Global attractor for 1D Dirac field coupled to nonlinear oscillator

TL;DR

This paper proves that solutions to a 1D nonlinear Dirac equation coupled with a nonlinear oscillator tend to nonlinear eigenfunctions over time, demonstrating a global attractor due to nonlinear energy transfer and dispersive effects.

Contribution

It establishes the long-time asymptotics and global attraction for all finite energy solutions of a 1D nonlinear Dirac equation with a nonlinear oscillator, introducing a spectrum inflation mechanism.

Findings

All finite energy solutions converge to nonlinear eigenfunctions.

The spectrum of omega-limit trajectories is confined within the spectral gap.

The spectrum inflation mechanism explains the global attraction phenomenon.

Abstract

The long-time asymptotics is analyzed for all finite energy solutions to a model $\mathbf{U}(1)$-invariant nonlinear Dirac equation in one dimension, coupled to a nonlinear oscillator: {\it each finite energy solution} converges as $t\to\pm\infty$ to the set of all `nonlinear eigenfunctions' of the form $(\psi_1(x)e^{-i\omega_1 t},\psi_2(x)e^{-i\omega_2 t})$. The {\it global attraction} is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. We justify this mechanism by the strategy based on \emph{inflation of spectrum by the nonlinearity}. We show that any {\it omega-limit trajectory} has the time-spectrum in the spectral gap $[-m,m]$ and satisfies the original equation. This equation implies the key {\it spectral inclusion} for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of $j$-th component of the omega-limit trajectory to a single harmonic $\omega_j\in[-m,m]$, $j=1,2$.

Equations

$$i\dot{\psi }(x,t)=D_m\psi (x,t)-D_m^{-1}\delta (x)F(\psi (0,t)),x\in \mathbb{R}.$$

$$\alpha=\left(\begin{array}[]{cc}0&1\\ -1&0\end{array}\right),\quad\beta=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right),$$

$$F(e^{i\theta }\zeta )=e^{i\theta }F(\zeta ),\zeta \in {\mathbb{C}}^2,\theta \in \mathbb{R}.$$

$$\psi (x,t)={\varphi }_{{\omega }_1}(x)e^{-i{\omega }_{1}t}+{\varphi }_{{\omega }_2}(x)e^{-i{\omega }_{2}t},({\omega }_1,{\omega }_2)\in {\mathbb{R}}^2.$$

$$\psi (\cdot ,t)⟶\mathcal{S},t\to \pm \infty ,$$

$$\left\{\begin{array}[]{l}i\dot{\psi}(x,t)=D_{m}\psi(x,t)-D_{m}^{-1}\delta(x)F(\psi(0,t)),\quad t\in{\mathbb R}\\ \psi|_{{}_{t=0}}=\psi_{0}(x),\end{array}\right|\quad x\in{\mathbb R}.$$

$$F_j(\zeta )=-{\partial }_{\overline{\zeta }{}_j}U(\zeta ),{\zeta }_j\in \mathbb{C},j=1,2,U\in C^2({\mathbb{C}}^2).$$

$$\dot{\psi }(t)=JD\mathcal{H}(\psi ),J=-i{D}_m^{-1},$$

$$\mathcal{H}(\psi )=\frac{1}{2}⟨\psi ,(-{\partial }_x^2+m^2)\psi ⟩+U(\psi (0)).$$

$$U(\zeta )\ge A-B|\zeta {|}^2,\zeta \in {\mathbb{C}}^2,A\in \mathbb{R},0\le B<m,$$

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

$$\Vert \psi (t){\Vert }_{H^1}\le C({\psi }_0),t\in \mathbb{R}.$$

$$U(\zeta )=U_1({\zeta }_1)+U_2({\zeta }_2),{\zeta }_j\in \mathbb{C},$$

$$U_j({\zeta }_j)=\sum _{n=0}^{N_j}{u}_{n,j}|{\zeta }_j{|}^{2n},u_{n,j}\in \mathbb{R},u_{N_j,j}>0,N_j\ge 2,j=1,2.$$

$$F_j({\zeta }_j)=-{\partial }_{{\overline{\zeta }}_j}{U}_j({\zeta }_j)=a_j(|{\zeta }_j{|}^2){\zeta }_j,j=1,2,$$

$$a_j(|{\zeta }_j{|}^2):=-\sum _{n=1}^{N_j}2n{u}_{n,j}|{\zeta }_j{|}^{2n-2}.$$

$$\psi (x,t)={\varphi }_{{\omega }_1}(x)e^{-i{\omega }_{1}t}+{\varphi }_{{\omega }_2}(x)e^{-i{\omega }_{2}t},({\omega }_1,{\omega }_2)\in {\mathbb{R}}^2,{\varphi }_{{\omega }_k}\in H^1,k=1,2.$$

$$\phi_{\omega_{1}}(x)=C_{1}\!\left(\!\!\begin{array}[]{cc}e^{-\varkappa_{1}|x|}+\frac{me^{-\varkappa_{1}|x|}-\varkappa_{1}e^{-m|x|}}{\omega_{1}}\\ \varkappa_{1}\mathop{\rm sgn}x\frac{e^{-\varkappa_{1}|x|}-e^{-m|x|}}{\omega_{1}}\end{array}\!\!\right)\!,~{}~{}\phi_{\omega_{2}}(x)=C_{2}\!\left(\!\!\begin{array}[]{cc}-\varkappa_{2}\mathop{\rm sgn}x\frac{e^{-\varkappa_{2}|x|}-e^{-m|x|}}{\omega_{2}}\\ e^{-\varkappa_{2}|x|}-\frac{me^{-\varkappa_{2}|x|}-\varkappa_{2}e^{-m|x|}}{\omega_{2}}\end{array}\!\!\right)\!,~{}~{}\omega_{j}\in(-m,m),~{}~{}j=1,2,$$

$$2C_{j}\varkappa_{j}=F_{j}\Big{(}C_{j}\Big{[}1+(-1)^{j+1}\frac{m-\varkappa(\omega_{j})}{\omega_{j}}\Big{]}\Big{)},\quad j=1,2.$$

$$\left\{\begin{array}[]{cc}\psi_{1}(x,t)=C_{1}\Big{(}e^{-\varkappa_{1}|x|}+\frac{me^{-\varkappa_{1}|x|}-\varkappa_{1}e^{-m|x|}}{\omega_{1}}\Big{)}e^{-i\omega_{1}t}-C_{2}\varkappa_{2}\mathop{\rm sgn}x\frac{e^{-\varkappa_{2}|x|}-e^{-m|x|}}{\omega_{2}}e^{-i\omega_{2}t}\\ \\ \psi_{2}(x,t)=C_{1}\varkappa_{1}\mathop{\rm sgn}x{\omega_{1}}\frac{e^{-\varkappa_{1}|x|}-e^{-m|x|}}{\omega_{1}}e^{-i\omega_{1}t}+C_{2}\Big{(}e^{-\varkappa_{2}|x|}-\frac{me^{-\varkappa_{2}|x|}-\varkappa_{2}e^{-m|x|}}{\omega_{2}}\Big{)}e^{-i\omega_{2}t}\end{array}\right|$$

$$\chi (x,t):=\psi (x,t)-i{D}_m^{-1}\dot{\psi }(x,t)={\chi }_{{\omega }_1}(x)e^{-i{\omega }_{1}t}+{\chi }_{{\omega }_2}(x)e^{-i{\omega }_{2}t},$$

$${\chi }_{{\omega }_k}={\varphi }_{{\omega }_k}-{\omega }_k{D}_m^{-1}{\varphi }_{{\omega }_k}=D_m^{-1}(D_m-{\omega }_k){\varphi }_{{\omega }_k},k=1,2.$$

$${\varphi }_{{\omega }_k}=D_m(D_m-{\omega }_k{)}^{-1}{\chi }_{{\omega }_k}=D_m(D_m+{\omega }_k)(D_m^2-{\omega }_k^2{)}^{-1}{\chi }_{{\omega }_k}={\chi }_{{\omega }_k}+({\omega }_k^2+{\omega }_k{D}_m)(D_m^2-{\omega }_k^2{)}^{-1}{\chi }_{{\omega }_k}.$$

$$D_m\chi (x,t)=D_m\psi (x,t)-i\dot{\psi }(x,t)=D_m^{-1}F(\psi (0,t))\delta (x).$$

$$e^{-i{\omega }_{1}t}{D}_m^2{\chi }_{{\omega }_1}(x)+e^{-i{\omega }_{2}t}{D}_m^2{\chi }_{{\omega }_2}(x)=F(\psi (0,t))\delta (x),D_m^2=-{\partial }_x^2+m^2.$$

$${\chi }_{{\omega }_k,j}(x)=C_{kj}\frac{e^{-m|x|}}{2m},k,j=1,2,$$

$$e^{-i{\omega }_{1}t}{C}_{1j}+e^{-i{\omega }_{2}t}{C}_{2j}=F_j({\varphi }_{{\omega }_1,j}(0)e^{-i{\omega }_{1}t}+{\varphi }_{{\omega }_2,j}(0)e^{-i{\omega }_{2}t}),j=1,2.$$

$$(D_m^2-{\omega }_k^2{)}^{-1}\frac{C_{kj}{e}^{-m|x|}}{2m}$$

$$\left(\begin{array}[]{cc}\phi_{\omega_{k},1}(x)\\ \phi_{\omega_{k},2}(x)\end{array}\right)=\left(\begin{array}[]{cc}C_{k1}\\ C_{k2}\end{array}\right)\frac{e^{-\varkappa_{k}|x|}}{2\varkappa_{k}}+\left(\begin{array}[]{cc}C_{k1}\\ -C_{k2}\end{array}\right)\Big{(}\frac{me^{-\varkappa_{k}|x|}-\varkappa_{k}e^{-m|x|}}{2\omega_{k}\varkappa_{k}}\Big{)}+\left(\begin{array}[]{cc}-C_{k2}\\ C_{k1}\end{array}\right)\mathop{\rm sgn}x\frac{e^{-\varkappa_{k}|x|}-e^{-m|x|}}{2\omega_{k}}.$$

$$\phi_{\omega_{k},j}(0)=\frac{C_{kj}}{2\varkappa_{k}}\big{(}1+(-1)^{j+1}\frac{m-\varkappa_{k}}{\omega_{k}}\big{)},\quad k,j=1,2.$$

Full text

We justify this mechanism by the strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time-spectrum in the spectral gap $[-m,m]$ and satisfies the original equation. This equation implies the key spectral inclusion for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of $j$-th component of the omega-limit trajectory to a single harmonic $\omega_{j}\in[-m,m]$, $j=1,2$.

Global attractor for 1D Dirac field

coupled to nonlinear oscillator

Elena Kopylova 111Research supported by the Austrian Science Fund (FWF) under Grant No. P27492-N25 and RFBR grant 18-01-00524.

*Faculty of Mathematics of Vienna University

Institute for Information Transmission Problems RAS *

Alexander Komech 222Research supported by the Austrian Science Fund (FWF) under Grant No. P28152-N35.

*Faculty of Mathematics of Vienna University

Institute for Information Transmission Problems RAS

Department Mechanics-Mathemcatics of Moscow State University *

1 Introduction

We prove a global attraction for a model 1D nonlinear Dirac equation

[TABLE]

Here $D_{m}$ is the Dirac operator $D_{m}:=\alpha\partial_{x}+m\beta$, where $m>0$, and

[TABLE]

$\psi(x,t)=(\psi_{1}(x,t),\psi_{2}(x,t))$ is a continuous $C^{2}$-valued wave function, and $F(\zeta)=(F_{1}(\zeta_{1}),F_{2}(\zeta_{2}))$, $\zeta=(\zeta_{1},\zeta_{2})\in{\mathbb C}^{2}$, is a nonlinear function. The dots stand for the derivatives in $t$. All derivatives and the equation are understood in the sense of distributions. Equation (1.1) describes the linear Dirac equation coupled to the nonlinear oscillator with the interaction concentrated in one point. We assume that equation (1.1) is $\mathbf{U}(1)$-invariant; that is,

[TABLE]

This condition leads to the existence of two-frequency solitary wave solutions of type

[TABLE]

We prove that indeed they form the global attractor for all finite energy solutions to (1.1). Namely, our main result is the following long-time asymptotics: In the case when polynomials $F_{j}$ are strictly nonlinear, any solution with initial data from $H^{1}({\mathbb R})\otimes C^{2}$ converges to the set $\mathcal{S}$ of all solitary waves:

[TABLE]

where the convergence holds in local $H^{1}$ - seminorms.

The asymptotics of type (1.4) was discovered first for linear wave and Klein-Gordon equations with external potential in the scattering theory [26, 22, 9, 8]. In this case, the attractor $\mathcal{S}$ consists of the zero solution only, and the asymptotics means well-known local energy decay.

The attraction to the set of all static stationary states with $\omega=0$ was established in [10]–[19] for a number nonlinear wave problems.

First results on the attraction to the set of all stationary orbits for nonlinear $\mathbf{U}(1)$-invariant Schrödinger equations were obtained in the context of asymptotic stability. This establishes asymptotics of type (1.4) but only for solutions with initial date close to some stationary orbit, proving the existence of a local attractor. This was first done in [3, 23, 24], and then developed in [2, 4, 5, 6, 16] and other papers.

The global attraction of type (1.4) to the solitary waves was established i) in [12, 15] for 1D Klein-Gordon equations coupled to nonlinear oscillators; ii) in [13, 14] for nD Klein-Gordon and Dirac equations with mean field interaction; iii) in [20, 21] for 3D wave and Klein-Gordon equations with concentrated nonlinearity.

The global attraction (1.4) for Dirac equation with concentrated nonlinearity was not considered previously as well as the attraction to solitary waves with two frequencies.

Let us comment on our methods. First we prove the omega-limit compactness. This means that for each sequence $s_{j}\to\infty$ the solutions $\psi(x,t+s_{j})$ contain an infinite subsequence which converges in energy seminorms for $|x|<R$ and $|t|<T$ for any $R,T>0$. Any limit function is called as the omega-limiting trajectory $\gamma(x,t)$. To prove the global convergence (1.4) is suffices to show that any omega-limiting trajectory lies on $\mathcal{S}$.

The proof relies on the study of the Fourier transform in time $\tilde{\psi}(x,\omega)$ and of its support which is the time-spectrum. The key role is played by the absolute continuity of the spectral densities $\tilde{\psi}(x,\omega)$ outside the spectral gap $[-m,m]$ for each $x\in{\mathbb R}$. The absolute continuity is a nonlinear version of Kato’s theorem on the absence of the embedded eigenvalues and provides the dispersion decay for the high energy component. Any omega-limit trajectory is the solution to (1.1).

This absolute continuity provides that the time-spectrum of $\tilde{\gamma}(x,\omega)=(\tilde{\gamma}_{1}(x,\omega),\tilde{\gamma}_{2}(x,\omega))$ is contained in the spectral gap $[-m,m]$ for each $x\in{\mathbb R}$. Finally, we apply the Titchmarsh convolution theorem (see [7, Theorem 4.3.3]) to conclude that each components $\tilde{\gamma}_{j}(x,\omega)$ of omega-limit trajectory is a singleton, i.e. it’s time-spectrum consists of a single frequency. The Titchmarsh theorem controls the inflation of spectrum by the nonlinearity. Physically, these arguments justify the following binary mechanism of the energy radiation, which is responsible for the attraction to the solitary waves: (i) nonlinear energy transfer from the lower to higher harmonics, and (ii) subsequent dispersion decay caused by the energy radiation to infinity.

The general scheme of the proof bring to mind the approach of [12, 13]. Nevertheless, the Dirac equation with nonlinear point interaction requires new ideas due to a more singular character. In particular, the equation with nonlinearity $F(\psi(0,t))\delta(x)$ seems not well posed. Indeed, the solution of the free 1D Dirac equation with the source $f(t)\delta(x)$ does not belong to $H^{1}({\mathbb R})$, so $\psi(0,t)$ is not well defined.

We found the novel model of Dirac equation with non-standard Hamilton structure, which provides good a priori estimates needed for the global well-posedness. As a consequence, the formulation of the problem and the techniques used are not a straightforward generalization of the one-dimensional result [12].

The plan of the paper is as follows. In Section 2 we state the main assumptions and results. In Section 3 we eliminate a dispersive component of the solution and construct spectral representation for the remaining part. In Section 4 we prove absolute continuity of high frequency spectrum of the remaining part. In Section 5 we exclude the second dispersive component corresponding to the high frequencies. In Section 6 we establish compactness for the remaining component with the bounded spectrum. In Section 7 we state the spectral properties of all omega-limit trajectories and apply the Titchmarsh Convolution Theorem. In Appendices we establish the global well-posedness for equation (1.1) and prove global attraction (1.4) in the case on linear $F(\psi)$.

2 Main results

Model

We consider the Cauchy problem for the Dirac equation coupled to a nonlinear oscillator:

[TABLE]

We will assume that the nonlinearity $F=(F_{1},F_{2})$ admits a real-valued potential:

[TABLE]

Then equation (2.1) formally can be written as a Hamiltonian system,

[TABLE]

where $D\mathcal{H}$ is the variational derivative of the Hamilton functional

[TABLE]

Global well-posedness

To have a priori estimates available for the proof of the global well-posedness, we assume that

[TABLE]

We will write $L^{2}$ and $H^{1}$ instead of $L^{2}({\mathbb R})\otimes C^{2}$ and instead of $H^{1}({\mathbb R})\otimes C^{2}$, respectively.

Theorem 2.1**.**

Let conditions (2.2) and (2.4) hold. Then:

For every $\psi_{0}\in H^{1}$ the Cauchy problem (2.1) has a unique solution $\psi(t)\in C({\mathbb R},H^{1})\cap C({\mathbb R},L^{2})$. 2. 2.

The map $W(t):\;\psi_{0}\mapsto\psi(t)$ is continuous in $H^{1}$ for each $t\in{\mathbb R}$. 3. 3.

The energy is conserved:

[TABLE] 4. 4.

The following a priori bound holds:

[TABLE]

We prove this theorem in Appendix A.

Solitary waves and the main theorem

We assume that the nonlinearity is polynomial. More precisely,

[TABLE]

where

[TABLE]

This assumption guarantees the bound (2.4) and it is crucial in our argument: it allow to apply the Titchmarsh convolution theorem. Equality (2.8) implies that

[TABLE]

[TABLE]

Definition 2.2**.**

(i) The solitary wave solution of equation (1.1) are solutions of the form

[TABLE]

(ii) The solitary manifold is the set: ${\cal S}=\left\{\phi_{\omega_{1}}+\phi_{\omega_{2}}{\rm:}\ ~{}~{}(\omega_{1},\omega_{2})\in{\mathbb R}^{2}\right\}$.

Note that for any $(\omega_{1},\omega_{2})\in{\mathbb R}^{2}$ there is a zero solitary wave with $\phi_{\omega_{1}}=\phi_{\omega_{2}}\equiv 0$, since $F(0)=0$. From (2.7) it follows that the set ${\cal S}$ is invariant under multiplication by $e^{i\theta}$, $\theta\in{\mathbb R}$.

Denote $\varkappa_{j}=\varkappa(\omega_{j})=\sqrt{m^{2}-\omega_{j}^{2}}>0$ for $\omega_{j}\in(-m,m)$.

Proposition 2.3**.**

(Existence of solitary waves). Assume that $F(\zeta)$ satisfies (2.9). Then nonzero solitary waves may exist only for $\omega_{j}\in(-m,m)$. The amplitudes of solitary waves are given by

[TABLE]

where $C_{j}$ are solutions to

[TABLE]

Corollary 2.4**.**

Substituting (2.12) into (2.11) we obtain the following representation for solitary wave solutions

[TABLE]

Proof of Proposition 2.3 We look for solution $\psi(x,t)$ to (1.1) in the form (2.11). Consider the function

[TABLE]

where

[TABLE]

Hence,

[TABLE]

Equation (1.1) implies, that

[TABLE]

Applying the operator $D_{m}$, we obtain by (2.15)

[TABLE]

Therefore, in the case $\omega_{1}\neq\omega_{2}$,

[TABLE]

where $C_{kj}$ are solutions to

[TABLE]

We can also assume this formulas in the case $\omega_{1}=\omega_{2}$ setting $\chi_{\omega_{2}}=0$. We will return to equation (2.18) later. First we derive the explicit formulas for $\phi_{\omega_{k}}(x)$, using (2.16) and (2.17) only. Applying [1, Formula 1.2.(11)], we get

[TABLE]

Substituting this into (2.16), we obtain

[TABLE]

Hence,

[TABLE]

Now we turn to the study of the equation (2.18). First, consider the case when $\omega_{1}=\omega_{2}=\omega$. We set $C_{1j}=C_{j}$, $C_{2j}=0$, $j=1,2$, and equation (2.18) becomes

[TABLE]

by (2.9). Using (2.20), we get after cancelation of exponential

[TABLE]

Here we denote $C_{j}^{\prime}=C_{j}/(2\varkappa)$. Finally, in the case $\omega_{1}=\omega_{2}=\omega$, equation (2.11) reads

[TABLE]

where, in accordance with (2.19),

[TABLE]

Now consider the case when $\omega_{1}\not=\omega_{2}$. Taking into account (2.9), we rewrite (2.18) as

[TABLE]

Lemma 2.5**.**

Let $\omega_{1}\neq\omega_{2}$. Then for solutions to (2.18) we have either $\phi_{\omega_{1},j}(0)=0$ or $\phi_{\omega_{2},j}(0)=0$ for each $j=1,2$.

Proof.

It suffices to consider the case $j=1$ and $\omega_{1}<\omega_{2}$ only. Denote $q_{1}:=\phi_{\omega_{1},1}(0)$, $q_{2}:=\phi_{\omega_{2},1}(0)$. We should prove that either $q_{1}=0$ or $q_{2}=0$. Assume, to the contrary, that $q_{1}\neq 0$ and $q_{2}\neq 0$. Then

[TABLE]

where ${q}_{1}\overline{q}_{2}\neq 0$ and $\overline{q}_{1}q_{2}\neq 0$. Hence, (2.10) implies

[TABLE]

where $b\neq 0$ since $a_{1}$ is a polynomial of degree $N_{1}-1\geq 1$ due to (2.8) and (2.10). Now the right hand side of (2.18) contains the terms $e^{-i[\omega_{1}t-(N_{j}-1)\delta]t}$ and $e^{-i[\omega_{2}t+(N_{j}-1)\delta]t}$ with nonzero coefficients, which are absent on the left hand side. This contradiction proves the lemma. ∎

This lemma and (2.20) imply

Corollary 2.6**.**

Let $\omega_{1}\neq\omega_{2}$. Then for solutions to (2.18) we have either $C_{1,j}=0$ or $C_{2,j}=0$ for each $j=1,2$.

Note that the cases $C_{21}=C_{22}=0$ and $C_{11}=C_{12}=0$ are exactly the case when $\omega_{1}=\omega_{2}$.

Suppose now, that $C_{12}=C_{21}=0$. Then (2.11) and (2.19) imply

[TABLE]

with $C_{1}=C_{11}$, $C_{2}=C_{22}$. Taking into account (2.20), we obtain equations for $C_{j}$:

[TABLE]

Equations (2.24) and (2.24) will coincide with equations (2.14) and (2.13) after the replacement $C_{j}$ by $2C_{j}\varkappa_{j}$.

It is easy to check that in the case $C_{11}=C_{22}=0$, we obtain the same formulas, interchanging $\omega_{1}$ and $\omega_{2}$.

Proposition is completely proved.

Remark 2.7*.*

(i) Equation (2.13) has generally discrete set of solutions $C_{j}$, while $\omega_{j}$ belongs generally to an open set.

(i) In the linear case, when $F_{j}(\psi_{k})=a_{j}\psi_{j}$ with $a_{j}\in{\mathbb R}$, the situation is contrary : we see from (2.13) that

[TABLE]

i.e., $\omega_{j}$ generally belongs to a discrete set, while $C_{j}\in{\mathbb C}$ is arbitrary.

Our main result is the following theorem.

Theorem 2.8** (Main Theorem).**

Let the nonlinearity $F(\psi)$ satisfy Assumption LABEL:assumption-a. Then for any $\psi_{0}\in H^{1}$ the solution $\psi(t)\in C({\mathbb R},H^{1})$ to the Cauchy problem (2.1) with $\psi(0)=\psi_{0}$ converges to ${\cal S}$ in the space $H^{1}_{loc}({\mathbb R})\otimes C^{2}$:

[TABLE]

3 Splitting of solutions

It suffices to prove Theorem 2.8 for $t\to+\infty$; We will only consider the solution $\psi(x,t)$ restricted to $t\geq 0$ and split it as

[TABLE]

Here $\phi(x,t)$ is a solution to the Cauchy problem for the free Dirac equation

[TABLE]

and $\psi_{S}(x,t)$ is a solution to the Cauchy problem for Dirac equation with the source

[TABLE]

The following lemma states well known local decay for the free Dirac equation.

Lemma 3.1**.**

(cf. [14, Proposition 4.3]) Let $\psi_{0}\in H^{1}$. Then $\phi\in C_{b}(\overline{{\mathbb R}^{+}},H^{1})$, and $\forall R>0$,

[TABLE]

Now (2.6) implies that

[TABLE]

Due to (3.3) it suffices to prove (2.26) for $\psi_{S}$ only.

Complex Fourier-Laplace transform

Let us analyse the complex Fourier-Laplace transform of $\psi_{S}(x,t)$:

[TABLE]

where $C^{+}:=\{z\in{\mathbb C}:\;{\rm Im\,}z>0\}$. Due to (3.4), $\tilde{\psi}_{S}(\cdot,\omega)$ is an $H^{1}$-valued analytic function of $\omega\in{\mathbb C}^{+}$.

Denote $f(t)=F(\psi(0,t)$. Then equation (3.2) for $\psi_{S}$ with zero initial data implies that

[TABLE]

It is easy to see that

[TABLE]

where $G(\cdot,\omega)\in H^{1}$ is the unique elementary solution to

[TABLE]

This solution is given by $\displaystyle G(x,\omega)=\frac{e^{ik(\omega)|x|}}{2ik(\omega)}$, where $k(\omega)$ stands for the analytic function

[TABLE]

which we extend to $\omega\in\mkern 2.0mu\overline{\mkern-2.0mu{\mathbb C}^{+}\mkern-5.0mu}\mkern 5.0mu$ by continuity. Thus,

[TABLE]

Note, that

[TABLE]

Therefore,

[TABLE]

and (3.7) becomes

[TABLE]

Here the last term vanishes for $x=0$. Denote $y(t):=\psi_{S}(0,t)\in C_{b}({\mathbb R})$. Then (3.8) implies

[TABLE]

Now (3.8) becomes

[TABLE]

Let us extend $\psi_{S}(x,t)$ and $f(t)$ by zero for $t<0$. Then

[TABLE]

by (3.4). The Fourier transform $\hat{\psi}_{S}(\cdot,\omega):=\mathcal{F}_{t\to\omega}[\psi_{S}(\cdot,t)]$ is a tempered $H^{1}$-valued distribution of $\omega\in{\mathbb R}$. The distribution $\hat{\psi}_{S}(\cdot,\omega)$ is the boundary value of the analytic $H^{1}$-valued function $\tilde{\psi}_{S}(\cdot,\omega)$, in the following sense:

[TABLE]

where the convergence holds in the space of tempered distributions $\mathscr{S}^{\prime}({\mathbb R},H^{1})$. Indeed,

[TABLE]

and $\psi_{S}(\cdot,t)e^{-\varepsilon t}\mathop{\longrightarrow}\limits_{\varepsilon\to 0+}\psi_{S}(\cdot,t)$, where the convergence holds in $\mathscr{S}^{\prime}({\mathbb R},H^{1})$ by (3.11). Therefore, (3.12) holds by the continuity of the Fourier transform $\mathcal{F}_{t\to\omega}$ in $\mathscr{S}^{\prime}({\mathbb R})$.

Similarly to (3.12), the distributions $\hat{f}(\omega)$ and $\hat{y}(\omega)$ of $\omega\in{\mathbb R}$ are boundary values of analytic in ${\mathbb C}^{+}$ functions $\tilde{f}(\omega)$ and $\tilde{y}(\omega)$, $\omega\in{\mathbb C}^{+}$, respectively:

[TABLE]

since the function $f(t)=F(\psi(0,t))$ is bounded for $t\geq 0$ and vanishes for $t<0$. The convergences hold in the space of tempered distributions $\mathscr{S}^{\prime}({\mathbb R})$. Let us justify that the representation (3.10) for $\hat{\psi}_{S}(x,\omega)$ is also valid when $\omega\in{\mathbb R}$.

Lemma 3.2**.**

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

[TABLE]

Here the multiplications are understood in the sense of quasimeasures (see [12, Appendix B]).

The proof follows from (3.10) similarly to [12, Proposition 3.1]. Namely, the convergence (3.13) holds in the space of quasimeasures, while $e^{ik(\omega)|x|}$ and $\frac{e^{ik(\omega)|x|}-e^{-m|x|}}{2\omega}$ are multiplicators in the space of quasimeasures.

4 Absolutely continuous spectrum

Denote

[TABLE]

Consider the functions

[TABLE]

Let $\hat{z}^{\,\pm}(\omega)$ be boundary values of $\tilde{z}^{\,\pm}(\omega)$ on $\Omega_{0}$:

[TABLE]

Now we rewrite (3.10) as

[TABLE]

We study the regularity of $\hat{z}^{\,\pm}(\omega)$. Note that the function $\omega k(\omega)$ is positive for $\omega\in\Omega_{0}$.

Proposition 4.1**.**

The distributions $\hat{z}^{\,\pm}(\omega)$ are absolutely continuous for $\omega\in\Omega_{0}$, i.e. and $\hat{z}^{\,\pm}\in L^{1}_{loc}(\Omega_{0})$. Moreover,

[TABLE]

Proof.

We use the arguments of Paley-Wiener type. Namely, the Parseval identity and (3.4) imply that

[TABLE]

Then (4.5) gives

[TABLE]

Evidently,

[TABLE]

Hence, (4.3) and (4.6) results in

[TABLE]

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

Lemma 4.2**.**

(cf. [12, Lemma 3.2])

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. 2.

For any $\delta>0$ there exists $\varepsilon_{\delta}>0$ such that

[TABLE]

Substituting (4.9) into (4.7), we get:

[TABLE]

with the same constant $C$ as in (4.7), and the region $\Omega_{\delta}$ defined in (4.1). We conclude that for each $\delta>0$ the set of functions

[TABLE]

defined for $\omega\in\Omega_{\delta}$, is bounded in the Hilbert space $L^{2}(\Omega_{\delta})$, and, by the Banach Theorem, is weakly compact. Hence, the convergence of the distributions (3.13) implies the following weak convergence in the Hilbert space $L^{2}(\Omega_{\delta})$:

[TABLE]

where the limit function $g^{\pm}_{\delta}(\omega)$ coincides with the distribution $\hat{z}^{\,\pm}(\omega)|\omega k(\omega)|^{1/2}$ restricted onto ${\Omega_{\delta}}$. It remains to note that the norms of all functions $g^{\pm}_{\delta}$, $\delta>0$, are bounded in $L^{2}(\Omega_{\delta})$ by (4.10), hence (4.4) follows. Finally, $\hat{z}^{\,\pm}(\omega)\in L^{1}_{loc}(\Omega_{0})$ by (4.4) and the Cauchy-Schwarz inequality. ∎

Lemma 4.3**.**

[TABLE]

Proof.

Denote the $2\times 2$ matrix $A^{\pm}(\omega)=I+\frac{m}{\omega}\beta\pm\frac{ik(\omega)}{\omega}\alpha$. Then $\hat{z}^{\,\pm}(\omega)=-A^{\pm}(\omega)\frac{\hat{f}(\omega)}{2ik(\omega)}$. For any $\omega\in\Omega_{0}$, the matrix $A^{\pm}(\omega)$ has two eigenvalues: $\lambda=0$ and $\lambda=2$ since

[TABLE]

The unit eigenvectors $\nu^{\pm}(\omega)$ of operators $A^{\pm}(\omega)$ corresponding to the eigenvalue $\lambda=2$ read

[TABLE]

Denote $g(\omega):=\frac{\hat{f}(\omega)}{2ik(\omega)}$, and $a^{\pm}(\omega):=\langle g(\omega),\nu^{\pm}(\omega)\rangle$. Then $\hat{z}^{\,\pm}(\omega)=-A^{\pm}(\omega)g(\omega)=-2a^{\pm}(\omega)\nu^{\pm}(\omega)$, and hence

[TABLE]

Taking into account (4.13), we get the system of equations for $g(\omega)=(g_{1}(\omega),g_{2}(\omega))$

[TABLE]

Therefore

[TABLE]

Hence,

[TABLE]

Now (4.12) follows from (4.4). ∎

5 Further decomposition of solutions

Denote

[TABLE]

and set

[TABLE]

where

[TABLE]

Consider

[TABLE]

We will show that $\psi_{d}(x,t)$ is a dispersive component of the solution $\psi(x,t)$, in the following sense.

Proposition 5.1**.**

(i) $\psi_{d}(\cdot,t)$ is a bounded continuous $H^{1}$-valued function:

[TABLE]

(ii) The local energy decay holds for $\psi_{d}(\cdot,t)$: for any $R>0$,

[TABLE]

Proof.

We split $\psi_{d}(x,t)$ as $\psi_{d}(x,t)=\varphi_{d}(x,t)+\chi_{d}(x,t)$, where

[TABLE]

First, consider $\chi_{d}(x,t)$. Note that

[TABLE]

by Lemma 4.3. Hence,

[TABLE]

Moreover,

[TABLE]

by Riemann-Lebesgue Theorem. Now consider $\varphi_{d}(x,t)$. Changing the variable $\omega\to k(\omega)=\sqrt{\omega^{2}-m^{2}}$, we rewrite $\varphi_{d}(x,t)$ as follows:

[TABLE]

Here $\omega(k)=\sqrt{k^{2}+m^{2}}$ is the branch analytic for ${\rm Im\,}k>0$ and continuous for ${\rm Im\,}k\geq 0$. Note that the function $\omega(k)$, $k\in{\mathbb R}\backslash 0$, is the inverse function to $k(\omega)$ defined on $\mkern 2.0mu\overline{\mkern-2.0mu{\mathbb C}^{+}\mkern-5.0mu}\mkern 5.0mu$ (see (3.6)) and restricted onto $\Omega_{0}$. Let us introduce the functions

[TABLE]

Both functions $\varphi^{\pm}(x,t)$ are solutions to the free Dirac equation (3.1) on the whole real line (see Appendix B). Moreover,

[TABLE]

Hence, the Parseval identity implies

[TABLE]

by (4.10). Hence, both $\varphi^{-}$ and $\varphi^{+}$ are bounded continuous $H^{1}$-valued functions:

[TABLE]

and for any $R>0$

[TABLE]

by Lemma 3.1. The function $\varphi_{d}(x,t)$ coincides with $\varphi^{+}(x,t)$ for $x\geq 0$ and with $\varphi^{-}(x,t)$ for $x\leq 0$:

[TABLE]

It remains to note that $\psi_{d}(x,t)=\varphi_{d}(x,t)+\chi_{d}(x,t)$ has no jump at $x=0$ and therefore $\partial_{x}\psi_{d}(x,t)$ is square-integrable over the whole $x$-axis. Hence,

[TABLE]

Finally, (5.5) follows from (5.14) and (5.9), and (5.6) follows from (5.15) and (5.10). ∎

Denote $y_{d}(t)=\psi_{d}(0,t)\in C_{b}(|{\mathbb R})$. Formulas (5.2) and (5.3) imply

[TABLE]

and (5.2) becomes

[TABLE]

6 Bound component

Spectral representation

We introduce the bound component of the solution $\psi(x,t)$ by

[TABLE]

Then (3.11) and (5.5) imply that

[TABLE]

In particular, $y_{b}(t):=\psi_{b}(0,t)=\psi_{S}(0,t)-\psi_{d}(0,t)\in C_{b}({\mathbb R})$. Hence, $\hat{y}_{b}(\omega):=\hat{\psi}_{b}(0,\omega)$ is a quasimeasure. Moreover, formulas (3.13) and (5.17) yield

[TABLE]

Here we denote

[TABLE]

Further, (5.1) implies, that

[TABLE]

Denote

[TABLE]

where $k(\omega)$ was introduced in (3.6). Let us note that $\varkappa(\omega)>0$ for $\omega\in(-m,m)$. Now we rewrite (6.3) as

[TABLE]

where

[TABLE]

Hence

[TABLE]

Now (3.14), (5.18), (6.1) and (6.8) imply the multiplicative relation

[TABLE]

where we denote

[TABLE]

From (6.8) and (6.10) it follows that $\hat{h}_{b}(x,\omega)$ for any fixed $x\in{\mathbb R}\setminus 0$ is a quasimeasure with the support $\mathop{\rm supp}\hat{h}_{b}(x,\omega)\subset[-m,m]$. Moreover, $e^{-\varkappa(\omega)|x|}$ and $(e^{-\varkappa(\omega)|x|}-e^{-m|x|})/\omega$ are multiplicators. Hence, function $\hat{\psi}_{b}(x,\omega)$ is quasimeasure for any fixed $x\in{\mathbb R}$ with supports in $[-m,m]$. Finally,

[TABLE]

where $\langle\cdot,\cdot\rangle$ is an extension of the scalar product $\langle f,g\rangle=\int f(\omega)\overline{g}(\omega)d\omega$.

Compactness

We are going to prove a compactness of the set of translations of the bound component, $\{\psi_{b,n}(x,s+t){\rm:}\ s\geq 0\}$, $n=1,2$.

Lemma 6.1**.**

(i) The function $\psi_{b}(x,t)$ is smooth for $x\neq 0$ and $t\in{\mathbb R}$. Moreover, for any fixed $x\not=0$, $t\in{\mathbb R}$, and any nonnegative integers $j,k$, the following representation holds

[TABLE]

where

[TABLE]

(ii) There is a constant $C_{j,k}>0$ so that

[TABLE]

The lemma follows similarly Proposition 4.1 from [12], since the factors $e^{-\varkappa(\omega)|x|}\zeta(\omega)$, $\frac{e^{-\varkappa(\omega)|x|}-e^{-m|x|}}{2\omega}\zeta(\omega)$, and $\frac{\varkappa^{j}(\omega)-m^{j}}{\omega}\zeta(\omega)$ are multiplicators in the space of quasimeasures. Here $\zeta(\omega)\in C_{0}^{\infty}({\mathbb R})$ is any cutoff function satisfying

[TABLE]

Corollary 6.2**.**

By the Ascoli-Arzelà Theorem, for any sequence $s_{l}\to\infty$ there exists a subsequence (which we also denote by $s_{l}$) such that for any nonnegative integers $j$ and $k$,

[TABLE]

for some $\gamma\in C_{b}({\mathbb R},H^{1})$. The convergence in (6.14) is uniform in $x$ and $t$ as long as $|x|+|t|\leq R$, for any $R>0$.

We call omega-limit trajectory any function $\gamma(x,t)$ that can appear as a limit in (6.14). Previous analysis demonstrates that the long-time asymptotics of the solution $\psi(x,t)$ in $H^{1}_{loc}$ depends only on the bound component $\psi_{b}(x,t)$. By Corollary 6.2, to conclude the proof of Theorem 2.8, it suffices to check that every omega-limit trajectory belongs to the set of solitary waves; that is,

[TABLE]

where $\psi_{\Omega^{+}}(x,t)=(e^{-i\omega_{1}^{+}t}\phi_{\omega_{1}^{+}}(x),e^{-i\omega_{2}^{+}t}\phi_{\omega_{2}^{+}}(x))$ with some $\omega_{1}^{+},\omega_{2}^{+}\in[-m,m]$.

Spectral identity for omega-limit trajectories

Here we study the time spectrum of the omega-limit trajectories.

Definition 6.3**.**

Let $\mu$ be a tempered distribution. By $\mathop{\rm Spec}\mu$ we denote the support of its Fourier transform:

[TABLE]

Proposition 6.4**.**

For any omega-limit trajectory $\gamma(x,t)$, the following spectral representation holds:

[TABLE]

where $\hat{p}(\omega)$ and $\hat{q}(x,\omega)=(\beta-\alpha\mathop{\rm sgn}x)\sigma^{-1}(\omega)\hat{p}(\omega)$ are quasimeasures for all $x\in{\mathbb R}$, and

[TABLE] 2. 2.

The following bound holds:

[TABLE]

Note that, according to (6.16), $\hat{p}(\omega)$ is the Fourier transform of the function $p(t):=\gamma(0,t)$, $t\in{\mathbb R}$.

Proof.

Formula (6.9) and representation (6.11) imply that

[TABLE]

Further, the convergence (6.14) and the bound (6.13) with $j=k=0$ imply that

[TABLE]

where $p\in C_{b}({\mathbb R})$. The convergence is uniform on $[-T,T]$ for any $T>0$. Hence,

[TABLE]

in the space of quasimeasures. Therefore,

[TABLE]

in the space of quasimeasures. Similarly,

[TABLE]

Hence, the representation (6.16) follows from (6.19), (6.22) and (6.23); and (6.17) follows from (6.5). Finally, the bound (6.18) follows from (6.2) and (6.14). ∎

The relation (6.16) implies the basic spectral identity:

Corollary 6.5**.**

For any omega-limit trajectory $\gamma(x,t)$,

[TABLE]

7 Nonlinear spectral analysis

Here we will derive (6.15) from the following identity:

[TABLE]

which will be proved in three steps.

Step 1

The identity (7.1) is equivalent to $\hat{p}_{j}(\omega)\sim\delta(\omega-\omega_{j}^{+})$, so we start with an investigation of $\mathop{\rm Spec}p_{j}:=\mathop{\rm supp}\hat{p}_{j}$.

Lemma 7.1**.**

For omega-limit trajectories the following spectral inclusion holds:

[TABLE]

Proof.

The convergence (6.14), Lemma 3.1 and Proposition 5.1 (ii) imply that the limiting trajectory $\gamma(x,t)$ is a solution to equation (1.1):

[TABLE]

Applying to both side operator $D_{m}$, we get

[TABLE]

Since $\gamma(x,t)$ is smooth function for $x\leq 0$ and $x\geq 0$, we get the following algebraic identities :

[TABLE]

The identities imply the spectral inclusion

[TABLE]

On the other hand, $\mathop{\rm Spec}\gamma^{\prime}_{j}(0+,\cdot)\cup\mathop{\rm Spec}\gamma^{\prime}_{k}(0-,\cdot)\subset\mathop{\rm Spec}p_{j}$ by (6.24). Therefore, (7.5) implies (7.2). ∎

Step 2

Proposition 7.2**.**

For any omega-limit trajectory, the following identity holds:

[TABLE]

Proof.

We are going to show that (7.6) follows from the key spectral relations (6.17), (7.2). Our main assumption (LABEL:f-is-such) implies that the function $F_{j}(t):=F_{j}(p_{j}(t))$ admits the representation (cf. (2.9))

[TABLE]

where, according to (LABEL:f-is-such),

[TABLE]

Both functions $p_{j}(t)$ and $a_{j}(t)$ are bounded continuous functions in ${\mathbb R}$ by Proposition 6.4 (ii). Hence, $p_{j}(t)$ and $a_{j}(t)$ are tempered distributions. Furthermore, $\hat{p}_{j}$ and $\hat{\overline{p}}_{j}$ have the supports contained in $[-m,m]$ by (6.17). Hence, $a_{j}$ also has a bounded support since it is a sum of convolutions of finitely many $\hat{p}_{j}$ and $\hat{\overline{p}}_{j}$ by (7.8). Then the relation (7.7) translates into a convolution in the Fourier space, $\hat{F}_{j}=\hat{a}_{j}\ast\hat{p}_{j}/(2\pi),$ and the spectral inclusion (7.2) takes the following form:

[TABLE]

Let us denote ${\bf F}_{j}=\mathop{\rm supp}\hat{F}_{j}$, ${\bf A}_{j}=\mathop{\rm supp}\hat{a}_{j}$, and ${\bf P}_{j}=\mathop{\rm supp}\hat{p}_{j}$. Then the spectral inclusion (7.9) reads as

[TABLE]

On the other hand, it is well-known that $\mathop{\rm supp}\hat{a}_{j}\ast\hat{p}_{j}\subset\mathop{\rm supp}\hat{a}_{j}+\mathop{\rm supp}\hat{p}_{j}$, or ${\bf F}_{j}\subset{\bf A}_{j}+{\bf P}_{j}.$ Moreover, the Titchmarsh convolution theorem (see [7, Theorem 4.3.3]) imply that

[TABLE]

Now (7.10) and (7.11) result in

[TABLE]

so that $\inf{\bf A}_{j}\geq 0\geq\sup{\bf A}_{j}$. Thus, we conclude that $\mathop{\rm supp}\hat{a}_{j}={\bf A}_{j}\subset\{0\}$, therefore the distribution $\hat{a}_{j}(\omega)$ is a finite linear combination of $\delta(\omega)$ and its derivatives. Then $a_{j}(t)$ are polynomial in $t$; since $a_{j}(t)$ is bounded by Proposition 6.4 (ii), we conclude that $a_{j}(t)$ is constant. Now the relation (7.6) follows since $a_{j}(t)$ is a polynomial in $|p_{j}(t)|$, and its degree is strictly positive by (7.8). ∎

Step 3

Now the same Titchmarsh arguments imply that $P_{j}:=\mathop{\rm Spec}p_{j}$ is a point $\omega_{j}^{+}\in[-m,m]$. Indeed, (7.6) means that $p_{j}(t)\overline{p}_{j}(t)\equiv C_{j}$, hence in the Fourier transform $\hat{p}_{j}\ast\hat{\overline{p}_{j}}=2\pi C_{j}\delta(\omega)$. Therefore, if $p_{j}$ is not identically zero, the Titchmarsh Theorem implies that

[TABLE]

Hence $\inf P_{j}=\sup P_{j}$ and therefore $P_{j}=\omega_{k}^{+}\in[-m,m]$, so that $\hat{p}_{j}(\omega)$ is a finite linear combination of $\delta(\omega-\omega_{j}^{+})$ and its derivatives. As the matter of fact, the derivatives could not be present because of the boundedness of $p_{j}(t)=\gamma_{j}(0,t)$ that follows from Proposition 6.4 (ii). Thus, $\hat{p}_{j}\sim\delta(\omega-\omega_{j}^{+})$, which implies (7.1).

Conclusion of the proof of Theorem 2.8

According to (7.1) and (6.23)

[TABLE]

where

[TABLE]

Then the representation (6.16) implies

[TABLE]

After simple evaluation, (7.13) becomes

[TABLE]

where we denote

[TABLE]

Therefore, $\gamma(x,t)$ is a solitary wave (2.14). Due to Lemma 3.1 and Proposition 5.6 it remains to prove that

[TABLE]

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

[TABLE]

for some $\delta>0$. According to Corollary 6.2, there exist a subsequence $s_{l_{n}}$ of the sequence $s_{l}$, $\omega_{1}^{+},\omega_{2}^{+}\in{\mathbb R}$ and vector-function $\gamma(x,t)$, defined in (7.13) such that the following convergence hold

[TABLE]

This implies that

[TABLE]

where

[TABLE]

The convergence (7.16) contradict to (7.15). This completes the proof of Theorem 2.8. $\Box$

Appendix A Global well-posedness

Here we prove Theorem 2.1. We first need to adjust the nonlinearity $F$ so that it becomes bounded, together with its derivatives. Define

[TABLE]

where $\psi_{0}\in H^{1}$ is the initial data from Theorem 2.1 and ${A}$, ${B}$ are constants from (2.4). Then we may pick a modified potential function $\widetilde{U}\in C^{2}({\mathbb C}^{2})$, so that

[TABLE]

$\widetilde{U}(\zeta)$ satisfies (2.4) with the same constants ${A}$, ${B}$ as $U(\zeta)$ does:

[TABLE]

and so that $|\widetilde{U}(\zeta)|$, $|\widetilde{U}^{\prime}(\zeta)|$, and $|\widetilde{U}^{\prime\prime}(\zeta)|$ are bounded for $\zeta\in C^{2}$. We define

[TABLE]

and consider the Cauchy problem of type (1.1) with the modified nonlinearity,

[TABLE]

This is a Hamiltonian system, with the Hamilton functional

[TABLE]

which is Fréchet differentiable in the space $H^{1}$. By the Sobolev embedding theorem, $\|\psi\|_{L^{\infty}}^{2}\leq\frac{1}{2}\|\psi\|_{H^{1}}^{2}$. Moreover,

[TABLE]

where $|\!\|\psi\|\!|^{2}:=\|\psi^{\prime}\|_{L^{2}}^{2}+m^{2}\|\psi\|_{L^{2}}^{2}$. Indeed, the Cauchy- Schwarz inequality and the Parseval identity imply

[TABLE]

Thus (A.3) leads to

[TABLE]

Taking into account (A.5), we obtain the inequality

[TABLE]

which implies

[TABLE]

Lemma A.1**.**

$\widetilde{\mathcal{H}}(\psi_{0})=\mathcal{H}(\psi_{0})$. 2. 2.

If $\psi\in H^{1}$ satisfies $\widetilde{\mathcal{H}}(\psi)\leq\widetilde{\mathcal{H}}(\psi_{0})$, then $\widetilde{U}(\psi(0))=U(\psi(0))$.

Proof.

According to (A.6), (A.7), and the choice of $\Lambda(\psi_{0})$ in (A.1),

[TABLE]

Thus, according to the choice of $\widetilde{U}$ (equality (A.2)), $\widetilde{U}(\psi_{0}(0))=U(\psi_{0}(0)),$ proving (i). 2. 2.

By (A.6), (A.7), the condition $\widetilde{\mathcal{H}}(\psi)\leq\widetilde{\mathcal{H}}(\psi_{0})$, and part (i) of this lemma, we have:

[TABLE]

Hence, (ii) follows by (A.2).

∎

Remark A.2*.*

We will show that if $\psi(t)$ solves (A.4), then $\widetilde{\mathcal{H}}(\psi(t))=\widetilde{\mathcal{H}}(\psi_{0})$, and therefore $\widetilde{U}(\psi(0,t))=U(\psi(0,t))$ by Lemma A.1 (ii). Hence, $\widetilde{F}(\psi(0,t))=F(\psi(0,t))$ for all $t\geq 0$, allowing us to conclude that $\psi(t)$ solves (1.1) as well as (A.4).

Local well-posedness

Denote by $e^{-iD_{m}t}$ the dynamical group of the free Dirac equation. Then the solution to the Cauchy problem (A.4) can be represented by

[TABLE]

The next lemma establishes the contraction principle for the integral equation (A.8).

Lemma A.3**.**

There exists a constant $C>0$ so that for any two functions $\psi_{k}(\cdot,t)\in C([-1,1],H^{1})$, $k=1,\,2$, one has:

[TABLE]

Proof.

It suffices to consider $t\geq 0$. In this case,

[TABLE]

where ${\cal G}(t)$ is the integral operator with the integral kernel

[TABLE]

Here $J_{0}$ is the Bessel function. According to (A.8) and (A.9),

[TABLE]

where

[TABLE]

First we prove the $L_{2}$ estimate for $I_{j}(x,t)$. By the Sobolev embedding theorem,

[TABLE]

where we took into account that $|\nabla\widetilde{F}(z)|$ is bounded due to the choice of $\widetilde{U}$. Similarly,

[TABLE]

Now, we derive the $L^{2}$ estimates for the derivatives $\partial_{x}I_{1}(x,t)$ and $\partial_{x}I_{2}(x,t)$. We have

[TABLE]

where

[TABLE]

Hence,

[TABLE]

Further,

[TABLE]

Hence,

[TABLE]

The $L^{2}$ norm of $J_{1}(x,t)$ is estimated similarly to the $L^{2}$ norm of $\partial_{x}I_{1}(x,t)$. Further, similarly to (A.10), we get

[TABLE]

[TABLE]

∎

For $E>0$, let us denote $H^{1}_{E}=\{\psi_{0}\in H^{1}{\rm:}\ \mathcal{H}(\psi_{0})\leq E\}$.

Corollary A.4**.**

For any $E>0$ there exists $\tau=\tau(E)>0$ such that for any $\psi_{0}\in H^{1}_{E}$ there is a unique solution $\psi(x,t)\in C([-\tau,\tau],H^{1})$ to the Cauchy problem (A.4) with the initial condition $\psi(0)=\psi_{0}$. 2. 2.

The maps $W(t):\;\psi_{0}\mapsto\psi(t)$, $t\in[-\tau,\tau]$ are continuous from $H^{1}_{E}$ to $H^{1}$.

Energy conservation and global well posedness

Lemma A.5**.**

For the solution to the Cauchy problem (A.4) with the initial data $\psi_{0}\in H^{1}$, the energy is conserved: $\widetilde{\mathcal{H}}(\psi(t))={\rm const}$, $t\in[-\tau,\tau]$.

Proof.

The Galerkin approximations provide a solution $\psi\in L^{\infty}({\mathbb R},H^{1})$ to (A.4) with energy estimate

[TABLE]

Moreover, estimates from the proof of Lemma A.3 imply that $\psi\in C({\mathbb R},H^{1})$. Therefore,

[TABLE]

since the inequality (A.14) also holds with $\psi(s)$ instead of $\psi_{0}$ for every $s\in[-\tau,\tau]$ by the uniqueness of solutions proved in Corollary A.4. ∎

Corollary A.6**.**

The solution $\psi$ to the Cauchy problem (A.4) with the initial data $\psi|_{{}_{t=0}}=\psi_{0}\in H^{1}$ exists globally: $\psi\in C_{b}({\mathbb R},H^{1})$. 2. 2.

The energy is conserved: $\widetilde{\mathcal{H}}(\psi(t))=\widetilde{\mathcal{H}}(\psi_{0}),\qquad t\in{\mathbb R}$.

Proof.

Corollary A.4 (i) yields a solution $\psi\in C([-\tau,\tau],H^{1})$ with a positive $\tau=\tau(E)$. However, the value of $\mathcal{H}(\psi(t))$ is conserved for $t\leq\tau$ by Lemma A.5. Corollary A.4 (i) allows then to extend $\psi$ to the interval $[-2\tau,2\tau]$, and eventually to all $t\in{\mathbb R}$. ∎

Conclusion of the proof of Theorem 2.1

The trajectory $\psi\in C_{b}({\mathbb R},H^{1})$ is a solution to (A.4), for which Corollary A.6 (ii) together with Lemma A.1 (i) imply the energy conservation (2.5). By Lemma A.1 (ii), $\widetilde{U}(\psi(0,t))=U(\psi(0,t))$, for all $t\in{\mathbb R}$. This tells us that $\psi(x,t)$ is a solution to (1.1). Finally, the a priori bound (2.6) follows from (A.7) and the conservation of $\mathcal{H}(\psi(t))$. This finishes the proof of Theorem 2.1.

Appendix B Free Dirac equation

Here we show that the function

[TABLE]

are the solutions to the free Dirac equation (3.1). It suffices to prove that for any $q=(q_{1},q_{2})\in{\mathbb C}^{2}$ the functions

[TABLE]

satisfy

[TABLE]

Indeed, the functions $u_{j}^{\pm}(x,t)=e^{\pm ikx}e^{-i\omega(k)t}q_{j}$ obviously satisfy

[TABLE]

Moreover,

[TABLE]

Hence,

[TABLE]

Appendix C Linear case

Here we consider the linear case, when

[TABLE]

Now equation (1.1) reads

[TABLE]

We restrict our consideration to the case when $a_{j}<2m$, $j=1,2$. It is in this case that condition (2.4) is satisfied, and then all conclusions of Theorem 2.1 on global well-posedness for equation (C.2) hold .

Let us calculate corresponding solitary waves. Now equations (2.13) become

[TABLE]

Cancelling the nonzero factors $\sqrt{m+\omega_{1}}$ and $\sqrt{m-\omega_{2}}$, we obtain

[TABLE]

Multiplying both sides of this equations by $\sqrt{m+\omega_{1}}+\sqrt{m-\omega_{1}}$ and $\sqrt{m+\omega_{2}}+\sqrt{m-\omega_{2}}$, respectively, we get

[TABLE]

where the left hand sides of both equalities are positive for $\omega_{j}\in(-m,m)$. Hence, there are no nonzero solitary waves for $a_{j}\leq 0$, $j=1,2$. For $0<a_{j}<2m$, the corresponding equation of (C.3) has the unique solution

[TABLE]

Finally, we conclude that for $0<a_{j}<2m$, $j=1,2$, the set of finite energy solitary waves is given by

[TABLE]

In the case $a_{1}<0$ the set ${\cal S}$ is given by (C.5) with $C_{1}=0$, while $C_{2}\in{\mathbb C}$ is arbitrary, and vice versa. .

Theorem C.1**.**

Assume that $F_{j}(\psi_{j})=a_{j}\psi_{j}$, where $a_{j}<2m$, $j=1,2$. Then for any $\psi_{0}\in H^{1}$ the solution $\psi(t)\in C({\mathbb R},H^{1})$ to the Cauchy problem (1.1) with $\psi(0)=\psi_{0}$ converges to ${\cal S}$ in the space $H^{1}_{loc}({\mathbb R})\otimes C^{2}$:

[TABLE]

Proof.

We proceed as in the proof of Theorem 2.8 until we get equation (7.3), which takes now the following form:

[TABLE]

In this case we cannot use the Titchmarsh arguments since the condition (2.8) fails. Now we should prove that

[TABLE]

for all solutions of (C.7) with the structure (6.16). In the Fourier transform $\hat{\gamma}(x,\omega)=\mathcal{F}_{t\to\omega}[\gamma(x,t)]$ the equation (C.7) becomes

[TABLE]

Applying the operator $D_{m}+\omega$, we get

[TABLE]

On the other hand, the representation (6.16) implies that

[TABLE]

Substituting this into (C.9) and equating coefficients with delta functions, we obtain

[TABLE]

where $\sigma^{-1}(\omega)$ is the diagonal matrix (6.8) with matrix elements $[\sigma^{-1}(\omega)]_{jj}:=\nu_{j}(\omega)=\varkappa(\omega)+m-(-1)^{j}\omega$.

Therefore, on the support of the distribution $\hat{p}_{j}(\omega)$, $j=1,2$, the identity hold

[TABLE]

Simplifying, we arrive at the following equation

[TABLE]

which are exactly equations (C.3) for soliton parameters $\omega_{j}$. Finally, we obtain that for $0<a_{j}<2m$

[TABLE]

where $\omega_{j}$ are given in (C.4). Hence, the inclusion (C.8) follows. This finishes the proof of Theorem C.1. ∎