\omega$ and $t$ due to (2.32) Hence, (2.47) implies

[TABLE]

uniformly in $\omega\in S_{1}$. Denote $r(t)=\omega\cdot q(t)$, $s=\omega\cdot y$, $\tilde{\rho}(q_{3})=\displaystyle\int dq_{1}dq_{2}\rho(q_{1},q_{2},q_{3})$ and decompose the $y$-integration in (2.48) along and transversal to $\omega$. Then

[TABLE]

Here $\theta=\theta(\tau)=\tau-r(\tau)$ is a nondegenerate diffeomorphism of ${\mathbb{R}}$ since $\dot{r}\leq\overline{r}<1$ due to (2.32), and

[TABLE]

Let us extend $q(t)=0$ for $t<0$. Then $\tilde{\rho}*g_{\omega}\,(t)$ is defined for all $t$, and coincides with $\overline{\pi}(\omega,t)$ for sufficiently large $t$. Hence, (2.49) reads as a convolution limit

[TABLE]

Moreover, $g^{\prime}_{\omega}(\theta)$ is bounded by (2.32). Therefore, (2.51) and the Wiener condition (2.34) imply

[TABLE]

by Pitt’s extension of the Wiener Tauberian Theorem, cf. [189, Thm. 9.7(b)]. Hence, (2.50) implies

[TABLE]

since $\theta(t)\to\infty$ as $t\to\infty$. Finally,

[TABLE]

since $|q(t)|\leq\overline{q}_{0}$ due to (2.32).

Remark 2.12**.**

*(i) We have used condition (2.28) in the proof of (2.46). However, (2.30) at this point is also sufficient. Hence, the relaxation (2.53) holds also under condition (2.30).

(ii) For point charge $\rho(x)=\delta(x)$, (2.51) implies (2.52) directly.

(iii) Condition (2.34) is necessary for the implication (2.52)$\Rightarrow$(2.53). Indeed, if (2.34) is violated, then $\hat{\rho}_{a}(\xi)=0$ for some $\xi\in{\mathbb{R}}$, and with the choice $g(\theta)=\exp(i\xi\theta)$ we have $\rho_{a}*g(t)\equiv 0$ whereas $g$ does not decay to zero.*

2.3.5 A compact attracting set

Here we show that the set

[TABLE]

is an attracting subset. It is compact in ${\cal E}_{F}$ since ${\cal A}$ is homeomorphic to a closed ball in ${\mathbb{R}}^{3}$.

Lemma 2.13**.**

The following attraction holds,

[TABLE]

Proof.

We need to check that for every $R>0$

[TABLE]

We estimate each summand separately.

i) $|p(t)|\to 0$ as $t\to\infty$ by (2.53).

ii) $\inf\limits_{|q|\leq\overline{q}_{0}}|q(t)-q|=0$ for any $t\in{\mathbb{R}}$ by (2.32).

iii) (2.39) implies for $t>R+T_{r}$ and $|x|<R$

[TABLE]

The integral in the RHS is bounded uniformly in $t>R+T_{r}$ and $x\in B_{R}$. Hence, $\|\pi_{r}(t)\|_{R}\to 0$ as $t\to\infty$ by (2.54). Then also $\|\pi(t)\|_{R}\to 0$.

iv) Obviously, we can replace $q$ with $q(t)$ in the last summand in (2.57). Then for $t>R+T_{r}$ and $|x|<R$, one has

[TABLE]

by (2.39). Moreover, $\rho(y-q(t-|x-y|))-\rho(y-q(t))\to 0$ as $t\to\infty$ uniformly in $x\in B_{R}$ due to (2.54). Hence, $\|\psi_{r}(t)-\psi_{q(t)}\|_{R}\to 0$ as $t\to\infty$. Then also $\|\psi(t)-\psi_{q(t)}\|_{R}\to 0$. Finally, $\|\nabla(\psi(t)-\psi_{q(t)})\|_{R}$ can be estimated in a similar way. ∎

2.3.6 Global attraction

Now we complete the proof of Theorem 2.8.

i) Let $Y(t)\in C({\mathbb{R}},{\cal E})$ be any finite energy solution to the system (2.22)–(2.23). If the attraction (2.36) does not hold, there is a sequence $t_{k}\to\infty$ for which

[TABLE]

Since ${\cal A}$ is a compact set in ${\cal E}_{F}$, (2.56) implies that

[TABLE]

for some subsequence $k^{\prime}\to\infty$. It remains to check that $\overline{Y}=S_{q_{*}}\in{\cal S}$ with some $q_{*}\in Z$, since this contradicts (2.58).

First, $\overline{Y}=S_{q}$ with some $|q|\leq\overline{q}_{0}$  by the definition (2.55). Similarly,  by the continuity of the map $W_{t}$ in ${\cal E}_{F}$,

[TABLE]

where $Q(\cdot)\in C^{2}({\mathbb{R}},{\cal E})$, since $W_{t}\overline{Y}\in C({\mathbb{R}},{\cal E})$ is a solution to the system (2.22)–(2.23). Finally, for $S_{Q(t)}$ to be a solution to the system (2.22)–(2.23), there must be $\dot{Q}(t)\equiv 0$. Therefore, $Q(t)\equiv q_{*}\in Z$ and $\overline{Y}=S_{q_{*}}\in\cal S$.

ii) If the set $Z$ is discrete in ${\mathbb{R}}^{3}$, then solitary manifold $\cal S$ is discrete in ${\cal E}_{F}$. $\Box$

2.4 Maxwell–Lorentz equations: radiation damping

In [133] global attraction to stationary states similar to (2.36), (2.37) was extended to the Maxwell–Lorentz equations with charged relativistic particle:

[TABLE]

Here $\rho(x-q)$ is the particle charge density, $\dot{q}\rho(x-q)$ is the corresponding current density, and $E^{\rm ext}=-\nabla\phi^{\rm ext}(x)$ and $B^{\rm ext}=-{\rm rot{\hskip 1.42262pt}}A^{\rm ext}(x)$ are external static Maxwell fields. Similarly to (2.28), we assume that effective scalar potential is confining:

[TABLE]

This system describes classical electrodynamics with “extended electron” introduced by Abraham [1, 2]. In the case of a point electron, when $\rho(x)=\delta(x)$, such system is not well defined. Indeed, in this case, any solutions $E(x,t)$ and $B(x,t)$ of Maxwell’s equations (the first line of (2.61)) are singular for $x=q(t)$, and, accordingly, the integral in the last equation (2.61) does not exist.

This system may be formally presented in Hamiltonian form, if the fields are expressed in terms of potentials $E(x,t)=-\nabla\phi(x,t)-\dot{A}(x,t)$, $B(x,t)=-{\rm rot{\hskip 1.42262pt}}A(x,t)$, [79]. The corresponding Hamilton functional reads

[TABLE]

The Hilbert phase space of finite energy states is defined as ${\cal E}:=L^{2}\oplus L^{2}\oplus{\mathbb{R}}^{3}\oplus{\mathbb{R}}^{3}$. Under the condition (2.62) a solution $Y(t)=(E(x,t),B(x,t),q(t),p(t))\in C({\mathbb{R}},{\cal E})$ of finite energy exists and is unique for any initial state $Y(0)\in{\cal E}$.

The Hamiltonian (2.63) is conserved along solutions, what provides a priori estimates, which play an important role in proving an attraction of the type (2.36), (2.37) in [133]. Key role in the proof of relaxation of an acceleration plays again (2.38), which is derived by a suitable generalization of our methods [134]: the expression of energy radiated to infinity via Liénard–Wiechert retarded potentials, its representation in the form of a convolution and the use of Wiener’s Tauberian theorem.

In classical electrodynamics the relaxation (2.38) known as radiation damping. It is traditionally derived from the Larmor and Liénard formulas for radiation power of a point particle (see formulas (14.22) and (14.24) of [86]), but this approach ignores field feedback although it plays the key role in the relaxation. The main problem is that this reverse field reaction for point particles is infinite. A rigorous sense of these classical calculations was first found in [134, 133] for the Abraham model of ‘‘extended electron’’ under the Wiener condition (2.34). Details can be found in [199].

2.5 Wave equation with concentrated nonlinearities

Here we prove the result of [180] on global attraction to solitary manifold for 3D wave equation with point coupling to an $\mathbf{U}(1)$-invariant nonlinear oscillator. This goal is inspired by fundamental mathematical problem of an interaction of point particles with the fields.

Point interaction models were first considered since 1933 in the papers of Wigner, Bethe and Peierls, Fermi and others (see [6] for a detailed survey) and of Dirac [42]. Rigorous mathematical results were obtained since 1960 by Zeldovich, Berezin, Faddev, Cornish, Yafaev, Zeidler and others [19, 34, 61, 214, 216], and since 2000 by Noja, Posilicano, Yafaev and others [180, 215, 4].

We consider real wave field $\psi(x,t)$ coupled to a nonlinear oscillator

[TABLE]

where $G(x)=\displaystyle\frac{1}{4\pi|x|}$ is the Green’s function of the operator $-\Delta$ in ${\mathbb{R}}^{3}$. Nonlinear function $F(\zeta)$ admits a potential:

[TABLE]

We assume that the potential is confining, i.e.,

[TABLE]

Еhe system (2.64) admits stationary solutions $\psi_{q}=qG(x)\in L^{2}_{loc}({\mathbb{R}}^{3})$, where $q\in Q:=\{q\in R:F(q)=0\}$. We assume that the set $Q$ is nonempty and does not contain intervals, i.e.,

[TABLE]

for any $a<b$.

As before, $\|\cdot\|$ and $\|\cdot\|_{R}$ denote the norms in $L^{2}=L^{2}({\mathbb{R}}^{3})$ and in $L^{2}(B_{R})$ respectively, and $\mathaccent 28695{H}^{1}=\mathaccent 28695{H}^{1}({\mathbb{R}}^{3})$ is the completion of the space $C_{0}^{\infty}({\mathbb{R}}^{3})$ in the norm $\|\nabla\psi(x)\|$. Denote

[TABLE]

We define the function sets

[TABLE]

and

[TABLE]

Obviously, $D\subset\dot{D}$.

Definition 2.14**.**

${\cal D}$* is the Hilbert manifold of states $\Psi=(\psi,\pi)\in D\times\dot{D}$.*

First, we prove global well-posedness for the system (2.64).

Theorem 2.15**.**

Let conditions (2.65) and (2.66) hold. Then

1. (i)

For every initial data $\Psi_{0}=(\psi_{0},\pi_{0})\in{\cal D}$ the system (2.64) has a unique solution $\Psi(t)=(\psi(t),\dot{\psi}(t))\in C({\mathbb{R}},{\cal D})$. 2. (ii)

The energy is conserved:

[TABLE] 3. (iii)

The following a priori bound holds

[TABLE]

Proof.

It suffices to prove the theorem for $t\geq 0$.

Step i) First we consider free wave equation with initial data from $\cal D$:

[TABLE]

where $(\psi_{0,reg},\pi_{0,reg})\in\mathaccent 28695{H}^{2}\oplus\mathaccent 28695{H}^{1}$.

Lemma 2.16**.**

There exists a unique solution $\psi_{f}(t)\in C([0;\infty),L^{2}_{loc})$ to (2.70). Moreover, for any $t>0$ there exists the limit

[TABLE]

and

[TABLE]

Proof.

We split $\psi_{f}(x,t)$ as

[TABLE]

where $\psi_{f,reg}$ and $g$ are solutions to free wave equation with initial data $(\psi_{0,reg},\pi_{0,reg})$ and $(\zeta_{0}G,\eta_{0}G)$, respectively. First, $\psi_{f,reg}\in C([0,\infty),\mathaccent 28695{H}^{2})$ by the energy conservation. Hence, $\lim\limits_{x\to 0}\psi_{f,reg}(x,t)$ exists for any $t\geq 0$ since $\mathaccent 28695{H}^{2}({\mathbb{R}}^{3})\subset C({\mathbb{R}}^{3})$.

Let us obtain an explicit formula for $g$. Note, that the function $h(x,t)=g(x,t)-(\zeta_{0}+\eta_{0}t)G(x)$ satisfies

[TABLE]

The unique solution to (2.72) is spherical wave :

[TABLE]

Here $\theta$ is the Heaviside function. Hence,

[TABLE]

and then

[TABLE]

Finally, $\dot{\psi}_{f,reg}(0,t)\in L^{2}_{loc}([0,\infty))$ by [180, Lemma 3.4]. Hence, (2.71) follows. ∎

Step ii) Now we prove local well-posedness. We modify the nonlinearity $F$ so that it becomes Lipschitz-continuous. Define

[TABLE]

We may pick a modified potential function $\tilde{U}(\zeta)\in C^{2}({\mathbb{R}})$, so that

[TABLE]

and the function $\tilde{F}(\zeta)=\tilde{U}^{\prime}(\zeta)$ is Lipschitz-continuous:

[TABLE]

The following lemma is trivial.

Lemma 2.17**.**

For small $\tau>0$ the Cauchy problem

[TABLE]

has a unique solution $\zeta\in C^{1}([0,\tau])$.

Denote

[TABLE]

with $\zeta$ from Lemma 2.17.

Lemma 2.18**.**

The function $\psi(x,t):=\psi_{f}(x,t)+\psi_{S}(x,t)$ is a unique solution to the system

[TABLE]

satisfying the condition

[TABLE]

Proof.

Initial conditions of (2.76) follow from (2.70). Further,

[TABLE]

Thus, the second equation of (2.76) is satisfied. At last,

[TABLE]

and $\psi$ solves the first equation of (2.76) then.

It remains to check (2.77). Note, that the function $\varphi_{reg}(x,t)=\psi(x,t)-\zeta(t)G_{1}(x)=\psi_{reg}(x,t)+\zeta(t)(G(x)-G_{1}(x))$, where $G_{1}(x)=G(x)e^{-|x|}$, satisfies

[TABLE]

with initial data from $H^{2}\oplus H^{1}$. Moreover, (2.71) and (2.75) imply that $\ddot{\zeta}\in L^{2}([0,\tau])$. Hence,

[TABLE]

by [180, Lemma 3.2]. Therefore,

[TABLE]

satisfies $(\psi_{reg}(t),\dot{\psi}_{reg}(t))\in\mathaccent 28695{H}^{2}\oplus\mathaccent 28695{H}^{1}$, $t\in[0,\tau]$, and (2.77) holds then.

It remains to prove the uniqueness. Suppose now that there exists another solution $\tilde{\psi}=\tilde{\psi}_{reg}+\tilde{\zeta}G$ to the system (2.76), with $(\tilde{\psi},\dot{\tilde{\psi}})\in{\cal D}$. Then, by reversing the above argument, the second equation of (2.76) implies that $\tilde{\zeta}$ solves the Cauchy problem (2.75). The uniqueness of the solution of (2.75) implies that $\tilde{\zeta}=\zeta$. Then, defining

[TABLE]

for $\tilde{\psi}_{f}=\tilde{\psi}-\psi_{S}$ one obtains

[TABLE]

i.e. $\tilde{\psi}_{f}$ solves the Cauchy problem (2.70). Hence, $\tilde{\psi}_{f}=\psi_{f}$ by the uniqueness of the solution to (2.70), and hence, $\tilde{\psi}=\psi$. ∎

According to [180, Lemma 3.7]

[TABLE]

Step iii) Now we are able to prove the globall well-posedness. First, note that

[TABLE]

Indeed, ${\cal H}_{F}(\Psi_{0})\geq U(\zeta_{0})$ by the definition of energy in (2.68). Therefore, $|\zeta_{0}|\leq\Lambda(\Psi_{0})$, and then $\tilde{U}(\zeta_{0})=U(\zeta_{0})$, ${\cal H}_{\tilde{F}}(\Psi_{0})={\cal H}_{F}(\Psi_{0})$. Further,

[TABLE]

and (2.74) implies that

[TABLE]

Now we can replace $\tilde{F}$ by $F$ in Lemma 2.18 and in (2.78). The solution $\Psi(t)=(\psi(t),\dot{\psi}(t))\in{\cal D}$ constructed in Lemma 2.18 exists for $0\leq t\leq\tau$, where the time span $\tau$ in Lemma 2.17 depends only on $\Lambda(\Psi_{0})$. Hence, the bound (2.80) at $t=\tau$ allows us to extend the solution $\Psi$ to the time interval $[\tau,2\tau]$. We proceed by induction to obtain the solution for all $t\geq 0$. Theorem 2.15 is proved. ∎

The main result of [180] is as follows.

Theorem 2.19**.**

Let $\Psi(x,t)=(\psi(x,t),\dot{\psi}(x,t))$ be a solution to (2.64) with initial data from ${\cal D}$. Then

[TABLE]

where $q_{\pm}\in Q$ and the convergence holds in $L^{2}_{loc}({\mathbb{R}}^{3})\oplus L^{2}_{loc}({\mathbb{R}}^{3})$.

Proof.

It suffices to prove this theorem for $t\to+\infty$ only. By Lemma 2.18, the solution $\psi(x,t)$ to (2.64) with initial data $(\psi_{0},\pi_{0})\in{\cal D}$, can be represented as the sum

[TABLE]

where dispersive component $\psi_{f}(x,t)$ is a unique solution to (2.70), and singular component $\psi_{S}(x,t)$ is a unique solution to the following Cauchy problem

[TABLE]

Here $\zeta(t)\in C^{1}_{b}([0,\infty))$ is a unique solution to

[TABLE]

Now we can prove local decay of $\psi_{f}(x,t)$.

Lemma 2.20**.**

For any $R>0$, the following convergence holds

[TABLE]

Here $B_{R}$ is the ball of radius $R$.

Proof.

We represent the initial data $(\psi_{0},\pi_{0})=(\psi_{0,reg},~{}\pi_{0,reg})+(\zeta_{0}G,~{}\eta_{0}G)\in{\cal D}$ as

[TABLE]

where a cut-of function $\chi\in C_{0}^{\infty}({\mathbb{R}}^{3})$ satisfies

[TABLE]

Let us show that

[TABLE]

Indeed,

[TABLE]

On the other hand,

[TABLE]

Now we split the dispersion component $\psi_{f}(x,t)$ as

[TABLE]

where $\varphi$ and $\varphi_{G}$ are defined as solutions to the free wave equation with initial data $(\varphi_{0},~{}p_{0})$ and $(\zeta_{0}\chi G,\eta_{0}\chi G)$, respectively, and study the decay properties of $\varphi_{G}$ and $\varphi$.

First, by the strong Huygens principle

[TABLE]

Indeed, $\varphi_{G}(x,t)=\zeta_{0}\dot{\psi}_{G}(x,t)+\eta_{0}\psi_{G}(x,t)$, where $\psi_{G}(x,t)$ is the solution to the free wave equation with initial data $(0,\chi G)\in H^{1}\oplus L^{2}$, and $\psi_{G}(x,t)$ satisfies the strong Huygens principle by [184, Theorem XI.87].

It remains to check that

[TABLE]

For $r\geq 1$ denote $\chi_{r}=\chi(x/r)$, where $\chi(x)$ is a cut-off function (2.85). Denote $\phi_{0}=(\varphi_{0},\pi_{0})$. Let $u_{r}(t)$ and $v_{r}(t)$ be solutions to free wave equations with the initial data $\chi_{r}\phi_{0}$ and $(1-\chi_{r})\phi_{0}$, respectively, so that $\varphi(t)=u_{r}(t)+v_{r}(t)$. By the strong Huygens principle

[TABLE]

To conclude (2.86), it remains to note that

[TABLE]

by the energy conservation for the free wave equation. We also use the embedding $\mathaccent 28695{H}^{1}({\mathbb{R}}^{3})\subset L^{6}({\mathbb{R}}^{3})$. The right-hand side of (2.87) could be made arbitrarily small if $r\geq 1$ is sufficiently large. ∎

Due to (2.81) and (2.84), for the proof of Theorem 2.19 it suffices to verify the convergence of $\psi_{S}(x,t)$ to stationary states:

Lemma 2.21**.**

Let $\psi_{S}(x,t)$ and $\zeta(t)$ be solutions to (2.82) and (2.83), respectively. Then

[TABLE]

where $q_{\pm}\in Q$ and the convergence holds in $L^{2}_{loc}({\mathbb{R}}^{3})\oplus L^{2}_{loc}({\mathbb{R}}^{3})$.

Proof.

The unique solution to (2.82) is the spherical wave

[TABLE]

cf. (2.72)–(2.73). Then a priori bound (2.69) and equation (2.83) imply that

[TABLE]

First, we prove the convergence of $\zeta(t)$. From (2.69) it follows that $\zeta(t)$ has the upper and lower limits:

[TABLE]

Suppose that $a<b$. Then the trajectory $\zeta(t)$ oscillates between $a$ and $b$. Assumption (2.67) implies that $F(\zeta_{0})\not=0$ for some $\zeta_{0}\in(a,b)$. For the concreteness, let us assume that $F(\zeta_{0})>0$. The convergence (2.84) implies that

[TABLE]

Hence, for sufficiently large $T$ we have

[TABLE]

Then for $t\geq T$ the transition of the trajectory from left to right through the point $\zeta_{0}$ is impossible by (2.83). Therefore, $a=b=q_{+}$, where $q_{+}\in Q$ since $F(q_{+})=0$ by (2.83). Hence (2.89) implies

[TABLE]

Further,

[TABLE]

uniformly in $|x|\leq R$. Then (2.88) and (2.91) imply that

[TABLE]

where the convergence holds in $L^{2}_{loc}({\mathbb{R}}^{3})$. It remains to verify the convergence of $\dot{\psi}_{S}(t)$. We have

[TABLE]

From (2.83), (2.90) and (2.91) it follows that $\dot{\zeta}(t)\to 0$ as $t\to\infty$. Then

[TABLE]

in $L^{2}_{loc}({\mathbb{R}}^{3})$ by (2.92). ∎

This completes the proof of Theorem 2.19. ∎

2.6 Remarks

All above results on global attraction to stationary states refer to “generic” systems with a trivial symmetry group, which are characterized by a suitable discreteness of attractors, by Wiener condition, etc.

Global attraction to stationary states (1.5) resembles similar asymptotics (1.1) for dissipative systems.  However, there are a number of fundamental differences:  

I. In dissipative systems attractor always consists of stationary states, the attraction (1.1) holds only as $t\to+\infty$, and this attraction is due to the absorption of energy and can be in global norms. Such attraction also holds for all finite-dimensional dissipative systems.

II. On the other hand, in Hamiltonian systems attractor may differ from the set of stationary states, as will be seen later. In addition, energy absorption in these systems is absent, and the attraction (1.5) to stationary states  is due to the radiation of energy to infinity, which plays the role of energy absorption. This attraction takes place both as $t\to\infty$, and as $t\to-\infty$, and it holds only in local seminorms.  Finally, it cannot hold for any finite-dimensional Hamiltonian systems (except for the case when the Hamiltonian is an identical constant).

3 Global attraction to solitons

As already mentioned in the introduction, the soliton asymptotics (1.7) with several solitons were discovered for the first time numerically in 1965 for KdV by Kruskal and Zabusky. Later on such asymptotics were proved by the method of inverse scattering problem for nonlinear integrable Hamiltonian translation-invariant equations by Ablowitz, Segur, Eckhaus, Van Harten and others (see [48]).

Here we present the results on global attraction to one soliton (1.6) for nonlinear translation-invariant non-integrable Hamiltonian equations. Such attraction was proved first in [132] and in [79] for charged relativistic particle coupled to the scalar wave field and to the Maxwell field respectively.

3.1 Translation-invariant wave-particle system

In [132] the system (2.22)–(2.23) was considered in the case of zero potential $V(x)\equiv 0$:

[TABLE]

which can be written in the Hamilton form (2.24). The Hamiltonian of this system is given by (2.25) with $V=0$, and it is conserved along trajectories. By Lemma 2.5 with $V(x)\equiv 0$, global solutions exist for all initial data $Y(0)\in{\cal E}$, and a priori estimates (2.32) hold.

This system is translation-invariant, so the corresponding full momentum

[TABLE]

is also conserved. Respectively, the system (3.1) admits traveling-wave type solutions (solitons)

[TABLE]

where $v,a\in{\mathbb{R}}^{3}$, and $|v|<1$. These functions are easily determined: for $|v|<1$ there is a unique function $\psi_{v}$ which makes (3.3) a solution to (3.1),

[TABLE]

where we set $\lambda=\sqrt{1-v^{2}}$ and $x=x_{\|}+x_{\bot}$, where $x_{\|}\|v$ and $x_{\bot}\bot v$ for $x\in{\mathbb{R}}^{3}$. Indeed, substituting (3.3) into the wave equation of (3.1), we get the stationary equation

[TABLE]

Through the Fourier transform

[TABLE]

which implies (3.4). The set of all solitons forms $6$ -dimensional soliton submanifold in the Hilbert phase space ${\cal E}$:

[TABLE]

where $\pi_{v}:=-v\nabla\psi_{v}$. Recall that the spaces ${\cal E}$ and ${\cal E}_{\sigma}$ are defined in Definition 2.4. The following theorem is the main result of [132].

Theorem 3.1**.**

Let the Wiener condition (2.34) hold and $\sigma>3/2$. Then for any initial state $Y(0)\in{\cal E}_{\sigma}$, the correspoding solution $Y(t)=(\psi(t),\pi(t),q(t),p(t))$ of the system (3.1) converges to the soliton manifold ${\cal S}$ in the following sense:

[TABLE]

[TABLE]

where the remainder decreases locally in the comoving frame: for each $R>0$

[TABLE]

The theorem means that, in particular,

[TABLE]

The proof [132] relies on a) relaxation of acceleration (2.38) in the case $V=0$ (see Remark 2.12 i)), and b) on the canonical change of variables to the comoving frame. The key role is played by the fact that the soliton $S_{v,a}$ minimizes the Hamiltonian (2.25) (in the case $V=0$) with a fixed total momentum (3.2), which implies orbital stability of solitons [62, 63]. In addition, the proof essentially relies on the strong Huygens principle for the three-dimensional wave equation.

Before entering into more precise and technical discussion, it may be useful to give general idea of our strategy. As was mentioned above, the total momentum (3.2) is conserved because of translation invariance.

We transform the system (3.1) to new variables $(\Psi(x),\Pi(x),Q,P)=(\psi(q+x),\pi(q+x),q,P(\psi,q,\pi,p))$. The key role in our strategy is played by the fact that this transformation is canonical, which is proved in Section 3.2. Through this canonical transformation one obtains the new Hamiltonian

[TABLE]

Since $Q$ is the cyclic coordinate (i.e., the Hamiltonian ${\cal H}_{P}$ does not depend on $Q$), we may regard $P$ as a fixed parameter and consider the reduced system for $(\Psi,\Pi)$ only. Let us define

[TABLE]

We will prove that $(\psi_{v},\pi_{v})$ is the unique critical point and global minimum of $H_{P(v)}$ . Thus, if initial data is close to $(\psi_{v},\pi_{v})$, then corresponding solution must remain close forever by conservation of energy, which translates into the orbital stability of the solitons. Here we follow the ideas of the Bambusi and Galgani paper [11], were the orbital stability of solitons for the Maxwell–Lorentz equations was proved for the first time. For a general class of nonlinear wave equations with symmetries such approach to orbital stability of the solitons was developed in the well known work [64].

However, the orbital stability by itself is not enough. It only ensures that initial states, close to a soliton, remain so, but does not yield the convergence of $\dot{q}(t)$ in (3.8), and even less the  asymptotics (3.9), (3.10). Thus we need an additional, not quite obvious argument which combines the relaxation (2.38) with the orbital stability in order to establish the soliton-like asymptotics (3.8), (3.9), (3.10). As one essential input we will use the strong Huygens principle for wave equation.

3.1.1 Canonical transformation and reduced system

Since the total momentum is conserved, it is natural to use $P$ as a new coordinate. To maintain the symplectic structure we have to complete this coordinate to a canonical transformation of the Hilbert phase space ${\cal E}$.

Definition 3.2**.**

Let the transform $T:{\cal E}\to{\cal E}$ be defined by

[TABLE]

where $P(\psi,q,\pi,p)$ is the total momentum (3.2).

Remarks 3.3**.**

*i) $T$ is continuous on ${\cal E}$ and Fréchet differentiable at points $Y=(\psi,q,\pi,p)$ with sufficiently smooth $\psi(x),\pi(x)$, but it is not everywhere differentiable.

ii) In the $T$-coordinates the solitons $Y_{v,a}(t)=(\psi_{v}(x-a-vt),\pi_{v}(x-a-vt),q=a+vt,p_{v})$ are stationary except for the coordinate $Q$,*

[TABLE]

with the total momentum $P(v)$ of the soliton defined in (3.12).

Denote ${\cal H}^{T}(Y)={\cal H}(T^{-1}Y)$ for $Y=(\Psi,\Pi,Q,P)\in{\cal E}$ . Then

[TABLE]

The functionals ${\cal H}^{T}$ and ${\cal H}$ are Fréchet-differentiable on the phase space ${\cal E}$.

Proposition 3.4**.**

Let $Y(t)\in C({\mathbb{R}},{\cal E})$ be a solution to the system (3.1). Then

[TABLE]

is a solution to the Hamiltonian system

[TABLE]

Proof.

The equations for $\dot{\Psi}$, $\dot{\Pi}$ and $\dot{Q}$ can be checked by direct computation, while the one for $\dot{P}$ follows from conservation of the total momentum (3.2) since the Hamiltonian ${\cal H}^{T}$ does not depend on $Q$. ∎

Remark 3.5**.**

Formally, Proposition 3.4 follows from the fact that $T$ is a canonical transform, see Section 3.2.

Recall that $Q$ is a cyclic coordinate. Hence, the system (3.15) is equivalent to a reduced Hamiltonian system for $\Psi$ and $\Pi$ only, which can be written as

[TABLE]

Due to (3.14), the soliton $(\psi_{v},\pi_{v})$ is a stationary solution to (3.16) with $P=P(v)$. Moreover, for every $P\in{\mathbb{R}}^{3}$, the functional ${\cal H}_{P}$ is Fréchet differentiable on the Hilbert space ${\cal F}=\mathaccent 28695{H}^{1}\oplus L^{2}$ . Hence, (3.16) implies that the soliton is a critical point of ${\cal H}_{P(v)}$ on ${\cal F}$ . The next lemma demonstrates that $(\psi_{v},\pi_{v})$ is a global minimum of ${\cal H}_{P(v)}$ on ${\cal F}$.

Lemma 3.6**.**

i) For every $v\in{\mathbb{R}}^{3}$ with $|v|<1$ the functional ${\cal H}_{P(v)}$ has the lower bound

[TABLE]

ii) ${\cal H}_{P(v)}$ has no other critical points on ${\cal F}$ except point $(\psi_{v},\pi_{v})$.

Proof.

Step i) Denoting $\Psi-\psi_{v}=\psi$ and $\Pi-\pi_{v}=\pi$, we have

[TABLE]

where $p_{v}=P(v)+\int\,d^{3}x\,\pi_{v}(x)\,\nabla\psi_{v}(x)$, and

[TABLE]

Taking into account that $v=(1+p_{v}^{2})^{-1/2}p_{v}$, we obtain

[TABLE]

It is easy to check that the expression in the third line is nonnegative. Then the lower bound (3.17) follows by using $|(1+p_{v}^{2})^{-1/2}p_{v}|=|v|$ .

Step ii) If $(\Psi,\Pi)\in{\cal F}$ is a critical point for ${\cal H}_{P(v)}$, then it satisfies

[TABLE]

where $\tilde{p}=P(v)+\displaystyle\int\,d^{3}x\,\Pi(x)\,\nabla\Phi(x)$ . This system is equivalent to equation (3.5) for solitons in the case of the velocity $\tilde{v}=(1+\tilde{p}^{2})^{-1/2}\tilde{p}$ . Hence, $\Psi=\psi_{\tilde{v}}$ , $\Pi=\pi_{\tilde{v}}$ and $P(\tilde{v})=P(v)$ .

It remains to check that $\tilde{v}=v$. Indeed, for the total momentum $P(v)$ of the soliton-like solution (3.3), the Parseval identity and (3.6) imply

[TABLE]

Hence, $P(v)=\kappa(|v|)v$ with $\kappa(|v|)\geq 0$, and for $v\not=0$ one has

[TABLE]

Since $|P(v)|=\kappa(|v|)|v|$ is a monotone increasing function of $|v|\in[0,1[$, we conclude that $v=\tilde{v}$. ∎

Remark 3.7**.**

Proposition 3.4 is not really needed for the proof of Theorem 3.1. However, the Proposition together with (3.14) and (3.16) show that $(\psi_{v},\pi_{v})$ is a critical point and suggest an investigation of the stability through a lower bound as in (3.17). In Section 3.2 we sketch the derivation of Proposition 3.4 for sufficiently smooth solutions based only on the invariance of symplectic structure. We expect that a similar proposition holds for other translation invariant systems similar to (3.1). **

3.1.2 Orbital stability of solitons

We follow [11] deducing orbital stability from the conservation of the Hamiltonian ${\cal H}_{P}$ together with its lower bound (3.17). For $|v|<1$ denote

[TABLE]

Lemma 3.8**.**

Let $Y(t)=(\psi(t),\pi(t),q(t),p(t))\in C({\mathbb{R}},{\cal E})$ be a solution to (3.1) with an initial state $Y(0)=Y^{0}=(\psi^{0},\pi^{0},q^{0},p^{0})\in{\cal E}$.Then for every $\varepsilon>0$ there exists a $\delta_{\varepsilon}>0$ such that

[TABLE]

provided $\delta\leq\delta_{\varepsilon}$.

Proof.

Denote by $P^{0}$ the total momentum of the considered solution $Y(t)$. There exists a soliton-like solution (3.3) corresponding to some velocity $\tilde{v}$ with the same total momentum $P(\tilde{v})=P^{0}$ . Then (3.19) implies that $|P^{0}-P(v)|=|P(\tilde{v})-P(v)|={\cal O}(\delta)$. Hence also $|\tilde{v}-v|={\cal O}(\delta)$ and

[TABLE]

Therefore, denoting $(\Psi^{0},Q^{0},\Pi^{0},P^{0})=TY^{0}$, we have

[TABLE]

Total momentum and energy conservation imply that for $(\Psi(t),Q(t),\Pi(t),P^{0})=TY(t)$

[TABLE]

Hence (3.21) and (3.17) with $\tilde{v}$ instead of $v$ imply

[TABLE]

uniformly in $t\in{\mathbb{R}}$ . On the other hand, total momentum conservation implies

[TABLE]

Therefore (3.22) leads to

[TABLE]

uniformly in $t\in{\mathbb{R}}$ . Finally (3.22), (3.23) together imply (3.20) because $|\tilde{v}-v|={\cal O}(\delta)$ . ∎

3.1.3 Strong Huygens principle and soliton asymptotics

We combine the relaxation of the acceleration and orbital stability with the Strong Huygens principle to prove Theorem 3.1.

Proposition 3.9**.**

*Let the assumptions of Theorem 3.1 be fulfilled. Then for every $\delta>0$ there exist a $t_{*}=t_{*}(\delta)$ and a solution $Y_{*}(t)=(\psi_{*}(x,t),\pi_{*}(x,t),q_{*}(t),p_{*}(t))\in C([t_{*},\infty),{\cal E})$ to the system (3.1) such that

i) $Y_{*}(t)$ coincides with $Y(t)$ in the future cone,*

[TABLE]

ii) $Y_{*}(t_{*})$ is close to a soliton $Y_{v,a}$ with some $v$ and $a$,

[TABLE]

Proof.

The Kirchhoff formula gives

[TABLE]

where

[TABLE]

Here $S_{t}(x)$ denotes the sphere $|y-x|=t$. Let us assume for simplicity that initial fields vanish. General case can be easily reduced to this situation using the strong Huygens principle. We will comment on this reduction at the end of the proof.

In the case of zero initial data the solution reduces to the retarded potential:

[TABLE]

We construct the solution $Y_{*}(t)$ as a modification of $Y(t)$. First, we modify the trajectory $q(t)$. The relaxation of acceleration (3.8) means that for any $\varepsilon>0$ there exist $t_{\varepsilon}>0$ such that

[TABLE]

Hence, the trajectory for large times locally tends to a straight line, i.e., for any fixed $T>0$

[TABLE]

Denote $\lambda_{\varepsilon}(t):=q(t_{\varepsilon})+\dot{q}(t_{\varepsilon})(t-t_{\varepsilon})$ and define modified trajectory as

[TABLE]

Then

[TABLE]

The next step we define the modified field as retarded potential of type (3.27)

[TABLE]

Lemma 3.10**.**

The right hand side of (3.30) depends on the trajectory $q_{*}(\tau)$ only from a bounded interval of time $\tau\in[t-T(x,t),t]$, where

[TABLE]

Here $\overline{v}=\sup\limits_{t\in{\mathbb{R}}}|\dot{q}(t)|<1$ by (2.32).

Proof.

This lemma is obvious geometrically, and its formal proof also is easy. The inegrand of (3.30) vanishes for $|y-q_{*}(t-|x-y|)|\geq R_{\rho}$ by (2.35). Therefore, the integral is spreaded over the region $|y-q_{*}(t-|x-y|)|\leq R_{\rho}$, which implies $|y-q_{*}(t)+q_{*}(t)-q_{*}(t-|x-y|)|\leq R_{\rho}$. Hence,

[TABLE]

On the other hand, $|x-y|\leq|x-q_{*}(t)|+|y-q_{*}(t)|$, and hence,

[TABLE]

Therefore,

[TABLE]

which implies

[TABLE]

Now the lemma is proved. ∎

The potential (3.30) satisfies the wave equation

[TABLE]

We should still prove equations for the trajectory $q_{*}(t)$:

[TABLE]

with sufficiently large $t_{*}\geq t_{\varepsilon}$. Let us note that the integral here is spreaded over the ball $|x-q_{*}(t)|\leq R_{\rho}$. Now Lemma 3.10 implies that $\psi_{*}(x,t)$ depends on the trajectory $q_{*}(\tau)$ only from a bounded interval $\tau\in[t-\overline{T},t]$, where

[TABLE]

Let us define $t_{*}:=t_{\varepsilon}+\overline{T}$. Then by Lemma 3.10

[TABLE]

since $q_{*}(t)\equiv q(t)$ for $t>t_{*}-\overline{T}=t_{\varepsilon}$ by (3.29). Hence, equations (3.32) hold for $q_{*}(t)$ as well as for $q(t)$.

It remains to prove (3.26). The key observation is that outside the cone $K_{\varepsilon}:=\{(x,t)\in{\mathbb{R}}^{4}:|x-q(t_{\varepsilon})|<t-t_{\varepsilon}\}$ the retarded potential (3.30) coincides with the soliton $\psi_{v,a}(x,t)$, where $v=\dot{q}(t_{\varepsilon})$ and $a=q(t_{\varepsilon})$ by our definition (3.29). In particular,

[TABLE]

In the ball $|x-q(t_{*})|<\overline{T}$ the coincidence generally does not hold, but the difference of the left hand side with the right hand side converges to zero as $\varepsilon\to 0$ uniformly for $|x-q(t_{*})|<\overline{T}$, and such uniform convergence holds for the gradient of the difference. This follows from the integral representation (3.30) by Lemma 3.10 since

[TABLE]

by the relaxation of acceleration (3.8). It is important that $T(x,t_{*})$ is bounded for $|x-q(t_{*})|<\overline{T}$ by (3.31). This proves Proposition 3.9 in the case of zero initial data.

The next step is the proof for initial data with bounded support:

[TABLE]

Now we apply the strong Huygens principle: in this case the potential (3.28) vanishes in a future cone,

[TABLE]

However, the estimate $|\dot{q}(t)|\leq\overline{v}<1$ implies that the trajectory $(q(t),t)$ lies in this cone for all $t>t_{0}$. Hence, the solution for $t>t_{0}$ again reduces to the retarded potential and the needed conclusion follows.

Finally, arbitrary finite energy initial data admits a splitting in two summands: the first vanishing for $|x|>R_{0}$ and the second vanishing for $|x|<R_{0}-1$. The energy of the second summand is arbitrarily small for large $R_{0}$, and the energy of the corresponding potential (3.28) is conserved in time since it is a solution to free wave equation. Hence, its role is negligible for sufficiently large $R_{0}$. ∎

Now we can prove our main result.

Proof of Theorem 3.1 For every $\varepsilon>0$ there exists $\delta>0$ such that (3.26) implies by Lemma 3.8,

[TABLE]

Therefore, (3.24) and (3.25) imply that for every $R>0$ and $t>t_{*}+\displaystyle\frac{R}{1-\overline{v}}$

[TABLE]

Since $\varepsilon>0$ is arbitrary, we conclude (3.10). Theorem 3.1 is proved.

3.2 Invariance of symplectic structure

The canonical equivalence of the Hamiltonian systems (3.1) and (3.15) can be seen from the Lagrangian viewpoint. We remain at the formal level. For a complete mathematical justification we would have to develop some theory of infinite dimensional Hamiltonian systems which is beyond the scope of this paper.

By definition we have ${\cal H}^{T}(\Psi,\Pi,Q,P)={\cal H}(\psi,\pi,q,p)$ with the arguments related through the transformation $T$. To each Hamiltonian we associate a Lagrangian through the Legendre transformation

[TABLE]

These Legendre transforms are well defined because the Hamiltonian functionals are convex in the momenta.

Lemma 3.11**.**

The following indentity holds,

[TABLE]

Proof.

Clearly we have to check the invariance of the canonical 1-form,

[TABLE]

For this purpose we substitute

[TABLE]

Then the left hand side of (3.33) becomes

[TABLE]

The lemma is proved. ∎

This lemma implies that he corresponding action functionals are identical when transformed by $T$. Hence, finally, the two Hamiltonian systems (3.1) and (3.15) are equivalent since dynamical trajectories are stationary points of the respective action functionals.

3.3 Translation-invariant Maxwell-Lorentz system

In [79] asymptotics of type (3.8)–(3.10) were extended to the Maxwell-Lorentz translation-invariant system (2.61) without external fields. In this case, the Hamiltonian coincides with (2.63) where $V(x)\equiv 0$. The extension of methods [132] to this case required a new detailed analysis of the corresponding Hamiltonian structure which is necessary for the canonical transformation. Now the key role in applying Huygens’ strong principle is played by new estimates of long-time decay for oscillations of energy and total momentum solutions for perturbed Maxwell-Lorentz system (estimates (4.24)–(4.25) in [79]).

3.4 The case of weak interaction

Soliton asymptotic of the type (3.8)–(3.10) for the system (2.22)–(2.23) was proved in a stronger form for the case of a weak coupling

[TABLE]

Namely, in [81] initial fields are considered with decay $|x|^{-5/2-\varepsilon}$, $\forall\varepsilon>0$ (condition (2.2) in [81]) provided that $\nabla V(q)=0$ for $|q|>{\rm const}$. Under these assumptions, more strong decay holds,

[TABLE]

for “outgoing” solutions that satisfy the condition

[TABLE]

With these assumptions asymptotics (3.8)–(3.10) can be significantly strengthen: now

[TABLE]

where ‘‘dispersion waves’’ $~{}W(t)\Phi_{\pm}$ are solutions of a free wave equation, and the remainder converges to zero in global energy norm:

[TABLE]

This progress compared with local decay (3.10) is due to the fact that we identified a dispersion wave $W(t)\Phi_{\pm}$ under the condition of smallness (3.34). This identification is possible due to the rapid decay (3.35), in difference with (2.38).

All solitons propagate with velocities $v<1$, and therefore they are spatially separated for large time from the dispersion waves $W(t)\Phi_{\pm}$, which propagate with unit velocity (Fig.  2).

The proofs rely on integral Duhamel representation and on rapid dispersion decay of solutions to free wave equation. Similar result was obtained in [78] for a system of type (2.22)–(2.23) with the Klein–Gordon equation and in [80] for the Maxwell-Lorentz system (2.61) with the same smallness condition (3.36) under assumption that $E^{\rm ext}(x)=B^{\rm ext}(x)=0$ for $|x|>{\rm const}$. In [82], this result was extended to the Maxwell-Lorentz system of type (2.61) with a rotating charge.

Remark 3.12**.**

The results of [81, 82] imply Soffer’s ‘‘Grand Conjecture’’ [194, p. 460] in a moving frame for translation-invariant systems under the condition of smallness (3.34). **

4 Adiabatic effective dynamics of solitons

The existence of solitons and the global attraction to solitons (1.6) are typical features of translation-invariant systems. However, if the deviation of a system from translational invariance is in some sense small, the system can admit solutions which are close forever to solitons with time-dependent parameters (velocity, etc.). Moreover, in some cases it is possible to identify an ‘‘effective dynamics’’ which describes the evolution of these parameters.

4.1 Wave-particle system with a slowly varying external potential

The solitons (3.3) are solutions to the system (3.1)–(2.23) with zero external potential $V(x)\equiv 0$. However, even for the system (2.22)–(2.23) with nonzero external potential soliton-like solutions of the form

[TABLE]

may exist if the potential is slowly changing:

[TABLE]

In this case, the total momentum (3.2) is generally not conserved, but its slow evolution and slow evolution of solutions (4.1) can be described in terms of some finite-dimensional Hamiltonian dynamics.

Namely, let $P=P_{v}$ be total momentum of the soliton $S_{v,Q}$ in the notation (3.7). It is important that the map ${\cal P}:v\mapsto P_{v}$ is an isomorphism of the ball $|v|<1$ on $R^{3}$. Therefore, we can consider $Q,P$ as global coordinates on the soliton manifold ${\cal S}$. We define effective Hamilton functional

[TABLE]

where ${\cal H}_{0}$ is unperturbed Hamiltonian (2.25) with $V=0$. This functional allows the splitting ${\cal H}_{\rm eff}(Q,\Pi)=E(\Pi)+V(Q)$ since the first integral in (2.25) does not depend on $Q$ while the last integral vanishes on the solitons. Hence, the corresponding Hamilton equations read

[TABLE]

The main result of [126] is the following theorem.

Theorem 4.1**.**

Let condition (4.2) hold, and the initial state $S_{0}=(\psi_{0},\pi_{0},q_{0},p_{0})\in{\cal S}$ is a soliton with a full momentum $P_{0}$. Then the corresponding solution $\psi(x,t),\pi(x,t),q(t),p(t)$ to the system (2.22)–(2.23) admits the following “adiabatic asymptotics”

[TABLE]

where $P(t)$ denotes total momentum (3.2), $v(t)={\cal P}^{-1}(\Pi(t))$, and $(Q(t),\Pi(t))$ is the solution to the effective Hamilton equations (4.4) with initial conditions

[TABLE]

Note that such relevance of effective dynamics (4.4) is due to the consistency of Hamiltonian structures:

1. The effective Hamiltonian (4.3) is a restriction of the Hamiltonian functional (2.25) with $V=0$ onto the soliton manifold ${\cal S}$.

2. As shown in [126], the canonical form of the Hamiltonian system (4.4) is also a restriction onto ${\cal S}$ of canonical form of the system (2.22)–(2.23): formally

[TABLE]

Therefore, the total momentum $P$ is canonically conjugate to the variable $Q$ on the soliton manifold ${\cal S}$. This fact justifies definition (4.3) of the effective Hamiltonian as a function of the total momentum $P_{v}$, and not of the particle momentum $p_{v}$.

One of the important results of [126] is the following ‘‘effective dispersion relation’’:

[TABLE]

It means that non-relativistic mass of a slow soliton increases due to an interaction with the field by the amount

[TABLE]

This increment is proportional to the field energy of a soliton in rest

[TABLE]

which agrees with the Einstein mass-energy equivalence principle (see below).

Remark 4.2**.**

The relation (4.7) gives only a hint that $m_{e}$ is an increment of the effective mass. The true dynamical justification for such an interpretation is given by the asymptotics (4.5)–(4.6) which demonstrate the relevance of the effective dynamics (4.4). **

Generalizations. After the paper [126] adiabatic effective asymptotics of type (4.5), (4.6) were obtained in [58, 57] for nonlinear Hartree and Schrödinger equations with slowly varying external potentials, and in [166, 204] - for nonlinear equations of Einstein’s–Dirac, Chern–Simon–Schrödinger and Klein–Gordon–Maxwell with small external fields.

Recently, similar adiabatic effective dynamics were established in [9] for an electron in second-quantized Maxwell field in the presence of a slowly changing external potential.

4.2 Mass–Energy equivalence

In [155], asymptotics (4.5), (4.6) were extended to solitons of the Maxwell–Lorentz equations (2.61) with small external fields. In this case the increment of nonrelativistic mass also turns out to be proportional to the energy of the static soliton’s own field.

Such equivalence of the self-energy of a particle with its mass was first discovered in 1902 by Abraham: he obtained by direct calculation that electromagnetic self-energy $E_{\rm own}$ of an electron at rest adds $m_{e}=\displaystyle\frac{4}{3}E_{\rm own}/c^{2}$ to its non-relativistic mass (see [1, 2], and also [102, p. 216–217]). It is easy to see that this self-energy is infinite for a point electron at the origin with a charge density $\delta(x)$, because in this case, the Coulomb electrostatic field $|E(x)|=C/|x|^{2}$ so the integral in (2.63) diverges around $x=0$. This means that the field mass for a point electron is infinite, which contradicts experiment. That’s why Abraham introduced the model of electrodynamics with ‘‘extended electron’’ (2.61), whose self-energy is finite.

At the same time, Abraham conjectured that the entire mass of an electron is due to its own electromagnetic energy; that is, $m=m_{e}$: ‘‘… matter disappeared, only energy remains … ’’, as philosophical-minded contemporaries wrote [76, p.   63, 87, 88] (smile :)  )

This conjecture was justified in 1905 by Einstein, who discovered the famous universal relation $E=m_{0}c^{2}$, suggested by Special Theory of Relativity [51]. Additional factor $\frac{4}{3}$ in the Abraham formula is due to nonrelativistic character of the system (2.61). According to modern view, about 80% of the electron mass is of electromagnetic origin   [54].

5 Global attraction to stationary orbits

Global attraction to stationary orbits (1.8) was first established in [101, 105, 106] for the Klein–Gordon equation coupled to nonlinear oscillator

[TABLE]

We consider complex solutions, identifying complex values $\psi\in{\mathbb{C}}$ with real vectors $(\psi_{1},\psi_{2})\in\mathbb{R}^{2}$, where $\psi_{1}={\rm Re{\hskip 1.42262pt}{\hskip 1.42262pt}}\psi$ and $\psi_{2}={\rm Im{\hskip 1.42262pt}{\hskip 1.42262pt}}\psi$. Suppose that $F\in C^{1}(\mathbb{R}^{2},\mathbb{R}^{2})$ and

[TABLE]

where $U$ is real function and $\nabla_{\overline{\psi}}:=(\partial_{1},\partial_{2})$. In this case, the equation (5.4) is formally equivalent to Hamiltonian system (2.2) in the Hilbert phase space ${\cal E}:=H^{1}(\mathbb{R})\oplus L^{2}(\mathbb{R})$, where $H^{1}:=H^{1}(\mathbb{R})$ and $L^{2}:=L^{2}(\mathbb{R})$. The Hamilton functional reads

[TABLE]

Let us write (5.1) in the vector form as

[TABLE]

where $Y(t)=(\psi(t),\dot{\psi}(t))$. We assume that

[TABLE]

In this case, finite energy solution $Y(t)\in C(\mathbb{R},{\cal E})$ exists and is unique for any initial state $Y(0)\in{\cal E}$. A priori bound

[TABLE]

holds due to conservation of energy (5.3). Note that the condition (2.10) is no longer necessary, since conservation of energy (5.3) with $m>0$ provides the boundedness of solutions.

Further, we assume $U(1)$ -invariance of the potential:

[TABLE]

Then differentiation (5.2) gives

[TABLE]

and therefore

[TABLE]

By ‘‘stationary orbits’’ we mean solutions of the form

[TABLE]

with $\omega\in{\mathbb{R}}$ and $\psi_{\omega}\in H^{1}({\mathbb{R}})$. Each stationary orbit corresponds to some solution to the equation

[TABLE]

which is the nonlinear eigenvalue problem. Solutions $\psi_{\omega}\in H^{1}({\mathbb{R}})$ of this equation have the form $\psi_{\omega}(x)\!=\!Ce^{-\kappa|x|}$, where $\kappa\!:=\!\sqrt{m^{2}\!-\!\omega^{2}}\!>\!0$, and the constant $C$ satisfies the nonlinear algebraic equation $2\kappa C=F(C).$ The solutions $\psi_{\omega}$ exist for $\omega$ from some set $\Omega\subset{\mathbb{R}}$, lying in the spectral gap $[-m,m]$. Denote the corresponding solitary manifold

[TABLE]

Finally, suppose the equation (5.4) be strictly nonlinear:

[TABLE]

For example, well known Ginzburg–Landau potential $U(\psi)=|\psi|^{4}/4-|\psi|^{2}/2$ satisfies all the conditions (5.5), (5.7) and (5.12).

Definition 5.1**.**

i) ${\cal E}_{F}\subset H^{1}_{loc}({\mathbb{R}}^{3})\oplus L^{2}_{loc}({\mathbb{R}}^{3})$ is the space ${\cal E}$ endowed with the seminorms

[TABLE]

ii) The convergence in ${\cal E}_{F}$ is equivalent to the convergence in every seminorm (5.13).

The convergence in ${\cal E}_{F}$ is equivalent to the convergence in the metric of type (2.13),

[TABLE]

Theorem 5.2**.**

Let the conditions (5.2), (5.5), (5.7) and (5.12) hold. Then any finite energy solution $Y(t)=(\psi(t),\dot{\psi}(t))\in C({\mathbb{R}},{\cal E})$ to equation (5.4) attracts to the solitary manifold (see Fig. 3):

[TABLE]

where the attraction holds in the sense (2.15).

Generalizations: The attraction (5.15) was extended in [109] to 1D Klein–Gordon equation with $N$ nonlinear oscillators

[TABLE]

and in [32, 108, 110] - to the Klein–Gordon and Dirac equations in ${\mathbb{R}}^{n}$ with $n\geq 3$ and   non-local interaction

[TABLE]

under Wiener’s condition (2.34), where $\alpha=(\alpha_{1},\dotsc,\alpha_{n})$ and $\beta=\alpha_{0}$ are Dirac matrices.

Recently, the attraction (5.15) was extended in [146] to 1D Dirac equation coupled to nonlinear oscillator, and in [140, 141, 145] to 3D wave and Klein-Gordon equations with concentrated nonlinearities.

In addition, attraction (5.15) was extended in [33] to non-linear discrete in space and time Hamiltonian equations that are discrete approximations of equations of the type (5.17), i.e. corresponding difference schemes. The proof relies on a novel version of the Titchmarsh theorem for distributions on a circle, obtained in   [111].

Open questions:

I. Global attraction (1.8) to orbits with fixed frequencies $\omega_{\pm}$ is not proved yet.

II. Global attraction to stationary orbits for nonlinear Schrödinger equations also is not proved. In particular, such attraction is not proved for the 1D Schrödinger equation associated with a nonlinear oscillator

[TABLE]

The main difficulty is the infinite ‘‘spectral gap’’ $(-\infty,0)$ (see Remark 5.15).

III. Global attraction to solitons (1.6) for nonlinear relativistically invariant  Klein–Gordon equations is an open problem. In particular, for one-dimensional equations

[TABLE]

The main difficulty is the presence of nonlinear interaction in every point $x\in{\mathbb{R}}$. Asymptotic stability of solitons (that is, local attraction to them) for such equations was first proved in [143, 144], see Section 6.3 below.

5.1 Method of omega-limit trajectories

The proof of Theorem 5.2 relies on a general strategy of omega-limit trajectories introduced first in [101] and developed further in [105, 106, 109, 107, 108, 110, 111, 140, 145, 146, 32, 33].

Definition 5.3**.**

An omega-limit trajectory for a given $Y(t)\in C({\mathbb{R}},{\cal E})$ is any limit function $Z(t)$ such that

[TABLE]

where $s_{j}\to\infty$.

Definition 5.4**.**

A function $Y(t)\in C({\mathbb{R}},{\cal E})$ is omega-compact if for any sequence $s_{j}\to\infty$ there exists such a subsequence $s_{j^{\prime}}\to\infty$ that (5.21) holds.

These concepts are useful due to the following lemma which lies in the basis our approach.

Lemma 5.5**.**

Let any solution $Y(t)\in C({\mathbb{R}},{\cal E})$ to (5.4) be omega-compact, and any omega-limit trajectory is a stationary orbit

[TABLE]

where $\omega\in{\mathbb{R}}$. Then the attraction (5.15) holds for each solution $Y(t)\in C({\mathbb{R}},{\cal E})$ to (5.4).

Proof.

We need to show that

[TABLE]

Assume by contradiction that there exists a sequence $s_{j}\to\infty$ such that

[TABLE]

According to the omega-compactness of the solution $Y$, the convergence (5.21) holds for some subsequence $s_{j^{\prime}}\to\infty$, and some stationary orbit (5.22):

[TABLE]

But this convergence with $t=0$ contradicts (5.23) since $Z(0)\in{\cal S}$ by definition (5.11). ∎

Now for the proof of Theorem 5.2 is suffices to check the conditions of Lemma 5.5:

I. Each solution $Y(t)\in C({\mathbb{R}},{\cal E})$ to (5.4) is omega-compact.

II. Any omega-limit trajectory is a stationary orbit (5.22).

We check these conditions analysing the Fourier transform in time of solutions. The main steps of the proof are as follows:

(1) Spectral representation for solutions to nonlinear equation (5.4):

[TABLE]

We call spectrum of a solution $\psi(t):=\psi(\cdot,t)$ the support of its spectral density $\tilde{\psi}(\cdot)$ which is a tempered distribution of $\omega\in{\mathbb{R}}$ with the values in $H^{1}$.

(2) Absolute continuity of the spectral density $\tilde{\psi}(\omega)$ on the continuous spectrum $(-\infty,-m)\cup(m,\infty)$ of the free Klein–Gordon equation, which is an analogue of the Kato theorem on the absence of embedded eigenvalues.

(3) Omega-limit compactness of each solution.

(4) Reduction of spectrum of each omega-limit trajectory to a subset of the spectral gap $[-m,m]$.

(5) Reduction of this spectrum to a single point using the Titchmarsh convolution theorem.

Below we follow this program, referring at some points to [101, 106] for technically important properties of quasi-measures.

5.2 Spectral representation and limiting absorption principle

It suffices to prove attraction (5.15) only for positive times. For the simplicity of exposition we consider the solution $\psi(x,t)$ to equation (5.1) corresponding to zero initial data only:

[TABLE]

General case of nonzero initial data can be reduced to this case by a trivial subtraction of the solution to the free Klein–Gordon equation with these initial data which is a dispersion wave [101, 106]. We extend $\psi(x,t)$ and $f(t):=F(\psi(0,t))$ by zero for $t<0$ and denote

[TABLE]

From (5.4) and (5.26) it follows that these functions satisfy the equation

[TABLE]

in the sense of distributions.

Fourier-Laplace transform in time. For tempered distributions $g(t)$ we denote by $\tilde{g}(\omega)$ their Fourier transform, which is defined for $g\in C_{0}^{\infty}(\mathbb{R})$ as

[TABLE]

A priori estimates (5.6) imply that  $\psi_{+}(x,t)$ and $f_{+}(t)$ are bounded functions of $t\in\mathbb{R}$ with values in the Sobolev space $H^{1}(\mathbb{R})$ and in ${\mathbb{C}}$, respectively. Therefore, their Fourier transforms are (by definition) quasi-measures with values in $H^{1}(\mathbb{R})$ and in ${\mathbb{C}}$, respectively [59]. Moreover, these Fourier transforms allow an extension from the real axis to analytic functions in the upper complex half-plane ${\mathbb{C}}^{+}:=\{\omega\in{\mathbb{C}}:~{}{\rm Im{\hskip 1.42262pt}{\hskip 1.42262pt}}\omega>0\}$ with values in $H^{1}(\mathbb{R})$ and in ${\mathbb{C}}$ respectively:

[TABLE]

Further, we have the following convergence of tempered distributions with values in $H^{1}$ and ${\mathbb{C}}$ respectively,

[TABLE]

Hence, also their Fourier transforms converge in the same sense,

[TABLE]

The analytic functions $\tilde{\psi}_{+}(x,\omega)$ and $\tilde{f}(\omega)$ grow (in the norm) not faster than $|{\rm Im{\hskip 1.42262pt}{\hskip 1.42262pt}}\omega|^{-1}$ as ${\rm Im{\hskip 1.42262pt}{\hskip 1.42262pt}}\omega\to 0+$ in view of (5.6). Hence, their boundary values at $\omega\in\mathbb{R}$ are tempered distributions of a small singularity: they are second order derivatives of continuous functions, as in the case of $\tilde{f}_{+}(\omega)=i/(\omega-\omega_{0})$ with $\omega_{0}\in{\mathbb{R}}$, which corresponds to $f_{+}(t)=\theta(t)e^{-i\omega_{0}t}$.

Limiting Absorption Principle. By (5.26) the equation (5.28) in the Fourier transform becomes stationary Helmholtz equation

[TABLE]

This equation has two linearly independent solutions, but only one of these solutions is analytic and bounded in ${\rm Im{\hskip 1.42262pt}{\hskip 1.42262pt}}\omega>0$ with values in $H^{1}(\mathbb{R})$:

[TABLE]

Here $k(\omega):=\sqrt{\omega^{2}-m^{2}}$, where the branch has a positive imaginary part for ${\rm Im{\hskip 1.42262pt}{\hskip 1.42262pt}}\omega>0$. For other branch, this function grows exponentially as $|x|\to\infty$. Such an argument in the selection of solutions to stationary Helmholtz equations is known as the ‘‘limiting absorption principle’’ in the diffraction theory [116, 131].

Spectral representation. We rewrite (5.31) in the form

[TABLE]

A nontrivial fact is that the identity of analytic functions (5.32) keeps its structure for their restrictions onto the real axis:

[TABLE]

where $\tilde{\psi}_{+}(\cdot,\omega+i0)$ and $\tilde{\alpha}(\omega+i0)$ are the corresponding quasi-measures with values in $H^{1}(\mathbb{R})$ and ${\mathbb{C}}$, respectively. The problem is that the factor $M_{x}(\omega):=e^{ik(\omega+i0)|x|}$ is not smooth in $\omega$ at the points $\omega=\pm m$. Respectively, the identity (5.33) requires a justification, based on the quasi-measure theory [106].

Finally, the inversion of the Fourier transform can be written as

[TABLE]

where$\langle\,\cdot{,}\,\cdot\rangle$ is a bilinear duality between distributions and smooth bounded functions. The right hand side exists by the Theorem 5.35, see below.

5.3 Nonlinear analogue of Kato’s theorem

It turns out that the properties of the quasimeasures $\tilde{\alpha}(\omega+i0)$ with $|\omega|<m$ and with $|\omega|>m$ significantly differ. This is due to the fact that the set $\{i\omega:|\omega|\geq m\}$ is the continuous spectrum of the generator

[TABLE]

which is the generator of the linearisation of equation (5.4). The following theorem plays a key role in the proof of Theorem 5.2. It is a nonlinear analogue of Kato’s theorem on the absence of embedded eigenvalues in the continuous spectrum, see Remark 5.9 below. Denote $\Sigma:=\{\omega\in\mathbb{R}:|\omega|>m\}$, and we will write below $\tilde{\alpha}(\omega)$ and $k(\omega)$ instead of $\tilde{\alpha}(\omega+i0)$ and $k(\omega+i0)$ for $\omega\in{\mathbb{R}}$.

Theorem 5.6**.**

([106, Proposition 3.2])* Let conditions (5.2), (5.5) and (5.7) hold, and $\psi(t)\in C({\mathbb{R}},{\cal E})$ is any finite energy solution to the equation (5.4). Then the corresponding tempered distribution $\tilde{\alpha}(\omega)$ is absolutely continuous on $\Sigma$. Moreover, $\alpha\in L^{1}(\Sigma)$ and*

[TABLE]

Proof.

First, let us first explain the main idea of the proof. By (5.34), the function $\psi_{+}(x,t)$ formally is a ‘‘linear combination’’ of the functions $e^{ik|x|}$ with the amplitudes $\hat{z}(\omega)$:

[TABLE]

For $\omega\in\Sigma$, the functions $e^{ik(\omega)|x|}$ are of infinite $L^{2}$-norm, while $\psi_{+}(\cdot,t)$ is of finite $L^{2}$-norm. This is possible only if the amplitude is absolutely continuous in $\Sigma$. This idea is suggested by the Fourier integral $f(x)=\displaystyle\int_{\mathbb{R}}e^{-ikx}g(k)dk$ which belongs to $L^{2}({\mathbb{R}})$ if and only if $g\in L^{2}({\mathbb{R}})$. For example, if one took $\hat{z}(\omega)=\delta(\omega-\omega_{0})$ with $\omega_{0}\in\Sigma$, then $\psi_{+}(\cdot,t)$ would be of infinite $L^{2}$-norm.

The rigorous proof relies on estimates of the Paley-Wiener type. Namely, the Parseval identity and (5.6) imply that

[TABLE]

On the other hand, we can estimate exactly the integral on the left-hand side of (5.36). Indeed, according to (5.34),

[TABLE]

Hence, (5.36) results in

[TABLE]

Here is a crucial observation about the asymptotics of the norm of $e^{ik(\omega+i\varepsilon)|x|}$ as $\varepsilon\to 0+$.

Lemma 5.7**.**

1. (i)

For $\omega\in{\mathbb{R}}$,

[TABLE]

where the norm in $H^{1}$ is chosen to be $\|\psi\|_{H^{1}}=\left(\|\psi^{\prime}\|_{L^{2}}^{2}+m^{2}\|\psi\|_{L^{2}}^{2}\right)^{1/2}.$ 2. (ii)

For any $\delta>0$ there exists such $\varepsilon_{\delta}>0$ that for $|\omega|>m+\delta$ and $\varepsilon\in(0,\varepsilon_{\delta})$,

[TABLE]

Proof.

Let us compute the $H^{1}$-norm using the Fourier space representation. Setting $k_{\varepsilon}=k(\omega+i\varepsilon)$, so that ${\rm Im{\hskip 1.42262pt}{\hskip 1.42262pt}}k_{\varepsilon}>0$, we get $F_{x\to k}\left[e^{ik_{\varepsilon}|x|}\right]=2ik_{\varepsilon}/(k_{\varepsilon}^{2}-k^{2})$ for $k\in{\mathbb{R}}$. Hence, by the Cauchy theorem

[TABLE]

Substituting here $k_{\varepsilon}^{2}=(\omega+i\varepsilon)^{2}-m^{2}$, we get

[TABLE]

Now the limits (5.38) follow since the function $k(\omega)$ is real for $|\omega|>m$, but is purely imaginary for $|\omega|<m$. Hence, the second statement of the Lemma also follows since $n(\omega)>0$ for $|\omega|>m$, and $n(\omega)\sim|\omega|^{2}$ for $|\omega|\to\infty$. ∎

Remark 5.8**.**

Obviously, $n(\omega)\equiv 0$ for $|\omega|<m$ without any calculations, since in that case the function $e^{ik(\omega)|x|}$ decays exponentially in $x$, and hence, the $H^{1}$-norm of $e^{ik(\omega+i\varepsilon)|x|}$ remains finite when $\varepsilon\to 0+$.

Substituting (5.39) into (5.37), we get:

[TABLE]

with the same $C$ as in (5.37), and the region $\Sigma_{\delta}:=\{\omega\in{\mathbb{R}}:|\omega|>m+\delta\}$. We conclude that for each $\delta>0$ the set of functions

[TABLE]

is bounded in the Hilbert space $L^{2}(\Sigma_{\delta})$, and, by the Banach Theorem, is weakly compact. Hence, the convergence of the distributions (5.29) implies the weak convergence in the Hilbert space $L^{2}(\Sigma_{\delta})$:

[TABLE]

where the limit function $g(\omega)$ coincides with the distribution $\hat{z}(\omega)|\omega k(\omega)|^{1/2}$ restricted onto ${\Sigma_{\delta}}$. It remains to note that the norms of $g$ in $L^{2}(\Sigma_{\delta})$ with all $\delta>0$ are bounded by (5.40), which implies (5.35). Finally, $\tilde{\alpha}(\omega)\in L^{1}(\bar{\Sigma})$ by (5.35) and the Cauchy-Schwarz inequality. ∎

Remark 5.9**.**

Theorem 5.6 is a nonlinear analogue of Kato’s theorem on the absence of embedded eigenvalues in the continuous spectrum. Indeed, solutions of type $\psi_{*}(x)e^{-i\omega_{*}t}$ become $\psi_{*}(x)[\pi i\delta(\omega-\omega_{*})+v.p.\frac{1}{i(\omega-\omega_{*})}]$ in the Fourier–Laplace transform that is forbidden for $|\omega_{*}|>m$ by Theorem 5.6.

5.4 Splitting onto dispersion and bound components

Theorem 5.6 suggests a splitting of the solutions (5.34) onto a ‘‘dispersion’’ and a ‘‘bound’’ components

[TABLE]

where

[TABLE]

Note that $\psi_{d}(x,t)$ is a dispersion wave, because

[TABLE]

according to the Riemann–Lebesgue theorem, since $\alpha\in L^{1}(\Sigma)$ by Theorem 5.6. Moreover, it is easy to prove that

[TABLE]

in the seminorms (2.12). Therefore, it remains to prove the attraction (5.15) for $Y_{b}(t):=(\psi_{b}(\cdot,t),\dot{\psi}_{b}(\cdot,t))$ instead of $Y(t)$:

[TABLE]

5.5 Omega-compactness

Here we establish the omega-compactness of the trajectrory $Y_{b}(t)$ that is necessary for the application of Lemma 5.5. First, we note that the bound component $\psi_{b}(x,t)$ is a smooth function for $x\neq 0$, and

[TABLE]

for any $j,l=0,1,\dots$. These formulas should be justified since the function $k(\omega)$ is not smooth at the points $\omega=\pm m$. The needed justification is done in [101, 106] by a suitable development of the theory of quasimeasures. These formulas imply the boundedness of each derivative:

Lemma 5.10**.**

([106, Proposition 4.1])* For all $j,l=0,1,2,\dotsc$ and $R>0$*

[TABLE]

Proof.

Note that the distribution $\tilde{\alpha}(\omega)$ generally is not a finite measure, since we only know that $\alpha(t):=\psi_{+}(0,t)$ is a bounded function by (5.32) and (5.6). To prove the lemma, it suffices to check that

[TABLE]

where the function $g_{x}(\cdot)$ belongs to a bounded subset of $L^{1}(\mathbb{R})$ for $x\neq 0$ and $t\in{\mathbb{R}}$. This implies the lemma, since the right-hand side of (5.44) equals, by the Parseval identity, to convolution

[TABLE]

where $\alpha(t)$ is a bounded function. ∎

Remark 5.11**.**

All needed properties of quasimeasures that we use, are justified in [101, 106] by similar arguments relying on the Parseval identity.

Now, by the Ascoli–Arzella theorem, for any sequence $s_{j}\to\infty$ there is such a subsequence $s_{j^{\prime}}\to\infty$, that

[TABLE]

for any $j,l=0,\dots,$ and this convergence is uniform on $|x|+|t|\leq R$. Estimates (5.45) imply that

[TABLE]

Corollary 5.12**.**

Each solution $Y(t)\in C({\mathbb{R}},{\cal E})$ to (5.4) is omega-compact. This follows from (5.41), (5.42) and (5.46).

5.6 Reduction of spectrum of omega-limit trajectories to spectral gap

The convergence of functions (5.46) implies the convergence of their Fourier transforms

[TABLE]

in the sense of temperate distributions of $\omega\in{\mathbb{R}}$.

Lemma 5.13**.**

For any $x\in{\mathbb{R}}$

[TABLE]

Proof.

Convergence (5.48) and representation (5.44) with $j=l=0$ imply that

[TABLE]

in the sense of temperate distributions of $\omega\in{\mathbb{R}}$. Moreover, this convergence takes place in a stronger Ascoli–Arzella topology in the space of quasimeasures [106]. In addition, $e^{-ik(\omega)|x|}$ is a multiplier in the space of quasimeasures with this topology by Lemma B.3 of [106]). Therefore, (5.50) implies that

[TABLE]

in the same topology of quasimeasures. Applying the same lemma again, we obtain

[TABLE]

Note that

[TABLE]

Finally, the key observation is that (5.51) and the theorem 5.6 imply

[TABLE]

by the Riemann – Lebesgue theorem. ∎

5.7 Reduction of spectrum of omega-limit trajectories to a single point

5.7.1 Equation for omega-limit trajectories and spectral inclusion

Now the question arises about available means for the proof of representation (5.22) for omega-limit trajectories. We have no formulas for solutions to equation (5.4), and so the only hope is to use the nonlinear equation itself.  The key observation, albeit simple, is that $\beta(x,t)$ is a solution to this nonlinear equation for all $t\in{\mathbb{R}}$, despite the fact that $\psi_{+}(x,t)$ is a solution to the equation (5.4) only for $t>0$ due to (5.27).

Lemma 5.14**.**

The function $\beta(x,t)$ satisfies the original equation (5.4):

[TABLE]

Proof.

This lemma follows by (5.42) and (5.46) in the limit $s_{j^{\prime}}\to\infty$ from the equation (5.4) for $\psi_{+}(x,s_{j^{\prime}}+t)=\psi_{d}(x,s_{j^{\prime}}+t)+\psi_{b}(x,s_{j^{\prime}}+t)$ with $s_{j^{\prime}}+t>0$. ∎

Now applying the Fourier transform to the equation (5.55), we get the corresponding ‘‘nonlinear stationary Helmholtz equation’’

[TABLE]

where we denote $f(t):=F(\beta(0,t))=F(\gamma(t))$ in accordance with (5.53). From (5.8), we get

[TABLE]

Finally, in the Fourier transform we get the convolution $\tilde{f}=\tilde{A}*\tilde{\gamma}$, which exists by (5.54). Respectively, (5.56) now reads

[TABLE]

This identity implies the key spectral inclusion

[TABLE]

because ${\rm supp{\hskip 1.42262pt}{\hskip 1.42262pt}}\tilde{\beta}(x,\cdot)\subset{\rm supp{\hskip 1.42262pt}{\hskip 1.42262pt}}\tilde{\gamma}$ and ${\rm supp{\hskip 1.42262pt}{\hskip 1.42262pt}}\tilde{\beta}^{\prime}(x,\cdot)\subset{\rm supp{\hskip 1.42262pt}{\hskip 1.42262pt}}\tilde{\gamma}$ by the representation ( 5.52). From this inclusion, we will derive below (5.22), using the fundamental result of Harmonic Analysis - Titchmarsh convolution theorem.

5.7.2 Titchmarsh convolution theorem

In 1926, Titchmarsh proved a theorem on the distribution of zeros of entire functions [162, p.119], [207], which implies, in particular, the following corollary [75, Theorem 4.3.3]:

Theorem. Let $f(\omega)$ and $g(\omega)$ be distributions of $\omega\in\mathbb{R}$ with bounded supports. Then

[TABLE]

where $[X]$ denotes convex hull of a set $X\subset\mathbb{R}$.

Note, that in our situation, ${\rm supp{\hskip 1.42262pt}{\hskip 1.42262pt}}\tilde{\gamma}$ is bounded by (5.54). Consequently, ${\rm supp{\hskip 1.42262pt}{\hskip 1.42262pt}}\tilde{A}$ is also bounded, since $A(t):=a(|\gamma(t)|)$ is a polynomial in $|\gamma(t)|^{2}$ according to (5.12). Now the spectral inclusion (5.57) and Titchmarsh theorem imply that

[TABLE]

whence it immediately follows that $[{\rm supp{\hskip 1.42262pt}{\hskip 1.42262pt}}\tilde{A}]=\{0\}$. Besides, $A(t):=a(|\gamma(t)|)$ is a bounded function due to (5.47), because $\gamma(t)=\beta(0,t)$. Therefore, $\tilde{A}(\omega)=C\delta(\omega)$. Hence,

[TABLE]

Now, strict nonlinearity condition (5.12) implies that

[TABLE]

This implies immediately that ${\rm supp{\hskip 1.42262pt}{\hskip 1.42262pt}}\tilde{\gamma}=\{\omega_{+}\}$ by the same Titchmarsh theorem for the convolution $\tilde{\gamma}*\overline{\tilde{\gamma}}=C_{3}\delta(\omega)$. Therefore, $\tilde{\gamma}(\omega)=C_{4}{\hskip 1.42262pt}\delta(\omega-\omega_{+})$, and now (5.22) follows from (5.52).

Remark 5.15**.**

In the case of the Schrödinger equation (5.19), the Titchmarsh theorem does not work. The fact is that the continuous spectrum of the operator $-d^{2}/dx^{2}$ is the half-line $[0,\infty)$, so now the role of the ‘‘spectral gap’’ plays unbounded interval $(-\infty,0)$. Respectively, in this case the spectral inclusion (5.58) gives only that ${\rm supp{\hskip 1.42262pt}{\hskip 1.42262pt}}\tilde{\beta}(x,\cdot)\subset(-\infty,0)$, while the Titchmarsh theorem applies only to distribution with bounded supports. **

5.8 Remarks on dispersion radiation and nonlinear energy transfer

Let us explain informal arguments for the attraction to stationary orbits behind formal proof of Theorem 5.2. Main part of the proof concerns the study of the spectrum of omega-limit trajectories

[TABLE]

Theorem 5.6 implies the spectral inclusion (5.54), which leads to

[TABLE]

Then the Titchmarsh theorem allows us to conclude that

[TABLE]

These two inclusions are suggested by the following two informal arguments:

A. Dispersion radiation in the continuous spectrum.

B. Nonlinear spreading of the spectrum and the energy transfer from lower to higher harmonics.

A. Dispersion radiation. Inclusion (5.58) is due to the dispersion mechanism, which can be illustrated by energy radiation in a wave field with harmonic excitation with a frequency lying in the continuous spectrum. Namely, let us consider one-dimensional linear Klein–Gordon equation with a harmonic source

[TABLE]

where $b\in L^{2}(\mathbb{R})$ and real frequency $\omega_{0}\neq\pm m$. Then the limiting amplitude principle holds [156, 175, 116]:

[TABLE]

For the equation (5.60), this follows directly from the Fourier–Laplace transform in time

[TABLE]

Namely, applying this transform to equation (5.60), we obtain

[TABLE]

where we assume zero initial data for the simplicity of exposition. Hence,

[TABLE]

where $R(\omega):=(H-\omega^{2})^{-1}$ is the resolvent of the Schrödinger operator $H:=-d^{2}/dx^{2}+m^{2}$. This resolvent is an operator of convolution with fundamental solution $-\frac{e^{ik(\omega)|x|}}{2ik(\omega)}$, where $k(\omega)=\sqrt{\omega^{2}-m^{2}}\in\overline{{\mathbb{C}}^{+}}$ for $\omega\in\overline{{\mathbb{C}}^{+}}$, as in (5.31). The last quotient of (5.63) is regular at $\omega=\omega_{0}$, and therefore its contribution is a dispersion wave which decays in local energy seminorms like (5.42). Hence, the long-time asymptotics of $\psi(x,t)$ is determined by the middle quotient of (5.63). Therefore, (5.61) holds with the limiting amplitude $a(x)=R(\omega_{0}+i0)b$. The Fourier transform of this limiting amplitude is equal to

[TABLE]

This formula shows that the properties of the limiting amplitude differ significantly in the cases $|\omega_{0}|<m$ and $|\omega_{0}|\geq m$: $a(x)\in H^{2}(\mathbb{R})$ for $|\omega_{0}|<m$, however,

[TABLE]

if $|\hat{b}(k)|\geq\varepsilon>0$ in the neighborhood of the ‘‘sphere’’ $|k|^{2}+m^{2}=\omega_{0}^{2}$ (which consists of two points in 1D case). This means the following:

I. In the case $|\omega_{0}|\geq m$ the energy of the solution $\psi(x,t)$ tends to infinity for large times according to (5.61) and (5.64). This means that energy is transmitted from the harmonic source to the wave field!

II. Contrary, for $|\omega_{0}|<m$ the energy of the solution remains bounded, so there is no radiation.

Exactly this radiation in the case of $|\omega_{0}|\geq m$ prohibits the presence of harmonics with such frequencies in omega-limit trajectories. Namely, any omega-limit trajectory cannot radiate at all since total energy is finite and bounded from below, and hence the radiation cannot last forever. These physical arguments make the inclusion (5.58) plausible, although its rigorous proof, as was seen above, requires special arguments.

Recall that the set $i\Sigma:=\{i\omega_{0}\in\mathbb{R}$, $|\omega_{0}|\geq m\}$ coincides with the continuous spectrum of the generator of the free Klein–Gordon equation. Radiation in the continuous spectrum is well known  in the theory of waveguides. Namely, waveguides can transmit only signals with a frequency $|\omega_{0}|>\mu$ where $\mu$ is a threshold frequency, which is an edge point of the continuous spectrum [163]. In our case, the waveguide occupies the “entire space” $x\in{\mathbb{R}}$ and is described by the nonlinear Klein–Gordon equation (5.1) with the threshold frequency $m$.

B. Nonlinear inflation of spectrum and the energy transfer from lower to higher harmonics. Let us show that the single spectrum (5.59) is due to the inflation of spectrum by nonlinear functions. For example, let us consider the potential $U(\psi)=|\psi|^{4}\!$. Respectively, $F(\psi)=-\nabla_{\overline{\psi}}U(\psi)=-4|\psi|^{2}\psi$. Consider the sum of two harmonics $\psi(t)=e^{i\omega_{1}t}+e^{i\omega_{2}t}$, which spectrum is shown on Fig. 4:

We substitute this sum into the nonlinearity:

[TABLE]

The spectrum of this expression contains harmonics with new frequencies $\omega_{1}-\Delta$ and $\omega_{2}+\Delta$. As a result, all frequencies $\omega_{1}-\Delta$, $\omega_{1}-2\Delta,\dots$ and $\omega_{2}+\Delta$, $\omega_{2}+2\Delta$, $\dots$ also will appear in the nonlinear dynamics (5.1) (see Fig. 5). Therefore, these frequencies will appear also in the nonlinear term with $\delta$-function.

As we already know, these frequencies lying in the continuous spectrum $|\omega|>m$ will surely cause energy radiation. This radiation will continue until the spectrum of the solution contains at least two different frequencies. Exactly this fact prohibits the presence of two different frequencies in omega-limit trajectories because total energy is finite, so the radiation cannot continue forever.

Let us emphasize that an exact meaning of the inflation of spectrum by nonlinearity is established by the Titchmarsh convolution theorem.

Remark 5.16**.**

The above arguments physically mean the following two-step nonlinear radiation mechanism:

i) Nonlinearity inflates the spectrum, which means energy transfer from lower to higher harmonics;

ii) The dispersion radiation transfers energy to infinity.

We have rigorously justified such nonlinear radiation mechanism for the first time for nonlinear $U(1)$ - invariant Klein–Gordon and Dirac equations (5.4) and (5.16)–(5.18). Our numerical experiments demonstrate similar radiation mechanism for nonlinear relativistic wave equations, see Remark 7.1. However, a rigorous proof is still missing. **

Remark 5.17**.**

Let us comment on the term generic equation in our conjecture (1.4).

i) Asymptotics (2.36), (2.37) hold under the Wiener condition (2.34), which defines some ‘‘open dense set’’ of functions $\rho$. This asymptotics may break down if the Wiener condition fails. For example, if $\rho(x)\equiv 0$, then the particle dynamics is independent from the fields, and hence, the attraction to stationary states can fail.

ii) Similarly, asymptotics (5.15) is valid for an open set of $U(1)$-invariant equations corresponding to polynomials (5.12) with $N\geq 2$. However, this asymptotics may break down for ‘‘exceptional’’ $U(1)$ - invariant equations. In particular, for linear equations, corresponding to polynomials (5.12) with $N=1$. The corresponding examples are constructed in [106].

iii) General situation is the following. Let a Lie group $g$ be a (proper) subgroup of some larger Lie group $G$. Then $G$-invariant equations form an ‘‘exceptional subset’’ among all $g$-invariant equations, and the corresponding asymptotics (1.4) may be completely different. For example, the trivial group $\{e\}$ is a subgroup in $U(1)$ and in ${\mathbb{R}}^{n}$, and asymptotics (1.6) and (1.8) may differ significantly from (1.5). **

6 Asymptotic stability of stationary orbits and solitons

Asymptotic stability of solitary manifolds means a local attraction, i.e. for states sufficiently close to the manifold. The main feature of this attraction is the instability of the dynamics of along the manifold. This follows directly from the fact that solitons move with different speeds and therefore run away for large times.

Analytically, this instability is caused by the presence of the eigenvalue $\lambda=0$ in spectrum of the generator of linearized dynamics. Namely, the tangent vectors to soliton manifolds are eigenvectors and associated vectors of the generator. They correspond to zero eigenvalue. Respectively, Lyapunov’s theory is not applicable to this case.

In a series of articles of Weinstein, Soffer and Buslaev, Perelman and Sulem 1985–2003 an original strategy was developed for proving asymptotic stability of solitary manifolds. This strategy relies on i) special projection of a trajectory onto the solitary manifold, ii) modulation equations for parameters of the projection, and iii) time-decay of transversal component. This approach is a far-reaching development of the Lyapunov stability theory.

6.1 Asymptotic stability of stationary orbits. Orthogonal projection

This strategy arose in 1985–1992 in the pioneering work of Soffer and Weinstein [195, 196, 212], see the review [194]. The results concern nonlinear $U(1)$ - invariant Schrödinger equations with real potential $V(x)$

[TABLE]

where $\lambda\in\mathbb{R}$, $p=3$ or $4$, $n=2$ or $n=3$, and $\psi(x,t)\in{\mathbb{C}}$. The corresponding Hamilton functional reads

[TABLE]

For $\lambda=0$, the equation (6.1) is linear. It is assumed that the discrete spectrum of the short range Schrödinger operator $H:=-\Delta+V(x)$ is a single point $\omega_{*}<0$, and the point zero is neither an eigenvalue nor a resonance for $H$. Let $\phi_{*}(x)$ denote the corresponding ground state:

[TABLE]

Then $C\phi_{*}(x)e^{-i\omega_{*}t}$ are periodic solutions for all complex constants $C$. Corresponding phase curves are circles, filling the complex plane.

For nonlinear equations (6.1) with a small real $\lambda\neq 0$, it turns out that a wonderful bifurcation occurs: small neighborhood of the zero of the complex plane turns into an analytic invariant soliton manifold ${\cal S}$ which is still filled with invariant circles which are trajectories of stationary orbits of type (5.10),

[TABLE]

whose frequencies $\omega$ are close to $\omega_{*}$.

Remark 6.1**.**

Now all these solutions $\psi_{\omega}(x)e^{-i\omega t}$ are called as ground states.

The main result of [195, 196] (see also [183]) is long-time attraction to one of these ground states for any solution with sufficiently small initial data:

[TABLE]

where the remainder decay in weighted norms: for $\sigma>2$

[TABLE]

where $\langle x\rangle:=(1+|x|)^{1/2}$. The proof relies on linearization of the dynamics and decomposition of solutions into two components

[TABLE]

with the orthogonality condition [195, (3.2) and (3.4)]:

[TABLE]

This orthogonality and dynamics (6.1) imply the modulation equations for $\omega(t)$ and $\gamma(t)$, where $\gamma(t):=\Theta(t)-\displaystyle\int_{0}^{t}\omega(s)ds$ (see (3.2) and (3.9a)–(3.9b) from [195]). The orthogonality (6.5) implies that the component $\phi(t)$ lies in the continuous spectral space of the Schrödinger operator $H(\omega_{0}):=-\Delta+V+\lambda|\psi_{\omega_{0}}|^{p}$, which leads to time-decay of $\phi(t)$ (see [195, (4.2a) and (4.2b)]). Finally, this decay implies the convergence $\omega(t)\to\omega_{\pm}$ and the asymptotics (6.4).

These results and methods were further developed in the numerous works for nonlinear Schrödinger, wave and Klein–Gordon equations with potentials under various spectral assumptions on linearized dynamics, [24, 22, 124, 195, 196, 183, 197, 198, 194, 212].

6.2 Asymptotic stability of solitons. Symplectic projection

Genuine breakthrough in the theory of asymptotic stability was achieved in 1990-2003 by Buslaev, Perelman and Sulem [25, 26, 27], who first generalised asymptotics of the type (6.4) for translation-invariant 1D Schrödinger equations

[TABLE]

which are also assumed to be $U(1)$-invariant. The latter means that the nonlinear function $F(\psi)=-\nabla_{\overline{\psi}}U(\psi)$ satisfies identities (5.7)–(5.9). Also the following condition is assumed

[TABLE]

which is caused probably by a failure of suitable technique. Under some simple additional conditions on the potential $U$ (see below), there exist stationary orbits which are finite energy solutions of the form

[TABLE]

with $\omega_{0}>0$. The amplitude $\psi_{0}(x)$ satisfies the corresponding stationary equation

[TABLE]

which implies the ‘‘conservation law’’

[TABLE]

where the ‘‘effective potential’’ $U_{e}(\psi)=U(\psi)+\omega_{0}\frac{|\psi|^{2}}{2}\sim\omega_{0}\frac{|\psi|^{2}}{2}$ as $\psi\to 0$ by (6.7).  For the existence of finite energy solution (6.8),  the graph of the effective potential $U_{e}(\psi)$ should be similar to Fig. 6. The finite energy solution is defined by (6.10)  with the constant $E=U_{e}(0)$ since for other $E$ the solutions to (6.10) do not converge to zero as $|x|\to\infty$. This equation with $E=U_{e}(0)$ implies that

[TABLE]

Hence, for finite energy solutions

[TABLE]

It is easy to verify that the following functions are also solutions (moving solitons)

[TABLE]

The set of all such solitons with parameters $\omega,v,a,\theta$ forms a 4-dimensional smooth submanifold ${\cal S}$ in the Hilbert phase space ${\cal X}:=L^{2}({\mathbb{R}})$. Moving solitons (6.13) are obtained from standing (6.8) by the Galilean transformation

[TABLE]

It is easy to verify that the Schrödinger equation (6.6) is invariant with respect to this group of transformations.

Linearization of the Schrödinger equation (6.6) at the stationary orbit (6.8) is obtained by substitution $\psi(x,t)=(\psi_{0}(x)+\chi(x))e^{-i\omega_{0}t}$ and retaining terms of the first order in $\chi$. This linearised equation contains $\chi$ and $\overline{\chi}$, and hence, it is not linear over the field of complex numbers. This follows from the fact that the nonlinearity of $F(\psi)$ is not complex-analytic due to the $U(1)$-invariance (5.7). Complexification of this linearized equation reads

[TABLE]

where $j$ is a real $2\times 2$ matrix, representing the multiplier $i$, $\Psi(x,t)\in{\mathbb{C}}^{2}$, and $H_{0}=-{d^{2}}/{dx^{2}}+\omega_{0}+V(x)$, where $V(x)$ is a real matrix potential, which decreases exponentially as $|x|\to\infty$ due to (6.12). Note that the operator $C_{0}=C_{\omega_{0},0,0,0}$ corresponds to the linearization at the soliton (6.13) with parameters $\omega=\omega_{0}$, and $a=v=\theta=0$. Similar operators $C_{\omega,a,v,\theta}$, corresponding to linearization at solitons (6.13) with various parameters $\omega,a,v,\theta$, are also connected by the linear Galilean transformation (6.14). Therefore, their spectral properties completely coincide. In particular, their continuous spectrum coincides with $(-i\infty,-i\omega_{0}]\cup[i\omega_{0},i\infty)$.

Main results of [25, 26, 27] are asymptotics of type (6.4) for solutions with initial data close to the solitary manifold ${\cal S}$:

[TABLE]

where $W(t)$ is the dynamical group of the free Schrödinger equation, $\Phi_{\pm}$ are some scattering states of finite energy, and $r_{\pm}$ are remainder terms which decay to zero in a global norm:

[TABLE]

These asymptotics were obtained under following assumptions on the spectrum of the generator $B_{0}$:

U1. The discrete spectrum of the operator $C_{0}$ consists of exactly three eigenvalues [math] and $\pm i\lambda$, and

[TABLE]

This condition means that the discrete mode can interact with the continuous spectrum already  in the first order of perturbation theory.

U2. The edge points $\pm i\omega_{0}$ of the continuous spectrum are neither eigenvalues, nor resonances of $C_{0}$.

U3. Furthermore, it is assumed the condition [27, (1.0.12)], which means a strong coupling of discrete and continuous spectral components, providing energy radiation, similarly to the Wiener condition (2.34). The condition [27, (1.0.12)] ensures that the interaction of discrete component with continuous spectrum does not vanish in the first order of perturbation theory. This condition is a nonlinear version of the Fermi Golden Rule [185], which was introduced by Sigal in the context of nonlinear PDEs [193].

Examples of potentials satisfying all these conditions are constructed in [121].

In 2001, Cuccagna extended results of [25, 26, 27] to nD translation-invariant Schrödinger equations in the dimensions $n\geq 2$, [35].

Method of symplectic projection in the Hilbert phase space. Novel approach [25, 26, 27] relies on symplectic projection of solutions onto the solitary manifold. This means that

[TABLE]

for the projection $S:=P\psi$. This projection is correctly defined in a small neighborhood of ${\cal S}$ because ${\cal S}$ is a symplectic manifold, i.e. the corresponding symplectic form is non-degenerate on the tangent spaces $T_{S}{\cal S}$. In particular, the approach [25, 26, 27] does not require a smallness of initial data.

Thus a solution $\psi(t)$ for each $t>0$ decomposes as $\psi(t)=S(t)+Z(t)$, where $S(t):=P\psi(t)$, and the dynamics is linearized on the soliton $S(t)$. Similarly, for each $t\in{\mathbb{R}}$ the total Hilbert phase space ${\cal X}:=L^{2}({\mathbb{R}})$ is splitted as ${\cal X}={\mathcal{T}}(t)\oplus{\cal Z}(t)$, where ${\mathcal{Z}}(t)$ is symplectic-orthogonal complement to the tangent space ${\mathcal{T}}(t):=T_{S(t)}{\cal S}$. The corresponding equation for the transversal component $Z(t)$ reads

[TABLE]

where $A(t)Z(t)$ is the linear part, and $N(t)={\cal O}(\|Z(t)\|^{2})$ is the corresponding nonlinear part.

The main difficulties in studying this equation are as follows i) it is non-autonomous, and ii) the generators $A(t)$ are not self-adjoint (see Appendix in [118]). It is important that $A(t)$ are Hamiltonian operators, for which the existence of spectral decomposition is provided by the Krein-Langer theory of $J$ -selfadjoint operators [152, 160]. In [118, 120] we have developed a special version of this theory providing the corresponding eigenfunction expansion which is necessary for the justification of the approach [25, 26, 27]. The main steps of this strategy are as follows.

$\bullet$ Modulation equations. The parameters of the soliton $S(t)$ satisfy modulation equations: for example, for the speed $v(t)$ we have

[TABLE]

where $M(\psi)={\cal O}(\|Z\|^{2})$ for small norms $\|Z\|$. This means that the parameters change ‘‘superslow’’ near the soliton manifold, like adiabatic invariants.

$\bullet$ Tangent and transversal components. The transversal component $Z(t)$ in the splitting $\psi(t)=S(t)+Z(t)$ belongs to the transversal subspace ${\cal Z}(t)$. The tangent space ${\mathcal{T}}(t)$ is the root space of the generator $A(t)$ and corresponds to the “unstable” spectral point $\lambda=0$. The key observation is that

i) the transversal subspace ${\mathcal{Z}}(t)$ is invariant with respect to the generator $A(t)$, since the subspace ${\mathcal{T}}(t)$ is invariant, and $A(t)$ is the Hamiltonian operator;

ii) moreover, the transversal subspace ${\mathcal{Z}}(t)$ does not contain ‘‘unstable’’ tangent vectors.

$\bullet$ Continuous and discrete components. The transversal component allows further splitting $Z(t)=z(t)+f(t)$, where $z(t)$ and $f(t)$ belong, respectively, to discrete and continuous spectral subspaces ${\cal Z}_{d}(t)$ and ${\cal Z}_{c}(t)$ of $A(t)$ in the space ${\cal Z}(t)={\cal Z}_{d}(t)+{\cal Z}_{c}(t)$.

$\bullet$ Poincare normal forms and Fermi Golden Rule. The component $z(t)$ satisfies a nonlinear equation, which is reduced to Poincare normal form up to higher order terms [27, Equations (4.3.20)]. The normal form allowed to obtain some ‘‘conditional decay’’ for $z(t)$ using the Fermi Golden Rule [27, (1.0.12)]. For the relativistic-invariant Ginzburg-Landau equation, a similar reduction done in [144, Equations (5.18)].

$\bullet$ Method of majorants. A skillful combination of the conditional decay for $z(t)$ with the superslow evolution of the soliton parameters allows us to prove the decay for $f(t)$ and $z(t)$ by the method of majorants. Finally, this decay implies the asymptotics (6.16)–(6.17).

6.3 Generalizations and Applications

$N$-soliton solutions. The methods and results of [27] were developed in [171, 173, 174, 181, 182, 187, 188] for $N$-soliton solutions for translation-invariant nonlinear Schrödinger equations.

Multiphoton radiation. In [37] Cuccagna and Mizumachi extended methods and results of [27] to the case when the inequality (6.18) is changed to

[TABLE]

with some natural $N>1$, and the corresponding analogue of condition U3 holds. It means, that the interaction of discrete modes with a continuous spectrum occurs only in the $N$-th order of perturbation theory. The decay rate of the remainder term (6.17) worsens with growing $N$.

Linear equations coupled to nonlinear oscillators and particles. The methods and results of [27] were extended i) in [24, 124] to the Schrödinger equation coupled to a nonlinear $U(1)$-invariant oscillator, ii) in [84, 85] to systems (3.1) and (2.61) with zero external fields, and iii) in [83, 112, 123] to similar translation-invariant systems of the Klein–Gordon, Schrödinger and Dirac equations coupled to a particle. The survey of these results can be found in [77].

For example, article [85] concerns solutions to the system (3.1) with initial data close to a soliton manifold (3.3) in weighted norm

[TABLE]

with sufficiently large $\sigma>0$. Namely, the initial state is close to soliton (3.3) with some parameters $v_{0},a_{0}$:

[TABLE]

where $\sigma>5$, and $\varepsilon>0$ is sufficiently small. Moreover, the Wiener condition (2.34) is assumed for $k\neq 0$. Additionally, let

[TABLE]

that is equivalent to equalities

[TABLE]

Under these conditions, the main results of [85] are the asymptotics

[TABLE]

(cf. (3.8) and (3.11)) and the attraction to solitons (3.9), where the remainder now decays in global weighted norms in the comoving frame (cf. (3.10)):

[TABLE]

Relativistic equations. In [16, 23, 139, 143, 144] methods and results [27] were extended for the first time to relativistic-invariant nonlinear equations. Namely, in [16] and [139, 143, 144] asymptotics of the type (6.16) were obtained for 1D relativistic-invariant nonlinear wave equations (5.20) with potentials of the Ginzburg–Landau type, and in [23] for relativistic-invariant nonlinear Dirac equations. In [121] we have constructed examples of potentials providing all spectral properties of the linearised dynamics imposed in [139, 143, 144].

In [118, 120] we have justified the eigenfunction expansions for nonselfadjoint Hamiltonian operators which were used in [139, 143, 144]. For the justification we have developed a special version of the Krein–Langer theory of $J$-selfadjoint operators [152, 160].

Vavilov-Cherenkov radiation. The article [56] concerns a system of type (3.1) with the Schrödinger equation instead of the wave equation (system (1.9)–(1.10) in [56]). This system is considered as a model of the Cherenkov radiation. The main result of [56] is long-time convergence to a soliton with the sonic speed for initial solitons with a supersonic speed in the case of a weak interaction (“Bogolubov limit”) and small initial field. Asymptotic stability of solitons for similar system was established in [112].

6.4 Further generalizations

The results on asymptotic stability of solitons were developed in different directions.

Systems with several bound states. Papers [10, 36, 209, 210, 211] concern asymptotic stability of stationary orbits (6.3) for the nonlinear Schrödinger, Klein–Gordon and wave equations in the case of several simple eigenvalues of the linearization. The typical assumptions are as follows:

i) the endpoint of continuous spectrum is neither an eigenvalue nor a resonance for linearized equation;

ii) the eigenvalues of the linearised equation satisfy several non-resonance conditions;

iii) a new version of the Fermi Golden Rule.

One typical difficulty is possible long stay of solutions near metastable tori which correspond to approximate resonances. Great efforts are being made to show that the role of metastable tori decreases as $t^{-1/2}$ as $t\to\infty$. The typical result is the long-time asymptotics “ground state + dispersion wave” in the norm $H^{1}({\mathbb{R}}^{3})$ for solutions close to the ground state.

General theory of relativity. The article [69] concerns so-called ‘‘kink instability’’ of self-similar and spherically symmetric solutions of the equations of the general theory of relativity with a scalar field, as well as with a ‘‘hard fluid’’ as sources. The authors constructed examples of self-similar solutions that are unstable to the kink perturbations.

The article [38] examines linear stability of slowly rotating Kerr solutions for the Einstein equations in vacuum. In [205] a pointwise damping of solutions to the wave equation is investigated for the case of stationary asymptotically flat space-time in the three-dimensional case.

In [7] the Maxwell equations are considered outside slowly rotating Kerr black hole. The main results are: i)   boundedness of a positive definite energy on each hypersurface $t={\rm const}$ and ii) convergence of each solution to a stationary Coulomb field.

In [43] the pointwise decay was proved for linear waves against the Schwarzschild black hole.

Method of concentration compactness. In [92] the concentration compactness method was used for the first time to prove global well-posedness, scattering and blow-up of solutions to critical focusing nonlinear Schrödinger equation

[TABLE]

in the radial case. Later on, these methods were extended in [44, 46, 93, 153] to general non-radial solutions and to nonlinear wave equations of the type

[TABLE]

One of the main results is splitting of the set of initial states, close to the critical energy level, into three subsets with certain long-term asymptotics: either a blow-up in a finite time, or an asymptotically free wave, or the sum of the ground state and an asymptotically free wave. All three alternatives are possible; all nine combinations with $t\to\pm\infty$ are also possible. Lectures [178] give excellent introduction to this area. The articles [45, 94] concern super-critical nonlinear wave equations.

Recently, these methods and results were extended to critical wave mappings [91, 153, 154]. The ‘‘decay onto solitons’’ is proved: every $1$-equivariant finite-energy wave mapping of exterior of a ball with Dirichlet boundary conditions into three-dimensional sphere exists globally in time and dissipates into a single stationary solution of its own topological class.

Weak convergence to equilibrium distributions in nonlinear Hamilton systems. The papers [148]–[151] concern the weak convergence to an equilibrium distribution in the Liouville, Vlasov and Schrödinger equations. In [151] the authors introduced the quantum Poincaré model.

6.5 Linear dispersion

The key role in all results on long-time asymptotic for nonlinear Hamiltonian PDEs is played by dispersion decay of solutions of the corresponding linearized equations. A huge number of publications concern this decay, so we choose only most important or recent.

Dispersion decay in weighted Sobolev norms. Dispersion decay for wave equations was first proved in linear scattering theory [161].

A powerful systematic approach to dispersion decay for the Schrödinger equation with potential was proposed by Agmon, Jensen and Kato [5, 87]. This theory was extended by many authors to wave, Klein–Gordon and Dirac equations and to the corresponding discrete equations, see [14, 15, 39, 40, 52, 65, 66, 49, 50] and [88, 113, 114, 116, 131, 117, 119, 122, 125, 137, 138, 142, 147] and references therein.

$L^{1}-L^{\infty}$** decay**

[TABLE]

for solutions of linear Schrödinger equation

[TABLE]

with $n\geq 3$ was proved for the first time by Journet, Soffer and Sogge [88] provided that $\lambda=0$ is neither an eigenvalue nor resonance for $H$. The potential $V(x)$ is sufficiently smooth and rapidly decays as $|x|\to\infty$. Here $P_{c}$ is an orthogonal projection onto continuous spectral space of the operator $H$. This result was generalised later by many authors, see below.

In [186] a decay of type (6.19) and Strichartz estimates were established for 3D Schrödinger equations (6.20) with ‘‘rough’’ and time-dependent potentials $V=V(x,t)$ (in stationary case $V(x)$ belongs to both the Rollnik class and the Kato class). Similar estimates were received in [14] for 3D Schrödinger and wave equations with (stationary) Kato class potentials.

In [52] the 4D Schrödinger equations (6.20) are considered for the case when there is a resonance or an eigenvalue at zero energy. In particular, in the case of an eigenvalue at zero energy, there is a time-dependent operator $F_{t}$ of rank $1$, such that $\|F_{t}\|_{L^{1}\to L^{\infty}}\leq 1/\log{\hskip 1.42262pt}t$ for $t>2$, and

[TABLE]

Similar dispersion estimates were proved also for solutions to 4D wave equation with a potential.

In [65, 66] the Schrödinger equation (6.20) is considered in ${\mathbb{R}}^{n}$ with $n\geq 5$ when there is an eigenvalue at the zero point of the spectrum. It is shown, in particular, that there is a time-dependent rank one operator $F_{t}$ such that $\|F_{t}\|_{L^{1}\to L^{\infty}}\leq C|t|^{2-n/2}$ for $|t|>1$, and

[TABLE]

With a stronger decay of the potential, the evolution admits an operator-valued expansion

[TABLE]

where $A_{-2}$ and $A_{-1}$ are finite rank operators $L^{1}({\mathbb{R}}^{n})\to L^{\infty}({\mathbb{R}}^{n})$, while $A_{0}$ maps weighted $L^{1}$ spaces to weighted $L^{\infty}$ spaces . Main members $A_{-2}$ and $A_{-1}$ equal to zero under certain conditions of the orthogonality of the potential $V$ to eigenfunction with zero energy. Under the same orthogonality conditions, the remainder term $|t|^{-n/2}A_{0}$ also maps $L^{1}({\mathbb{R}}^{n})$ to $L^{\infty}({\mathbb{R}}^{n})$, and therefore, the group $e^{itH}P_{c}(H)$ has the same dispersion decay as free evolution, despite its eigenvalue at zero.

$L^{p}-L^{q}$** decay** was first established in [172] for solutions of the free Klein–Gordon equation $\ddot{\psi}=\Delta\psi-\psi$ with initial state $\psi(0)=0$:

[TABLE]

where $1\leq p\leq 2$, $1/p+1/q=1$, and $d\geq 0$ is a piecewise-linear function of $(1/p,1/q)$. The proofs use the Riesz interpolation theorem.

In [13], the estimates (6.21) were extended to solutions of perturbed Klein–Gordon equation

[TABLE]

with $\psi(0)=0$. The authors show that (6.21) holds for $0\leq 1/p-1/2\leq 1/(n+1)$. The smallest value of $p$ and the fastest decay rate $d$ occurs when $1/p=1/2+1/(n+1)$, $d=(n-1)/(n+1)$. The result is proved under the assumption that the potential $V$ is smooth and small in a suitable sense. For example, the result true when $|V(x)|\leq c(1+|x|^{2})^{-\sigma}$, where $c>0$ is sufficiently small. Here $\sigma>2$ for $n=3$,  $\sigma>n/2$ for odd $n\geq 5$, and $\sigma>(2n^{2}+3n+3)/4(n+1)$ for even $n\geq 4$. The results also apply to the case when $\psi(0)\neq 0$.

The seminal article [88] concerns $L^{p}-L^{q}$ decay of solutions to the Schrödinger equation (6.20). It is assumed that $(1+|x|^{2})^{\alpha}V(x)$ is a multiplier in the Sobolev spaces $H^{\eta}$ for some $\eta>0$ and $\alpha>n+4$, and the Fourier transform of $V$ belongs to $L^{1}({\mathbb{R}}^{n})$. Under this conditions,  the main result of [88] is the following theorem: if $\lambda=0$ is neither an eigenvalue nor a resonance for $H$, then

[TABLE]

where $1\leq p\leq 2$ and $1/p+1/q=1$. Proofs are based on $L^{1}-L^{\infty}$ decay (6.19) and the Riesz interpolation theorem.

In [213] estimates (6.22) were proved for all $1\leq p\leq 2$ under suitable conditions on decay of $V(x)$ if $\lambda=0$ is neither an eigenvalue nor a resonance for $H$, and for all $3/2<p\leq 2$ otherwise.

The Strichartz estimates were extended i) in [40] to the Schrödinger magnetic equations in $R^{n}$ with $n\geq 3$, ii) in [39] - to wave equations with a magnetic potential in $R^{n}$ for $n\geq 3$, and iii) in [15] - to wave equation in ${\mathbb{R}}^{3}$ with potentials of the Kato class.

7 Numerical Simulation of Soliton Asymptotics

Here we describe the results of joint work with Arkady Vinnichenko (1945-2009) on numerical simulation of i) global attraction to solitons (1.6) and (1.7), and ii) adiabatic effective dynamics of solitons (4.6) for relativistic-invariant one-dimensional nonlinear wave equations. In [127] can be found an additional information.

7.1 Kinks of relativistic-invariant Ginzburg–Landau equations

First, let us describe numerical simulations of solutions to relativistic-invariant 1D nonlinear wave equations with polynomial nonlinearity

[TABLE]

Since $F(\psi)=0$ for $\psi=0,\pm 1$, there are three equilibrium state : $S(x)\equiv 0,+1,-1$. This equation formally is equivalent to a Hamiltonian system (2.2) with the Hamiltonian

[TABLE]

where the potential $U(\psi)=\frac{\psi^{4}}{4}-\frac{\psi^{2}}{2}+\frac{1}{4}$. This Hamiltonian is finite for functions $(\psi,\pi)$ from the space ${\cal E}_{c}$, defined in (2.3)– (2.5) with $C_{\pm}=\pm 1$, for which the convergence

[TABLE]

is sufficiently fast.

The corresponding potential $U(\psi)=\frac{\psi^{4}}{4}-\frac{\psi^{2}}{2}+\frac{1}{4}$ has minima at $\psi=\pm 1$ and a maximum at $\psi=0$. Respectively, two finite energy solutions $\psi={\pm 1}$ are stable, and the solution $\psi=0$ with infinite energy is unstable. Such potentials with two wells are called potentials of Ginzburg-Landau type.

Besides the constant stationary solutions $S(x)\equiv 0,+1,-1$, there is also a non-constant one $S(x)=\tanh{\hskip 1.42262pt}{x}/{\sqrt{2}}$, which is called ‘‘kink’’. Its shifts and reflections $\pm S(\pm x-a)$ are also stationary solutions as well as their Lorentz transforms

[TABLE]

These are uniformly moving ‘‘travelling waves’’ (i.e. solitons). The kink is strongly compressed when the velocity $v$ is close to $\pm 1$. This compession is known as the ‘‘Lorentz contraction’’.

Numerical Simulation. Our numerical experiments show a decay of finite energy solutions to a finite set of kinks and dispersion waves outside the kinks, that corresponds to the asymptotics of (1.7). One of the experiments is shown on Fig. 7: a finite energy solution to the equation (7.1) decays to three kinks. Here the vertical line is the time axis, and the horizontal line is the space axis. The spatial scale redoubles at $t=$ 20 and $t=$ 60.

Red color corresponds to $\psi>1+\varepsilon$ values, blue color to $\psi<-1-\varepsilon$ values, and the yellow one, to intermediate values $-1+\varepsilon<\psi<1-\varepsilon$, where $\varepsilon>0$ is sufficiently small. Thus, the yellow stripes represent the kinks, while the blue and red zones outside the yellow stripes are filled with dispersion waves.

For $t=0$, the solution begins with a rather chaotic behavior, when there are no visible kinks. After 20 seconds, three separate kinks appear, which subsequently move almost uniformly.

The Lorentz contraction. The left kink moves to the left at a low speed $v_{1}\approx$ 0.24, the central kink is almost standing, because its velocity $v_{2}\approx 0.02$ is very small, and the right kink moves very fast with the speed $v_{3}\approx$ 0.88. The Lorentz spatial contraction $\sqrt{1-v_{k}^{2}}$ is clearly visible in this picture: the central kink is wide, the left is a bit narrower, and the right one is quite narrow.

The Einstein time delay. Also, the Einstein time delay is also very pronounced. Namely, all three kinks oscillate due to the presence of nonzero eigenvalue in the linearized equation at the kink. Indeed, substituting $\psi(x,t)=S(x)+\varepsilon\varphi(x,t)$ in (7.1), we get in the first order approximation the linearized equation

[TABLE]

where the potential $V(x)=3S^{2}(x)-3=-\frac{3}{\cosh^{2}{x}/{\sqrt{2}}}$ decays exponentially for large $|x|$. It is a great joy that for this potential the spectrum of the corresponding Schrödinger operator $H:=-\frac{d^{2}}{dx^{2}}+2+V(x)$ is well known [158]. Namely, the operator $H$ is non-negative, and its continuous spectrum coincides with $[2,\infty)$. It turns out that $H$ still has a two-point discrete spectrum: the points $\lambda=0$ and $\lambda=\frac{3}{2}$. Exactly this nonzero eigenvalue is responsible for the pulsations that we observe for the central slow kink, with  the frequency $\omega_{2}\approx\sqrt{\frac{3}{2}}$ and period $T_{2}\approx 2\pi/\sqrt{\frac{3}{2}}\approx 5$. On the other hand, for fast kinks, the ripples are much slower, i.e., the corresponding period is longer. This time delay agrees numerically with the Lorentz formulas, that confirms the relevance of these results of numerical simulation.

Dispersion waves. An analysis of dispersion waves provides additional confirmation. Namely, the space outside the kinks in Fig. 7 is filled with dispersion waves, whose values are very close to $\pm 1$, with an accuracy $0.01$. These waves satisfy with high accuracy, the linear Klein–Gordon equation, which is obtained by linearization of the Ginzburg–Landau equation (7.1) at the stationary solutions $\psi_{\pm}\equiv\pm 1$:

[TABLE]

The corresponding dispersion relation $\omega^{2}=k^{2}+2$ determines the group velocities of high-frequency wave packets:

[TABLE]

These wave packets are clearly visible in Fig. 7 as straight lines, whose propagation speeds converge to $\pm 1$. This convergence is explained by the high-frequency limit $\omega^{\prime}(k)\to\pm 1$ as $\omega\to\pm\infty$. For example, for dispersion waves emitted by central kink, the frequencies $\omega=\pm n\omega_{2}\to\pm\infty$ are generated by the polynomial nonlinearity in (7.1) in accordance with Fig. 5.

Remark 7.1**.**

These observations of dispersion waves agree with the radiation mechanism from Remark 5.16.**

The nonlinearity in (7.1) is chosen exactly because of well-known spectrum of the linearized equation (7.3). In numerical experiments [127] also more general nonlinearities of the Ginzburg - Landau type were considered . The results were qualitatively the same: for “any” initial data, the solution decays for large times to a sum of kinks and dispersion waves. Numerically, this is clearly visible, but rigorous justification remains an open problem.

7.2 Numerical observation of soliton asymptotics

Besides the kinks the numerical experiments [127] also resulted in the soliton-type asymptotics (1.7) and adiabatic effective dynamics of type (4.6) for complex solutions to the 1D relativistically-invariant nonlinear wave equations (5.20). Polynomial potentials of the form

[TABLE]

were considered with $a,b>0$ and $m>n=2,3,\dotsc$. Respectively,

[TABLE]

The parameters $a,b,m,n$ were taken as follows:

[TABLE]

Various ‘‘smooth’’ initial functions $\psi(x,0),\dot{\psi}(x,0)$ with supports on the interval $[-20,20]$ were considered. The second order finite-difference scheme with $\Delta x,\Delta t\sim 0.01,0.001$ was employed. In all cases the asymptotics of type (1.7) were obsereved with the numbers of solitons $0,1,3,5$ for $t>100$.

7.3 Adiabatic effective dynamics of relativistic solitons

In the numerical experiments [127] was also observed the adiabatic effective dynamics of type (4.6) for soliton-like solutions of type (4.1) to the 1D equations (5.20) with a slowly varying external potential (4.2):

[TABLE]

This equation formally is equivalent to the Hamilton system (2.2) with the Hamilton functional

[TABLE]

The soliton-like solutions are of the form (cf. (4.1))

[TABLE]

Numerical experiments [127] qualitatively confirm the adiabatic effective Hamilton dynamics for the parameters $\Theta,\omega,q$, and $v$, but its rigorous justification is still not established. Figure 8 represents solutions to equation (7.7) with the potential (7.5), where $a=10$, $m=6$ and $b=8.75$, $n=5$. The potential $V(x)=-0.2\cos(0.31x)$ and the initial conditions read

[TABLE]

where $v_{0}=0$, $\omega_{0}=0.6$ and $q_{0}=5.0$. Note, that the initial state does not belong to solitary manifold. An effective width (half-amplitude) of the solitons is in the range $[4.4,5.6]$. It is quite small when compared with the spatial period of the potential $2\pi/0.31\sim 20$. The results of numerical simulations are shown on Figure 8:

$\bullet$ Blue and green colors represent a dispersion wave with values $|\psi(x,t)|<0.01$, while red color represents a soliton with values $|\psi(x,t)|\in[0.4,0.8]$.

$\bullet$ The soliton trajectory (‘‘red snake’’) corresponds to oscillations of a classical particle in the potential $\!V(x)\!$.

$\bullet$ For $0\!<t<\!140$, the solution is rather distant from the solitary manifold, and the radiation is intense.

$\bullet$ For $3020\!<\!t\!<\!3180$, the solution approaches the solitary manifold, and the radiation weakens. The oscillation amplitude of the soliton is almost unchanged for a long time, confirming Hamilton type of effective dynamics.

$\bullet$ However, for $5260\!<\!t\!<\!5420$, the amplitude of the soliton oscillation is halved. This suggests that at a large time scale the deviation from the Hamilton effective dynamics becomes essential. Consequently, the effective dynamics gives a good approximation only on an adiabatic time scale of type $t\sim\varepsilon^{-1}$.

$\bullet$ The deviation from the Hamilton effective dynamics is due to radiation, which plays the role of dissipation.

$\bullet$ The radiation is realized as dispersion waves which bring the energy to the infinity. The dispersion waves combine into uniformly moving wave packets with discrete set of group velocities, as in Fig. 7. The magnitude of the solution is of order $\sim 1$ on the trajectory of the soliton, while the values of the dispersion waves is less than $0.01$ for $t>200$, so that their energy density does not exceed $0.0001$. The amplitude of the dispersion waves decays at large times.

$\bullet$ In the limit $t\to\pm\infty$, the soliton should converge to a static position corresponding to a local minimum of the potential $V(x)$. However, the numerical observation of this ‘‘ultimate stage’’ is hopeless since the rate of the convergence decays with the decay of the radiation.

Appendix A Attractors and Quantum Mechanics

The foregoing results on attractors of nonlinear Hamilton equations were suggested by fundamental postulates of quantum theory, primarily Bohr’s postulate on transitions between quantum stationary orbits. Namely, in 1913 Bohr suggested the following two postulates which gives the ‘‘Columbus’’ solution of the problem of stability and radiation of atoms and molecules [20]:

[TABLE]

[TABLE]

Both these postulates should become theorems in discovered later quantum theory of Schrödinger and Heisenberg. However, this did not happen, and both postulates are still actively used in quantum theory. This lack of theoretical clarity hinders the progress in the theory (e.g., in superconductivity and in nuclear reactions), and in numerical simulation of many engineering processes (e.g., of laser radiation and quantum amplifiers) since a computer can solve dynamical equations but cannot take postulates into account.

A.1 On dynamical interpretation of quantum jumps

The simplest dynamic interpretation of the postulate B1 is the attraction to stationary orbits (1.8) for any finite energy quantum trajectory $\psi(t)$. This means that stationary orbits form a global attractor of the corresponding quantum dynamics. However, this attraction contradicts the Schrödinger linear equation due to the superposition principle. Thus, Bohr’s transitions B1 in the linear theory do not exist.

It is natural to suggest that the attraction (1.8) holds for a nonlinear modification of the linear Schrödinger theory. On the other hand, it turns out that even the original Schrödinger theory is nonlinear, because it involves interaction with the Maxwell field. The corresponding nonlinear Maxwell–Schrödinger system is contained essentially in the first Schrödinger’s articles [191]:

[TABLE]

where the units are chosen so that $\hbar=e=m=c=1$. Maxwell’s equations are written here in the 4-dimensional form, where $A=(A_{0},{\bf A})=(A_{0},A_{1},A_{2},$ $A_{3})$ denotes 4-dimensional potential of the Maxwell field with the Lorentz gauge $\dot{A}_{0}+\nabla\cdot{\bf A}=0$. Further, $A^{\rm ext}=(A_{0}^{\rm ext},{\bf A}^{\rm ext})$ is an external 4-potential, and $J=(\rho,j_{1},j_{2},j_{3})$ is the 4-dimensional current. To make these equations a closed system, we must also express the density of charges and currents via the wave function:

[TABLE]

where $k=1,2,3$, and ‘‘$\cdot$’’ denotes the scalar product of two-dimensional real vectors corresponding to complex numbers. In particular, these expressions satisfy the continuity equation $\dot{\rho}+{\rm div{\hskip 1.42262pt}}j=0$ for any solution of the Schrödinger equation with arbitrary potentials [102, Section 3.4].

System (A.1) is nonlinear in $(\psi,A)$ although the Schrödinger equation is formally linear in $\psi$. Now the question arises: what should be ‘‘stationary orbits’’ for the nonlinear hyperbolic system (A.1)? It is natural to suggest that these are solutions of type

[TABLE]

in the case of static external potentials $A^{\rm ext}(x,t)=A^{\rm ext}(x)$.

Indeed, in this case functions (A.3) give stationary distributions of charges and currents (A.2). Moreover, these functions are the trajectories of one-parameter subgroups of the symmetry group $U(1)$ of the system (A.1). Namely, for any solution $(\psi(x,t),A(x,t))$ and $\theta\in{\mathbb{R}}$ the functions

[TABLE]

are also solutions. The same remarks apply to the Maxwell—Dirac system introduced by Dirac in 1927:

[TABLE]

where $\nabla_{0}:=\partial_{t}$. Thus, Bohr’s transitions B1 for the systems (A.1) and (A.5) with a static external potential $A^{\rm ext}(x,t)=A^{\rm ext}(x)$ can be interpreted as the long-time asymptotics

[TABLE]

for every finite energy solution, where the asymptotics hold in local energy norms. The maps $U_{\theta}$ form a group isomorphic to $U(1)$, and the functions (A.3) are the trajectories of its one-parametric subgroups. Hence, the asymptotics (A.6) correspond to our general conjecture (1.4) with the symmetry group $G=U(1)$.

Furthermore, in the case of zero external potentials these systems are translation-invariant. Respectively, for their solutions one should expect the soliton asymptotics of type (1.7) in global energy norms as $t\to\pm\infty$:

[TABLE]

Here $\Phi_{\pm}^{k}(x,t)$ are suitable phase functions, and each soliton $(\psi_{\pm}^{k}(x-v^{k}_{\pm}t)e^{i\Phi_{\pm}^{k}(x,t)},\,A_{\pm}^{k}(x-v^{k}_{\pm}t))$ is a solution to the corresponding nonlinear system, while $\varphi_{\pm}(x,t)$ and $A_{\pm}(x,t)$ represent some dispersion waves which are solutions to the free Schrödinger and Maxwell equations respectively. The existence of the solitons to the Maxwell–Schrödinger and Maxwell–Dirac systems was established in [31] and [53] respectively.

The asymptotics (A.6) and (A.7) are not proved yet for the Maxwell–Schrödinger and Maxwell–Dirac equations (A.1) and (A.5). One could expect that these asymptotics should follow by suitable modification of the arguments from Section 5. Namely, let the time spectrum of an omega-limit trajectory $\psi(x,t)$ contain at least two different frequencies $\omega_{1}\neq\omega_{2}$: for example, $\psi(x,t)=\psi_{1}(x)e^{-i\omega_{1}t}+\psi_{2}(x)e^{-i\omega_{2}t}$. Then the currents $J_{\alpha}(x,t)$ in the systems (A.1) and (A.5) contains the terms with the harmonics $e^{in\Delta t}$ with $n\in{\mathbb{Z}}$, where $\Delta:=\omega_{1}-\omega_{2}\neq 0$. Thus the nonlinearity inflates the spectrum as in $U(1)$-invariant equations, considered in Section 5.

In it own turn, these harmonics $e^{in\Delta t}$ with $n\neq 0$ on the right hand side of the Maxwell equations induce the radiation of electromagnetic waves with the frequencies $n\Delta$ according to the limiting amplitude principle (5.61) since the continuous spectrum of the Maxwell generator is ${\mathbb{R}}\setminus 0$. Finally, this radiation brings the energy to infinity which is impossible for omega-limit trajectories. This contradiction suggests the validity of the one-frequency asymptotics (A.6).

Methods of Section 5 give a rigorous justification of similar arguments for $U(1)$-invariant equations (5.4) and (5.16)–(5.18). However, a rigorous justification for the systems (A.1) and (A.5) is still an open problem.

A.2 Bohr’s postulates by perturbation theory

The remarkable success of the Schrödinger theory was the explanation of the Bohr’ postulates in the case of static external potentials by perturbation theory applied to the coupled Maxwell–Scrödinger equations (A.1). Namely, as a first approximation, the time-dependent fields ${\bf A}(x,t)$ and $A^{0}(x,t)$ in the Schrödinger equation of the system  (A.1) can be neglected:

[TABLE]

For ‘‘sufficiently good’’ external potentials and initial conditions any finite energy solution can be expanded in eigenfunctions

[TABLE]

where integration is performed over the continuous spectrum of the Schrödinger operator $H$, and the integral decays as $t\to\infty$ in each bounded domain $|x|\leq R$, see, for example, [116, Theorem 21.1]. The substitution of this expansion into the expression for currents (A.2) gives the series

[TABLE]

where $J_{c}(x,t)$ has a continuous frequency spectrum. Therefore, the currents on the right hand side of the Maxwell equation from (A.1) contains, besides the continuous spectrum, only discrete frequencies $\omega_{nn^{\prime}}$. Hence, the discrete spectrum of the corresponding Maxwell field also contains only these frequencies $\omega_{nn^{\prime}}$. This proves the Bohr rule B2 in the first order of perturbation theory, since this calculation ignores the inverse effect of radiation onto the atom.

Moreover, these arguments also clarify the asymptotics (A.6). Namely, the currents (A.11) on the right hand of the Maxwell equation from (A.1) produce the radiation when nonzero frequencies $\omega_{nn^{\prime}}$ are present. However, this radiation cannot last forever since the total energy is finite. Hence, in the long-time limit should remain only $\omega_{nn^{\prime}}=0$ which means exactly one-frequency asymptotics (A.6) and the limiting stationary Maxwell field.

A.3 Conclusion

The discussion above suggests that Bohr’s postulates cannot be explained by linear Schrödinger equation alone but admit a hypothetical explanation in the framework of the coupled Maxwell–Schrödinger equation.

This fact was the cause of heated discussions by Einstein with Bohr and other physicists [21]. In [71, 72], Heisenberg began developing a nonlinear theory of elementary particles.