KdV-like solitary waves in two-dimensional FPU-lattices

TL;DR

This paper proves the existence of solitary waves in two-dimensional FPU lattices in the KdV limit, extending previous results to include arbitrary directions and non-unidirectional profiles through asymptotic analysis.

Contribution

It generalizes prior work by allowing for arbitrary propagation directions and complex wave profiles in 2D FPU lattices.

Findings

Existence of solitary waves in 2D FPU lattices established.

Generalization to arbitrary directions and non-unidirectional profiles.

Use of asymptotic analysis of integral equations.

Abstract

We prove the existence of solitary waves in the KdV limit of two-dimensional FPU-type lattices using asymptotic analysis of nonlinear and singularly perturbed integral equations. In particular, we generalize the existing results by Friesecke and Matthies since we allow for arbitrary propagation directions and non-unidirectional wave profiles.

Equations

$$\left(\begin{array}{c}{r}_{\ast }i\\ r_{\ast }j\end{array}\right)+q_{ij}(t),$$

$$\begin{array}{rl}{\ddot{q}}_{ij}=& +(\nabla {\varphi }_1)(q_{i+1j}-q_{ij})-(\nabla {\varphi }_1)(q_{ij}-q_{i-1j})\\ & +(\nabla {\varphi }_2)(q_{ij+1}-q_{ij})-(\nabla {\varphi }_2)(q_{ij}-q_{ij-1})\\ & +(\nabla {\varphi }_3)(q_{i+1j+1}-q_{ij})-(\nabla {\varphi }_3)(q_{ij}-q_{i-1j-1})\\ & +(\nabla {\varphi }_4)(q_{i+1j-1}-q_{ij})-(\nabla {\varphi }_4)(q_{ij}-q_{i-1j+1}),\end{array}$$

$${\varphi }_1(x_1,x_2)$$

$${\varphi }_3(x_1,x_2)$$

$$\displaystyle q_{ij}(t)=\epsilon Q_{\epsilon}\Big{(}\epsilon\big{(}\kappa_{1}i+\kappa_{2}j-c_{\epsilon}t\big{)}\Big{)}\in\mathbb{R}^{2},$$

$$\displaystyle\epsilon^{3}c_{\epsilon}^{2}Q_{\epsilon}^{\prime\prime}=\sum_{m=1}^{M}k_{m}\epsilon\delta_{-k_{m}\epsilon}F^{m}\big{(}k_{m}\epsilon^{2}\delta_{+k_{m}\epsilon}Q_{\epsilon}\big{)}.$$

$$k_1={\kappa }_1,k_2={\kappa }_2,k_3={\kappa }_1+{\kappa }_2,k_4={\kappa }_1-{\kappa }_2,$$

$$F^m(x_1,x_2)=\nabla {\varphi }_m(x_1,x_2)-\nabla {\varphi }_m(0,0)$$

$${\delta }_{+\eta }Y:=\frac{Y(\cdot +\eta )-Y(\cdot )}{\eta },{\delta }_{-\eta }Y:=\frac{Y(\cdot )-Y(\cdot -\eta )}{\eta },\eta >0,$$

$$W_{ϵ}=Q_{ϵ}^{\prime }.$$

$$F_{i}^{m}(x_{1},x_{2})=\alpha_{i,1}^{m}x_{1}+\alpha_{i,2}^{m}x_{2}+\tfrac{1}{2}\big{(}\beta_{i,11}^{m}x_{1}^{2}+2\beta_{i,12}^{m}x_{1}x_{2}+\beta_{i,22}^{m}x_{2}^{2}\big{)}+\Psi_{i}^{m}(x_{1},x_{2})$$

$$W_{ϵ}:=W_0+{ϵ}^2{V}_{ϵ},$$

$${\sigma }_{ϵ}:=c_{ϵ}^2={\sigma }_0+{ϵ}^2$$

$${\mathcal{B}}_{ϵ}{W}_{ϵ}={\mathcal{Q}}_{ϵ}[W_{ϵ}]+{ϵ}^2{\mathcal{P}}_{ϵ}[W_{ϵ}].$$

$${\mathcal{B}}_0{W}_0={\mathcal{Q}}_0[W_0],$$

$$W_{0,1}=W_{\ast },W_{0,2}=\lambda W_{\ast },$$

$$W_{\ast }^{\prime \prime }=d_1{W}_{\ast }-d_2{W}_{\ast }^2.$$

$$d_1{\partial }_{t}w+d_2{\partial }_{\xi }{w}^2+{\partial }_{\xi }^{3}w=0.$$

$$V_{ϵ}={\mathcal{F}}_{ϵ}[V_{ϵ}],$$

$$({\mathcal{A}}_{\eta }Y)(\xi ):=\frac{1}{\eta }{\int }_{\xi -\frac{\eta }{2}}^{\xi +\frac{\eta }{2}}Y(\tilde{\xi })d\tilde{\xi }\text{with}Y\in {\mathsf{L}}^2(\mathbb{R}).$$

$$z\mapsto \mathrm{sinc}(\eta z/2)=\frac{\sin (\eta z/2)}{\eta z/2},$$

$${\mathcal{A}}_{\eta }=\mathrm{id}+\frac{{\eta }^2}{24}{\partial }^2+O({\eta }^4).$$

$$1+\sum _{m=1}^M{\gamma }_m\frac{1-\mathrm{sinc}(k_{m}z/2{)}^2}{{ϵ}^2},$$

$$\Big{(}\operatorname*{id}+\sum_{m=1}^{M}\gamma_{m}\frac{\operatorname*{id}-\mathcal{A}_{k_{m}}^{2}}{\epsilon^{2}}\Big{)}^{-1}.$$

$${ϵ}^2{\sigma }_{ϵ}{W}_{ϵ}$$

$$||{\mathcal{A}}_{\eta }W|{|}_{\infty }\le {\eta }^{-1/2}||W|{|}_2,||{\mathcal{A}}_{\eta }W|{|}_2\le ||W|{|}_2.$$

$$||{\mathcal{A}}_{\eta }W-W|{|}_2\le C{\eta }^2||W^{\prime \prime }|{|}_2,||{\mathcal{A}}_{\eta }W-W|{|}_{\infty }\le C{\eta }^2||W^{\prime \prime }|{|}_{\infty }$$

$$||{\mathcal{A}}_{\eta }W-W-\frac{{\eta }^2}{24}{W}^{\prime \prime }|{|}_2\le C{\eta }^4||W^{\prime \prime \prime \prime }|{|}_2,||{\mathcal{A}}_{\eta }W-W-\frac{{\eta }^2}{24}{W}^{\prime \prime }|{|}_{\infty }\le C{\eta }^4||W^{\prime \prime \prime \prime }|{|}_{\infty }$$

$${\mathcal{A}}_{\eta }W\to W\text{strongly in}{\mathsf{L}}^2(\mathbb{R})$$

$${\alpha }_{1,2}^m={\alpha }_{2,1}^m,{\beta }_{1,22}^m={\beta }_{2,12}^m,{\beta }_{1,12}^m={\beta }_{2,11}^m,$$

$${\mathcal{B}}_{ϵ}^{\text{can}}:=\frac{1}{{ϵ}^2}\left(\begin{array}{cc}{\sigma }_{ϵ}-{\sum }_{m=1}^M{k}_m^2{\alpha }_{1,1}^m{\mathcal{A}}_{k_mϵ}^2& -{\sum }_{m=1}^M{k}_m^2{\alpha }_{1,2}^m{\mathcal{A}}_{k_mϵ}^2\\ & \\ -{\sum }_{m=1}^M{k}_m^2{\alpha }_{2,1}^m{\mathcal{A}}_{k_mϵ}^2& {\sigma }_{ϵ}-{\sum }_{m=1}^M{k}_m^2{\alpha }_{2,2}^m{\mathcal{A}}_{k_mϵ}^2\end{array}\right),$$

Full text

Keywords:

two-dimensional FPU-lattices, KdV limit of lattice waves,

asymptotic analysis of singularly perturbed integral equations

MSC (2010):

37K60, 37K40, 74H10

1. 1 Introduction

2. 1.1 Setting of the problem

3. 1.2 Proof strategy and main result

4. 2 Abstract existence result

5. 2.1 Reformulation as integral equation

6. 2.2 Linear main part

7. 2.3 Leading order problem and KdV wave

8. 2.4 Fixed-point equation for the corrector $V_{\varepsilon}$

9. 2.5 Inverting the operator $\mathcal{B}_{\epsilon}$

10. 2.6 Further auxiliary results

11. 2.7 Inverting the operator $\mathcal{L}_{\epsilon}$

12. 2.8 Nonlinear fixed-point argument

13. 3 Applications to different lattices

14. 3.1 Square lattice

15. 3.2 Diamond lattice

16. 3.3 Triangle lattice

1 Introduction

Since the pioneering work [3] a lot of mathematical research has been devoted to one-dimensional FPU lattices, of which the only nontrivial completely integrable type is the so-called Toda lattice, that admits explicit traveling wave solutions in terms of Jacobian elliptic functions, see for instance [6]. Among these works [28] was the first to reveal a connection between solitary waves with long-wave-lengths and small amplitudes and the Korteweg-de Vries (KdV) equation. By diminishing the lattice spacing the authors were able to pass to a continuum described by a KdV equation, which is a completely integrable PDE. However, since their analysis was purely formal, it didn’t lead to a rigorous proof. The existence of KdV-like traveling waves in FPU chains was justified in a rigorous way much later, in the first of a series of papers [7], [8], [9] and [10], of which the other three deal with the stability theory. The proof was based on the introduction of a new framework for passing to the continuum and concerned only lattices with interactions between nearest neighbors, see also [6] for related results concerning periodic waves. The existence result from [7] has been extended in [15] to 1D lattices with nonlocal interactions, replacing the original analysis of Fourier poles in the complex plane by direct estimates on the real line. The KdV approximation can also be used to derive modulation equations for the large-scale dynamics of well-chosen initial data. We refer to [23, 17, 18, 19] for results in the classical FPU setting, and to [12] for more complex atomic systems.

Existence results on various types of waves in FPU chains can be found in the literature. The paper [21] provides a precise description of all subsonic and supersonic waves with small amplitudes via the center manifold reduction. In [11] the existence of solitary waves with large amplitudes was proved for the first time by applying the concentration-compactness principle, see also [1] for similar results under slightly different conditions and [4, 13] for another variational approach yielding unimodal waves. In all these large-amplitude results the wave speed is in principle unknown. However, using mountain pass techniques as in [25, 22] one also finds periodic and solitary waves with prescribed wave speed above a critical threshold. Finally, for certain potentials there even exist several types of heteroclinic waves, see [16, 26, 24].

The investigations mentioned above concern exclusively 1D FPU lattices. In the practice of physics, two or more dimensional FPU-lattices are more relevant, since they provide simplified models for crystals and solids. However, to the best of our knowledge, no existence result for KdV-like waves in such FPU lattices has been obtained except for a degenerate case [5], where it is assumed that the wave is unidirectional and longitudinal as it moves along the horizontal direction. This assumption reduces the original two-dimensional system to a one-dimensional one and the geometric nonlinearity, which arises from the linear stress-strain relation due to the introduction of diagonal springs, enables one to apply the results from [7].

In the present paper we show an existence result for 2D FPU lattices in a general framework, which covers different lattice geometries and allows for arbitrary propagation directions. Our asymptotic approach adapts some arguments developed in [15] but is more sophisticated due to the two-dimensional setting. The discussion of the 2D case in the present paper hints at similar results for 3D lattices, which have to be left for further investigations. Besides, the study of stability of the resulting solutions requires ideas lying beyond the scope of the present paper and will be completely omitted.

1.1 Setting of the problem

In this subsection we study the square lattice — see the left panel of Figure 1 — as prototypical example and show that any lattice wave complies with the nonlocal advance-delay-differential equation formulated in (3). Other lattice geometries produce basically the same equation but the number of coupling terms $M$, the coupling parameters $k_{m}$, and the effective nonlinearities $F^{m}_{i}$ will depend on the mechanical details in the lattice, see §3.

We assume that the lattice is indexed by $(i,j)\in\mathbb{Z}^{2}$ and the position of the $(i,j)^{\text{th}}$ particle is given by

[TABLE]

where $r_{*}>0$ is a reference lattice parameter and $q_{ij}(t)=(q_{ij,1}(t),q_{ij,2}(t))^{T}\in\mathbb{R}^{2}$ represents the displacement of the $(i,j)^{\text{th}}$ particle at time $t$. By summing up the forces exerted on a given particle by its eight neighbors and applying the second Newton’s law we obtain

[TABLE]

where

[TABLE]

are the four effective potentials corresponding to the horizontal, vertical, and diagonal springs, respectively. In this paper we are solely interested in traveling wave solutions of the following type

[TABLE]

where $\kappa:=(\kappa_{1},\kappa_{2})^{T}$ is the wave vector prescribing the propagating direction of the wave and $c_{\epsilon}>0$ the wave speed that depends on the scaling parameter $\epsilon>0$. By a suitable scaling we can assume that $\kappa$ is normalized with respect to its length, and denoting the angle between $\kappa$ and the positive axis by $\alpha$, we write $\kappa=(\cos(\alpha),\sin(\alpha))^{T}$.

Substituting the ansatz (2) into (1), we arrive at a system of two difference-differential equations for the unknown function $Q_{\epsilon}$, which can be written in abstract form as

[TABLE]

Here $M=4$ denotes the number of interacting neighbors (divided by two), the coupling constants depend on $\alpha$ via

[TABLE]

and

[TABLE]

abbreviates the (normalized) gradient of the effective potential related to $k_{m}$. Moreover, the right hand side of (3) is formulated in terms of the difference operators

[TABLE]

which reflect that the problem involves for each $m=1,...,M$ precisely one advance and one delay term and allow us to reformulate (3) in §2 as an integral equation for the velocity profile

[TABLE]

Recall that the special case $\alpha=0$ has been studied in [5]. In particular, it has been shown that any wave propagating along the horizontal lattice direction must be unidirectional and longitudinal. This means that any reasonable solution to (3) must satisfy $W_{\epsilon,2}=0$ and the original 2D problem can hence be reduced to a one-dimensional one. Moreover, the identities $k_{2}=0$ and $k_{1}=k_{3}=k_{4}$ imply that the different coupling terms can be subsumed into a single one. For general $\alpha$, however, both simplifications are no longer possible and the lattice waves are asymptotically unidirectional in the sense that $W_{\epsilon,2}(\xi)/W_{\epsilon,1}(\xi)=\mathrm{const}$ manifests only in the limit $\epsilon\to 0$, see Figure 2.

It has also been observed in [5] that the diagonal springs lead to the nonlinearity that is necessary for the existence of solitary waves. This holds even in the case of linear spring forces, i.e., if the physical interactions potentials $V_{i}$ are quadratic (geometric nonlinearity). Without diagonal springs there would be no resistance against the sheer forces and the mechanical structure would become unstable.

1.2 Proof strategy and main result

In this paper we follow the fundamental ideas behind KdV limits and construct solutions to (3) in the asymptotic regime of small $\epsilon>0$, i.e., we suppose that the lattice waves have small amplitudes and large wave-lengths. This is illustrated in Figure 3 and enables us to employ Taylor expansion and to regard the difference terms as approximate differential operators. To this end we assume that the nonlinearities $F_{i}^{m}$, which take over the role of the partial derivatives of effective potentials in the lattices, have the form

[TABLE]

for all $i=1,2$ and $m=1,...,M$, where $\Psi_{i}^{m}(x_{1},x_{2})$ represent the higher order terms and are hence asymptotically small perturbations of the linear and the quadratic contributions.

Following [15] we consider the vector-valued velocity profile $W_{\epsilon}$ from (4) and decompose it as the sum of a limit profile $W_{0}$ and an $O(\epsilon^{2})$-corrector via

[TABLE]

where the small parameter $\epsilon>0$ quantifies via

[TABLE]

how much the squared wave speed exceeds a given reference value $\sigma_{0}$, which is completely determined by the linear coefficients $\alpha_{i,j}^{m}$ from (5), see formula (21). Applying (7) and gathering linear terms with respect to $W_{\epsilon}$ in (16) on the left hand side and the nonlinear terms on the right, we arrive at the equation

[TABLE]

Here, $\mathcal{B}_{\epsilon}$ is a linear integral operator introduced in (23), $\mathcal{Q}_{\epsilon}$ depends quadratically on $W_{\epsilon}$, and $\mathcal{P}_{\epsilon}$ stems from the higher order terms $\Psi_{i}^{m}$, see (24) and (25). The key observation for small $\epsilon>0$ is that the formal limit equation

[TABLE]

which will be identified in (26) and (27), is equivalent to

[TABLE]

where the non-trivial and decaying function $W_{*}$ is the unique even and homoclinic solution to the ODE

[TABLE]

The constants $d_{1}$ and $d_{2}$ are positive and can be computed explicitly, see (23) and (28). They depend on the linear and quadratic coefficients $\alpha_{i,j}^{m}$ and $\beta_{i,jk}^{m}$, as well as on the scalar quotient $\lambda=W_{0,2}/W_{0,1}$. Moreover, $W_{*}$ defines via $w(t,\xi):=W_{*}(\xi-t)$ the solitary wave for the KdV equation

[TABLE]

At the next level, we turn (11) into a fixed-point equation with respect to the corrector $V_{\epsilon}$ by inserting (6) and rearranging the terms in such a way, that the linear terms with respect to $V_{\epsilon}$ stand on the left hand side, which yield automatically a linear operator $\mathcal{L}_{\epsilon}$. After the uniform inversion of $\mathcal{L}_{\epsilon}$ on $(\mathsf{L}_{\rm{even}}^{2}(\mathbb{R}))^{2}$, see Theorem 15, we finally deduce the fixed point equation

[TABLE]

where the operator $\mathcal{F}_{\epsilon}$ will be identified in (33). The contraction property of $\mathcal{F}_{\epsilon}$ will be shown in a sufficiently large ball in $(\mathsf{L}_{\rm{even}}^{2}(\mathbb{R}))^{2}$, and the Banach fixed-point theorem provides us with a solution, which is at the same time unique in the chosen ball. Notice that uniqueness of large-amplitude traveling waves is a notoriously difficult and open problem in the mathematical theory of Hamiltonian lattice waves.

The sketched asymptotic approach shares many similarities with the discussion in [15], but all arguments have to be adapted to the two-dimensional situation. For instance, our version of $\mathcal{B}_{\epsilon}$ is not symmetric and its inversion involves the study of a matrix of operators instead of a single operator on $\mathsf{L}^{2}(\mathbb{R})$. Moreover, the fundamental operators and scalar quantities depend on the propagation angle $\alpha$.

Our main result guarantees that the abstract advance-delay-differential equation (3) admits KdV-like solutions for sufficiently small $\epsilon>0$ provided that the nonlinearities $F^{m}_{i}$ and the parameters $k_{m}$ satisfies certain requirements. These conditions, which can be checked as described in §3, encode the particular lattice geometry, the propagation direction of the wave, and the choice of the physical potentials.

Theorem 1**.**

Under Assumptions 4, 5, 6, and 7 there exists a sound speed $\sigma_{0}$ along with a KdV traveling wave $W_{0}$ as in (10) and (11) such that for any sufficiently small $\epsilon>0$ and wave speed $c_{\epsilon}=\sqrt{\sigma_{0}+\epsilon^{2}}$, equation (3) has a solution whose velocity profile $W_{\epsilon}=Q_{\epsilon}^{\prime}\in(\mathsf{L}_{\rm{even}}^{2}(\mathbb{R}))^{2}$ satisfies $||W_{\epsilon}-W_{0}||_{2}\leq C\epsilon^{2}$.

This paper is organized as follows: Within the sections §2.1–§2.4 we reformulate (3) first as a nonlinear eigenvalue equation for $W_{\epsilon}$, and afterwards as fixed problem for the corrector $V_{\epsilon}$, see (12). We also study the leading order equation (9) and specify our assumptions concerning the Taylor coefficients from (5) and their dependence on the propagation direction. Using the auxiliary results from §2.5 and §2.6 we prove in §2.7 the key asymptotic result, i.e., we show that the operator $\mathcal{L}_{\epsilon}$, which stems from the linearization of (8) around $W_{0}$, is uniformly invertible in the $\mathsf{L}^{2}$-space of even functions. The contraction property of the fixed point operator $\mathcal{F}_{\epsilon}$ is then verified in §2.8 and concludes the proof of our main result. Finally, in §3 we discuss the three different lattices from Figure 1 as applications of Theorem 1.

2 Abstract existence result

In this section we write (3) as a fixed point equation with respect to the corrector $V_{\epsilon}$ in (6) and prove under certain assumptions on the nonlinearities the existence of KdV-type solutions. To this end we denote by $C$ any generic constant that is independent of $\epsilon$ but can depend on $M$, the constants $k_{m}$, and the properties of the functions $F_{i}^{m}$. Moreover, we write $||\cdot||_{p}$ and $||\cdot||_{l,p}$ for the usual norms in the Lebesgue and Sobolev spaces $\mathsf{L}^{p}(\mathbb{R})$ and $\mathsf{W}^{l,p}(\mathbb{R})$, respectively, and $\langle\cdot,\cdot\rangle$ means the usual dual paring of functions. The same symbols also abbreviate the corresponding norms of vector-valued functions.

2.1 Reformulation as integral equation

For an arbitrary constant $\eta>0$, we define the integral operator $\mathcal{A}_{\eta}$ by

[TABLE]

This operator corresponds in Fourier space to the multiplication with the symbol function

[TABLE]

and can be formally expanded as follows

[TABLE]

In particular, $\mathcal{A}_{\eta}$ is a singular perturbation of the identity due to the remainder terms with higher derivatives and this complicates the asymptotic analysis. Fortunately, the derivation of the decisive properties of the auxiliary operator $\mathcal{B}_{\epsilon}$, see (23), only involves the inversion of the Fourier symbol function

[TABLE]

which corresponds to

[TABLE]

The asymptotic properties of such an operator are well understood, see for instance [15] and the discussion in §2.5.

As already mentioned in §1, the operator $\mathcal{A}_{\epsilon}$ allows our problem to be reformulated as one concerning the velocity profile $W_{\epsilon}$ and this leads to a new framework for solving (3).

Lemma 2** (reformulation of traveling wave equation).**

Suppose $W_{\epsilon}$ from (4) belongs to $(\mathsf{L}^{2}(\mathbb{R}))^{2}$. Then the traveling wave equation (3) is equivalent to the nonlinear eigenvalue problem

[TABLE]

with scaled eigenvalue $\sigma_{\epsilon}=c_{\epsilon}^{2}$ as in (7) and unknown eigenfunction $W_{\epsilon}$.

Proof.

Suppose (16) is satisfied and let $Q_{\epsilon}(\xi):=\int_{0}^{\xi}W_{\epsilon}(\zeta)d\zeta$. Then we can replace any term $(\mathcal{A}_{\eta}W_{\epsilon})(\xi)$ in the argument of the nonlinearities by $(\delta_{+\eta}Q_{\epsilon})(\xi-\eta/2)$ and differentiating both sides of (16) we obtain (3) due to $(\mathcal{A}_{\eta}Y)^{\prime}(\xi)=(\delta_{-\eta}Y)(\xi+\eta/2)$. Conversely, integrating (3) we obtain immediately (16) plus a constant vector $\in\mathbb{R}^{2}$, but this constant of integration vanishes due to $W_{\epsilon}\in(\mathsf{L}^{2}(\mathbb{R}))^{2}$ and $F^{m}((0,0))=(0,0)$ for all $m$.∎

In our subsequent analysis we use the following properties of $\mathcal{A}_{\eta}$, where (17) justifies the formal expansion (15) of $\mathcal{A}_{\eta}$ in a rigorous way.

Lemma 3** (properties of the integral operator $\mathcal{A}_{\eta}$).**

For each $\eta>0$, the integral operator $\mathcal{A}_{\eta}$ has the following properties:

For any $W\in\mathsf{L}^{2}(\mathbb{R})$ we have $\mathcal{A}_{\eta}W\in\mathsf{L}^{2}(\mathbb{R})\cap\mathsf{L}^{\infty}(\mathbb{R})$ with

[TABLE]

Moreover, $\mathcal{A}_{\eta}W$ admits a weak derivative with $||(\mathcal{A}_{\eta}W)^{\prime}||_{2}\leq\eta^{-1/2}||W||_{2}$. 2. 2.

For any $W\in\mathsf{L}^{\infty}(\mathbb{R})$, we have $||\mathcal{A}_{\eta}W||_{\infty}\leq||W||_{\infty}$. 3. 3.

$\mathcal{A}_{\eta}$* respects the even-odd parity and the non-negativity of functions.* 4. 4.

$\mathcal{A}_{\eta}$* diagonalizes in Fourier space and corresponds to the symbol function from (14).* 5. 5.

$\mathcal{A}_{\eta}$* is self-adjoint in the $\mathsf{L}^{2}$-sense.* 6. 6.

The operators $\mathcal{A}_{\eta_{1}}$ and $\mathcal{A}_{\eta_{2}}$ commute with each other for any $\eta_{1},\eta_{2}>0$. 7. 7.

There exists a constant $C>0$, which does not depend on $\eta$, such that the estimates

[TABLE]

and

[TABLE]

hold for any sufficiently regular $W$. In particular, we have

[TABLE]

for any $W\in\mathsf{L}^{2}(\mathbb{R})$.

Proof.

Equation (13) can be written as $\mathcal{A}_{\eta}W=\chi_{\eta}\ast W$, where $\chi_{\eta}$ is the normalized indicator function of an interval and satisfies $\|\chi_{\eta}\|_{1}=1$. Moreover, the function in (14) is just the Fourier transform of $\chi_{\eta}$. All assertions thus follow from standard arguments concerning convolutions and their Fourier representations, see [13, Lemma 2.5], [15, Lemma 5] and also [1, 2] for the details.∎

2.2 Linear main part

Since for traveling waves the functions $F_{i}^{m}$ represent the partial derivatives of the effective potentials, see for instance (1), a first natural condition can be formulated as follows.

Assumption 4** (linear and quadratic coefficients ).**

The Taylor coefficients of $F_{i}^{m}$, see (5), satisfy

[TABLE]

for all $m=1,...,M$.

As already explained in the introduction, our strategy is to collect the linear and the quadratic terms from (16) into two operators $\mathcal{B}_{\epsilon}$ and $\mathcal{Q}_{\epsilon}$, respectively, as in (8). A first natural choice for the linear part would be

[TABLE]

but this operator does not converge as $\epsilon\to 0$. In fact, in view of (7) the dominant part is given by

[TABLE]

with

[TABLE]

and diverges as $\epsilon\to 0$. In what follows we therefore define the operator $\mathcal{B}_{\epsilon}$ in a slightly different way. Moreover, we choose $\sigma_{0}$ such that $\mathcal{B}_{\epsilon}^{\text{sing}}$ has a nontrivial kernel because otherwise any solution to the leading order equation (9) would vanish identically. The latter condition reads

[TABLE]

and provides two possible solution branches. However, it turns out that only the larger value of $\sigma_{0}$ is admissible for our asymptotic analysis, see the remarks at the beginning of §3, and thus we set

[TABLE]

Using this particular choice of $\sigma_{0}$ we now define

[TABLE]

with

[TABLE]

which implies that the corresponding leading order part is given by the non-invertible matrix

[TABLE]

Using this definition of $\mathcal{B}_{\epsilon}$, the nonlinear integral equation (16) can be reformulated as (8), where the quadratic and higher terms correspond to operators $\mathcal{Q}_{\epsilon}$ and $\mathcal{P}_{\epsilon}$ with components

[TABLE]

and

[TABLE]

2.3 Leading order problem and KdV wave

By straightforward computations we verify the formal expansion

[TABLE]

with leading order terms

[TABLE]

and

[TABLE]

Here, the constants are given by

[TABLE]

as well as

[TABLE]

and

[TABLE]

The first and the second component of the leading order equation (9) can therefore be written as (10) and (11), respectively, where

[TABLE]

denote the constants in (11). This ODE for the KdV wave is a planar Hamiltonian system with potential energy $E[W]:=\frac{1}{3}d_{2}W^{3}-\frac{1}{2}d_{1}W^{2}$, see Figure 4, and its coefficients $d_{1}$, $d_{2}$ are completely determined by the linear and quadratic terms of $F_{i}^{m}$. A simple phase plane analysis reveals that (11) admits a non-trivial homoclinic orbit with unstable equilibrium at [math] if and only if $d_{1}>0$ and $d_{2}\neq 0$. Moreover, we readily verify that the corresponding even solution $W_{*}\in\mathsf{L}_{\rm{even}}^{2}(\mathbb{R})$ is given by

[TABLE]

and corresponds to the shape parameters

[TABLE]

In this paper we always assume that the leading order problem (11) admits the homoclinic solution (29). To avoid technicalities also suppose that $c_{2}$ does not vanish and restrict our considerations to the case $\sigma_{0}>0$, which provides a uniform lower bound for the speed $\sqrt{\sigma_{\varepsilon}}$ of the solitary lattice waves according to (7). The one-dimensional analogue to the latter condition is a nonvanishing (and real-valued) sound speed, see [15]. In summary, we rely on the following assumption.

Assumption 5** (genericity and well-definedness of KdV waves).**

The constants defined in (19), (21), and (28) satisfy

$\sigma_{0}>0$* and $c_{2}\neq 0$,* 2. 2.

$d_{1}>0$* and $d_{2}\neq 0$.*

2.4 Fixed-point equation for the corrector $V_{\varepsilon}$

Having determined $\sigma_{0}$ and $W_{0}$, we turn to the discussion of the equation for the corrector term $V_{\epsilon}$. Substituting the ansatz (6) into (8) we obtain

[TABLE]

where

[TABLE]

Moreover, the linear operator $\mathcal{M}_{\epsilon}$ depends on the KdV wave $W_{*}$ from (29) and is given by

[TABLE]

with constants

[TABLE]

The uniform invertibility of $\mathcal{L}_{\epsilon}$ will be crucial in our existence proof, for it allows us to reformulate (31) as the fixed point equation (12), where the operator

[TABLE]

turns out to be contractive in some bounded set provided that $\epsilon$ is sufficiently small. Before we discuss the construction and the properties of $\mathcal{F}_{\epsilon}$ in greater detail, we formulate a further assumption which guarantees that the higher order terms are negligible compared to the linear and quadratic terms. In the simplest case we have $\Psi_{i}^{m}\equiv 0$.

Assumption 6** (regularity of higher order terms).**

The remainder terms $\Psi_{i}^{m}$ in the Taylor expansion (5) satisfy

[TABLE]

and there exists constants $\gamma_{i}^{m}$ such that

[TABLE]

for all $(x_{1},x_{2}),(y_{1},y_{2})\in\mathbb{R}^{2}$, $i=1,2$ and $m=1,...,M$.

2.5 Inverting the operator $\mathcal{B}_{\epsilon}$

A particular challenge in the existence proof for $\mathcal{L}_{\epsilon}^{-1}$ is to invert the operator $\mathcal{B}_{\epsilon}$ from (22) for small values of $\epsilon>0$. Since all operators $\mathcal{A}_{k_{m}\epsilon}$ commute with each another, it is quite natural to consider the ‘determinant’ of $\mathcal{B}_{\epsilon}$ defined by

[TABLE]

In fact, if this operator is invertible on $\mathsf{L}^{2}(\mathbb{R})$, a straightforward calculation reveals that the inverse of $\mathcal{B}_{\epsilon}$ can be written as

[TABLE]

In order to prove the uniform invertibility of $\det\mathcal{B}_{\epsilon}$, we introduce two auxiliary functions

[TABLE]

for $k\in\mathbb{R}$ and $z\in\mathbb{R}$, see Figure 5 and (14). Moreover, we rely on the following assumption, which will be checked numerically in §3 for several lattice geometries.

Assumption 7** (conditions for the inversion of $\mathcal{B}_{\epsilon}$).**

There exists a constant $\delta_{0}>0$, such that

[TABLE]

holds for all $z\in\mathbb{R}$.

The function $T$ is independent of $\epsilon$ and can be locally expanded as $T(z)=\tau z^{2}+O(z^{4})$ near $z=0$, where \tau:=\frac{1}{24}\sum_{m=1}^{M}k_{m}^{4}\big{(}(\sigma_{0}-c_{3})\alpha_{1,1}^{m}+(\sigma_{0}-c_{1})\alpha_{2,2}^{m}+2c_{2}\alpha_{1,2}^{m})\big{)}. If $\tau>0$, then the required condition is satisfied in the vicinity of $z=0$ for any $0<\delta_{0}<\tau$.

Lemma 8** (existence of $\mathcal{B}_{\epsilon}^{-1}$).**

Assumption 7 implies for all sufficiently small $\epsilon>0$ that the symbol function $\widehat{\det(\mathcal{B}_{\epsilon})}$ of the operator from (34) satisfies

[TABLE]

for all $z\in\mathbb{R}$ and some constant $\delta>0$ independent of $\epsilon$, where $2\sigma_{0}-(c_{1}+c_{3})>0$ holds according to (21). In particular, $\det\mathcal{B}_{\epsilon}$ is uniformly invertible on $\mathsf{L}^{2}(\mathbb{R})$.

Proof.

We set

[TABLE]

and notice that for any $m=1,..,M$ we have $|S_{k_{m}}(z)|\leq C_{m}(\min\{|z|,2\})^{2}$ for some constant $C_{m}$. This implies

[TABLE]

for all $z\in\mathbb{R}$ and a sufficiently large constant $\delta_{1}>0$. The Fourier symbols of the components of $\mathcal{B}_{\epsilon}^{\text{can}}$ are given by

[TABLE]

Now we compute the Fourier symbol function $\widehat{\det(\mathcal{B}_{\epsilon})}$ of the determinant $\det(\mathcal{B}_{\epsilon})$ by substituting these expressions. More precisely, combining the above formulas with (20) and (36) we get

[TABLE]

and thanks to (21), Assumption 7, and (37) we show that

[TABLE]

holds for all sufficiently small $\epsilon>0$. This implies

[TABLE]

for all $z\in\mathbb{R}$, and we conclude that $\det(\mathcal{B}_{\epsilon})$ is uniformly invertible as its Fourier symbol is bounded from below by a positive constant independent of $\epsilon$. ∎

In order to characterize the asymptotic properties of $\mathcal{B}_{\epsilon}$ we introduce a cut-off operator

[TABLE]

by defining the corresponding symbol function in Fourier space via

[TABLE]

This gives rise to the following estimates for $\mathcal{B}_{\epsilon}^{-1}$, which below remove the main difficulty in the inversion of $\mathcal{L}_{\epsilon}$.

Lemma 9** (properties of $\mathcal{B}_{\epsilon}^{-1}$).**

There exists a constant $C$ independent of $\epsilon$ such that

[TABLE]

hold for any $G\in\mathsf{L}^{2}(\mathbb{R})$ and all sufficiently small $\epsilon>0$.

Proof.

Notice that for each $k>0$ there exists a constant $C_{k}>0$ such that the estimate $S_{k}(z)\leq C_{k}(\min\{|z|,2\})^{2}$ for all $z\in\mathbb{R}$, see Figure 5 and (36). Combining this with the Fourier representation of $\mathcal{A}_{k_{m}\epsilon}$ from Lemma 3, Formula (35), and Lemma 8 we get

[TABLE]

and similarly we find

[TABLE]

where we did not write the argument $z$ on the left hand side to ease the notation. Theses identities combined with (38) and (39) imply

[TABLE]

as well as

[TABLE]

for all $G\in\mathsf{L}^{2}(\mathbb{R})$ and $i=1,2$. In particular, we showed the first claimed estimate. Towards the second one, we observe that

[TABLE]

and

[TABLE]

hold for all $z\in\mathbb{R}$, and estimate

[TABLE]

for all $G\in\mathsf{L}^{2}(\mathbb{R})$. ∎

2.6 Further auxiliary results

We proceed with bounding the components of the operator $\mathcal{M}_{\epsilon}$ from (32).

Lemma 10** (properties of $\mathcal{M}_{\epsilon}$).**

For any $0<\epsilon\leq 1$ we have

[TABLE]

for some constant $C$ independent of $\epsilon$ and all $G\in(\mathsf{L}^{2}(\mathbb{R}))^{2}$.

Proof.

Lemma 3 yields

[TABLE]

for all $G\in\mathsf{L}^{2}(\mathbb{R})$ and the claim follows from the definition of $\mathcal{M}_{\epsilon}$ in (32). ∎

We next observe that $\mathcal{L}_{\epsilon}$ from (31) is a bounded linear operator on $(\mathsf{L}_{\rm{even}}^{2}(\mathbb{R}))^{2}$ and admits the adjoint

[TABLE]

Passing formally to the limit $\epsilon\to 0$ in (31) we find the limit operator $\mathcal{L}_{0}$ with

[TABLE]

thanks to (23) and (32), and the corresponding adjoint $\mathcal{L}_{0}^{*}$ reads

[TABLE]

Notice that both $\mathcal{L}_{0}$ and $\mathcal{L}_{0}^{*}$ are unbounded operators and that the above formulas are only well-defined if $\phi$ is sufficiently smooth. Moreover, for the computation of $\mathcal{L}_{0}^{*}$ we supposed that $\phi$ decays sufficiently fast.

The operators $\mathcal{L}_{\epsilon}^{*}$ and $\mathcal{L}_{0}^{*}$ play a prominent role in the proof of the invertibility of $\mathcal{L}_{\epsilon}$. The next lemma shows that $\mathcal{L}_{\epsilon}^{*}\phi$ converges to $\mathcal{L}_{0}^{*}\phi$ provided that $\phi$ is sufficiently nice.

Lemma 11** (pointwise convergence of $\mathcal{L}_{\epsilon}$).**

For any function $\phi\in(\mathsf{W}^{4,2}(\mathbb{R}))^{2}$ we have

[TABLE]

strongly in $(\mathsf{L}^{2}(\mathbb{R}))^{2}$.

Proof.

The operator $\mathcal{L}^{*}_{\epsilon}$ consists of linear combinations of $\epsilon^{-2}(\operatorname*{id}-\mathcal{A}_{k\epsilon}^{2})$ and $\mathcal{A}_{k_{m}\epsilon}(\mathcal{A}_{k_{m}\epsilon}W_{*})\mathcal{A}_{k_{m}\epsilon}$. Hence it suffices to show the pointwise $\mathsf{L}^{2}$-convergence of these operators for any test function $\phi\in\mathsf{W}^{4,2}(\mathbb{R})$. Using Lemma 3 we have

[TABLE]

and

[TABLE]

The assertion follows immediately. ∎

Another key ingredient to our asymptotic analysis is that the kernel of the limit operator $\mathcal{L}_{0}$ is simple. We also mention that a similar argument appears in all rigorous results on the KdV limit of FPU-type lattices and that the entire spectrum of $\mathcal{L}_{0}$ can be computed explicitly, see for instance [7, 10] and the references therein.

Lemma 12** (nullspace of $\mathcal{L}_{0}$).**

Under Assumption 5, the kernel of $\mathcal{L}_{0}$ in $(\mathsf{L}^{2}(\mathbb{R}))^{2}$ is spanned by $W^{\prime}_{0}=(W^{\prime}_{*},\lambda W^{\prime}_{*})^{T}\in(\mathsf{L}_{odd}^{2}(\mathbb{R}))^{2}$, where $W_{*}$ represents the KdV wave from (11) and (29).

Proof.

Let $(\phi_{1},\phi_{2})$ be in the kernel of $\mathcal{L}_{0}$. The second component of $\mathcal{L}_{0}\phi=0$ gives $\phi_{2}=\lambda\phi_{1}$, and substituting this back into the first one, we obtain

[TABLE]

where $d_{1}$, $d_{2}$ are given by (28). Equation (40) can be viewed as the linearization of (8) and is solved by $\phi_{1}=W^{\prime}_{*}\in\mathsf{L}_{odd}^{2}(\mathbb{R})$. The uniqueness of $\phi_{1}$ (and hence of $\phi_{2}$) up to normalization follows from a classical Wronski argument for linear and planar ODEs, see [15, Lemma 3.1] for more details.∎

The next lemma shows that $R_{\epsilon}[W_{0}]$ and $\mathcal{P}_{\epsilon}[W_{0}]$ are uniformly bounded for small $\epsilon$.

Lemma 13** (residuals of the KdV wave).**

There exists a constant $C$ independent of $\epsilon$ such that

[TABLE]

for all $0<\epsilon\leq 1$.

Proof.

Because $W_{*}$ and all its derivatives belong to $\mathsf{L}^{2}(\mathbb{R})\cap\mathsf{L}^{\infty}(\mathbb{R})$ according to (29), the assertions can be verified by means of Lemma 3 and arguing as in the proof of Lemma 11.∎

2.7 Inverting the operator $\mathcal{L}_{\epsilon}$

Since $\mathcal{L}_{0}$ has a nontrivial kernel by Lemma 12, we cannot expect to bound $\mathcal{L}_{\epsilon}^{-1}$ uniformly on $(\mathsf{L}^{2}(\mathbb{R}))^{2}$. We can, however, do so in the space of even functions.

Lemma 14** (parity properties).**

The subspace $(\mathsf{L}_{\rm{even}}^{2}(\mathbb{R}))^{2}$ is invariant in $(\mathsf{L}^{2}(\mathbb{R}))^{2}$ under the linear mapping $\mathcal{L}_{\epsilon}$.

Proof.

According to the definitions in (22), (31), and (32), all components of $\mathcal{L}_{\epsilon}$ are linearly spanned by $\mathcal{A}_{k_{m}\epsilon}^{2}$ and \big{(}\mathcal{A}_{k_{m}\epsilon}(\mathcal{A}_{k_{m}\epsilon}W_{*})\big{)}\mathcal{A}_{k_{m}\epsilon}. We know $\mathcal{A}_{k_{m}\epsilon}$ respects the even-odd parity and $W_{*}$ is an even function. The assertion follows. ∎

Now we are in a position to state and prove the main asymptotic result.

Theorem 15** (invertibility of $\mathcal{L}_{\epsilon}$).**

There exists a constant $\epsilon_{*}>0$ such that for any $\epsilon\in(0,\epsilon_{*})$ the operator $\mathcal{L}_{\epsilon}$ is invertible on $\mathsf{L}_{\rm{even}}^{2}(\mathbb{R})$. More precisely, there exists a constant $C$ which does not depend on $\epsilon$ such that

[TABLE]

for all $\epsilon\in(0,\epsilon_{*})$ and all $G\in(\mathsf{L}_{\rm{even}}^{2}(\mathbb{R}))^{2}$.

Proof.

Preliminaries: We will show that there is a constant $c>0$ such that

[TABLE]

for all $V\in(\mathsf{L}_{\rm{even}}^{2}(\mathbb{R}))^{2}$ and all sufficiently small $\epsilon$. This implies that $\mathcal{L}_{\epsilon}$ is injective and has closed image. The same holds for the operator $\mathcal{T}_{\epsilon}^{-1}\circ\mathcal{L}_{\epsilon}$, which is symmetric by Assumption 4, and we conclude that both $\mathcal{T}_{\epsilon}^{-1}\circ\mathcal{L}_{\epsilon}$ and $\mathcal{L}_{\epsilon}$ are surjective. Moreover, (41) implies that the inverse of $\mathcal{L}_{\epsilon}$ is a continuous operator.

Antithesis: We suppose that a constant $c$ as in (41) does not exist. Then we find a sequence $(\epsilon_{n})_{n\in\mathbb{N}}$ with $\lim_{n\rightarrow\infty}\epsilon_{n}=0$ as well as sequences $(V_{n})_{n\in\mathbb{N}}$ and $(G_{n})_{n\in\mathbb{N}}\subset(\mathsf{L}_{\rm{even}}^{2}(\mathbb{R}))^{2}$ such that

[TABLE]

In view of the invertibility of $\mathcal{B}_{\epsilon}$ we have

[TABLE]

where $(\mathcal{M}_{\epsilon_{n}}V_{n}+G_{n})_{1}$, $(\mathcal{M}_{\epsilon_{n}}V_{n}+G_{n})_{2}$ denote the components of $\mathcal{M}_{\epsilon_{n}}V_{n}+G_{n}$. Lemma 10 gives

[TABLE]

and

[TABLE]

where the right hand side in (45) vanishes as $n\to\infty$.

Weak convergence to 0: Since the sequence $(V_{n})_{n\in\mathbb{N}}$ is bounded, there exists a $V_{\infty}\in(\mathsf{L}_{\rm{even}}^{2}(\mathbb{R}))^{2}$ such that

[TABLE]

for a not relabelled subsequence. Due to Lemma 11 we have

[TABLE]

for any test function $\phi$ that is sufficiently smooth and localized. Let $\phi_{1}$, $V_{\infty,1}$ denote the first components of $\phi$, $V_{\infty}$ and $\phi_{2}$, $V_{\infty,2}$ the second ones. Putting $\phi_{1}=0$ we obtain

[TABLE]

for any $\phi_{2}$, and thanks to (23) we get

[TABLE]

Substituting this back into the equation $\langle V_{\infty},\mathcal{L}_{0}^{*}\phi\rangle=0$ and letting $\phi_{2}=0$ we obtain

[TABLE]

for any $\phi_{1}$, and this gives

[TABLE]

We conclude that $V_{\infty,1}$ belongs to $\mathsf{W}^{2,2}(\mathbb{R})$, and (47) ensures $V_{\infty}\in(\mathsf{W}^{2,2}(\mathbb{R}))^{2}$. We can hence apply $\mathcal{L}_{0}$ to $V_{\infty}$ and conclude in view of (46) that

[TABLE]

holds for any sufficiently nice test function $\phi$. In particular, the even function $V_{\infty}$ is contained in the kernel of $\mathcal{L}_{0}$ and

[TABLE]

is a direct consequence of Lemma 12.

Further notations: In what follows we consider a constant $0<K<\infty$, denote by $\chi_{K}$ the characteristic function of the interval $I_{K}:=[-K,+K]$, and set

[TABLE]

where $k_{\max}=\max_{m=1..M}k_{m}$. Moreover, in order to estimate $V_{n}$ we break it up via

[TABLE]

with

[TABLE]

where $V_{n,1}$ and $V_{n,2}$ denote the components of $V_{n}$. Notice that

[TABLE]

holds by construction for all $n$ and $j=1,2,3$.

Choice of $K$: Since $V_{n}^{(2)}$ is supported in $\mathbb{R}\setminus I_{K}$, the properties of the convolution operators $\mathcal{A}_{k_{m}\epsilon_{n}}$ imply

[TABLE]

for $i=1,2$, $m=1..M$ and all $0<\epsilon_{n}\leq 1$, and exploiting (32) and Lemma 9 we obtain

[TABLE]

for some constant $C$ independent of $K$. Consequently, choosing $K$ sufficiently large we can guarantee that

[TABLE]

holds for all $n$.

Strong convergence of $V_{n}^{(1)}$ and $V_{n}^{(3)}$: Lemma 9 combined with (43) yields

[TABLE]

so (44) and (45) provide

[TABLE]

To decompose $V_{n}^{(1)}$ we define

[TABLE]

such that

[TABLE]

By contruction, both $U_{n}^{(1)}$ and $U_{n}^{(2)}$ are supported in $I_{K}$, and from Lemma 9 we deduce the estimates

[TABLE]

and

[TABLE]

thanks to (44) and (45), where the latter inequality shows that

[TABLE]

Moreover, since $V_{n}^{(2)}$ is supported in $\mathbb{R}\setminus I_{K}$ we have

[TABLE]

and our results derived so far guarantee that

[TABLE]

Combing this with

[TABLE]

and using that $\mathsf{W}^{2,2}(I_{K})$ is compactly embedded into $\mathsf{L}^{2}(I_{K})$, we get

[TABLE]

and hence

[TABLE]

This finally implies

[TABLE]

since $V_{n}^{(1)}$ is supported $I_{K}$.

Derivation of the contradiction: By (43) and (48) we have

[TABLE]

and passing to the limit $n\to\infty$ gives

[TABLE]

thanks to the Lemmas 9 and 10, and according to (49), (50), and (51). This, however, contradicts the normalization condition in (42) and the antithesis must be false.∎

2.8 Nonlinear fixed-point argument

To conclude the existence proof for lattice waves it remains to verify the conditions of the Banach fixed-point theorem.

Theorem 16** (existence and uniqueness of lattice waves).**

Under Assumptions 4, 5, 7 and 6 there exist constants $D>0$ and $\epsilon_{*}>0$, such that for any $\epsilon<\epsilon_{*}$ the operator $\mathcal{F}_{\epsilon}$ has a unique fixed point in the ball $B_{D}:=\{V\in(\mathsf{L}_{\rm{even}}^{2}(\mathbb{R}))^{2}:||V||_{2}\leq D\}$.

Proof.

We show that the operator $\mathcal{F}_{\epsilon}$ is a contraction in a sufficiently large ball $B_{D}$ for any sufficiently small $\epsilon$, thus satisfying all the conditions of the Banach Fixed Point Theorem. The value of $D$ will be chosen later.

Estimates for the quadratic terms: For arbitrary $V_{1},\ V_{2}\in B_{D}$, Assumption 6 gives

[TABLE]

as well as

[TABLE]

Applying this to $\mathcal{Q}_{\epsilon}$ yields

[TABLE]

for $V_{1},V_{2}\in B_{D}$, and setting $V_{1}=V$ and $V_{2}=0$ we get

[TABLE]

for any $V\in B_{D}$. We emphasize that $C$ is a constant, which does not depend on $D$.

Estimates for the higher order terms: Assumption 6 combined with Lemma 3 provides

[TABLE]

for $V_{1},V_{2}\in B_{D}$. This implies

[TABLE]

for $V_{1},V_{2}\in B_{D}$, and also

[TABLE]

for all $V\in B_{D}$ thanks to $\mathcal{N}_{\epsilon}[W_{0};0]=0$.

Concluding arguments: Lemma 13 finally gives

[TABLE]

and this implies

[TABLE]

for any $V\in B_{D}$. Moreover, the above estimates also ensure that

[TABLE]

for all $V_{1},V_{2}\in B_{D}$. Now we choose $D$ to be $2C$ and find afterwards a value $\epsilon_{*}>0$ such that

[TABLE]

The claim is now granted by Banach’s Contraction Mapping Principle.∎

3 Applications to different lattices

In this section we discuss the implication of our abstract existence result from §2 for the three lattice geometries sketched in Figure 1. The existence of a solitary solution as given by Theorem 1 involves the verification of Assumptions 4, 5, 7 and 6. Notice that since all the relevant effective potentials are smooth at the point $(0,0)$, Assumptions 4 and 6 are automatically satisfied. Thus only two assumptions remain to be verified, and this will be done numerically. For an intuitive picture we also give the behavior of $W_{0}$ and $\lambda$ with respect to the propagation direction, i.e., the angle $\alpha$. All simulations are performed with $r_{*}=0.8047$ and $V_{1}(r)=V_{2}(r)=V(r)=\tfrac{1}{2}(r-1)^{2}$.

We first remind that (20) gives two possible candidates for $\sigma_{0}$, but in our approach we always chose the larger solution, see (21). The reason is that although the other solution branch might give rise to a well-defined KdV wave in the limit $\epsilon\to 0$, we cannot expect the corresponding operator $\mathcal{B}_{\epsilon}$ to be invertible. To see this, we observe that $\mathcal{B}_{\epsilon}$ is invertible if and only if $\sigma_{\epsilon}$ does not belong to the spectrum of the operator $\mathcal{J}_{\epsilon}:=\sigma_{\epsilon}\operatorname*{id}-\epsilon^{2}\mathcal{B}_{\epsilon}^{\text{can}}$, see formulas (18) and (22). Since the Fourier symbols of the components of $\mathcal{J}_{\epsilon}$ are given by

[TABLE]

we conclude that

[TABLE]

where the functions $\mu_{1}$, $\mu_{2}$ can be computed pointwise in $\epsilon z$ and are completely determined by the coefficients $k_{m}$ and $\alpha_{i,j}^{m}$, see Figure 6 for an illustration. Moreover, by condition (20) we have $\sigma_{0}\in\{\mu_{1}(0),\,\mu_{2}(0)\}$, where our choice in (21) implies $\sigma_{0}=\max\{\mu_{1}(0),\,\mu_{2}(0)\}$ and Assumption 7 guarantees via

[TABLE]

the desired invertibility of $\mathcal{B}_{\epsilon}$. For $\sigma_{0}=\min\{\mu_{1}(0),\,\mu_{2}(0)\}$, however, Assumption 7 can generically not be satisfied. In this case we expect that KdV-type waves still exist, but exhibit oscillatory tails due to resonances with the continuous spectrum. A proof of this fact would involve sophisticated arguments lying beyond the scope of the present paper, but we refer to [14, 27, 20] for similar rigorous results on certain generalizations of one-dimensional FPU chains.

3.1 Square lattice

We continue the discussion of the square lattice and refer to §1.1 for the expressions of the effective potentials and the parameters $k_{m}$.

Test of Assumption 5: The purpose of Assumption 5 is to ensure the existence of KdV traveling waves in the formal limit $\epsilon\rightarrow 0$. This is actually a minimal condition for the existence of a solitary wave for $\epsilon>0$. In Figure 7 we see that $\sigma_{0}$ is always positive but non-constant and that both $\lambda$ and the shape parameters $p_{i}$ from (30) are well-defined and non-vanishing for all values of $\alpha$ except for $\alpha=\pm\pi/2$, and this implies that the parameters $d_{i}$ from (28) satisfy the second part of Assumption 5. The singularities at $\alpha=\pm\pi/2$, however, have no geometric meaning but reflect that the corresponding wave profiles extend via $W_{1}=0\neq W_{2}$ only in the vertical direction, see (10).

Test of Assumption 7: We list in Figure 8 the numerically computed function $T$ for $\alpha\in\{0,\,\tfrac{\pi}{12},\,\tfrac{\pi}{6},\tfrac{\pi}{4}\}$, and recall that this function is quadratic near $z=0$, see the remark to Assumption 7. These graphs indicate that the required condition is satisfied for all values of $\alpha$.

Velocity profiles: To illustrate the direction dependence of the lattice waves, we display in Figure 9 the two components of the limiting velocity profiles $W_{0}=\lim_{\epsilon\to 0}W_{\epsilon}$ for the same set of values of $\alpha$ as in Figure 8. For $\alpha=\frac{\pi}{4}$ we observe the coincidence of the two components of $W_{0}$, which is consistent with the axial symmetry of the lattice with respect to the diagonal direction. We also recall that the case $\alpha=0$ has been studied in [5].

3.2 Diamond lattice

The diamond lattice from the right panel in Figure 1 arises by taking out the springs on one diagonal of the square lattice and then rotating by $45$ degrees. For each particle three pairs of forces come into question, and thus we have $M=3$. Moreover, similarly to the discussion in §1.1, the values for $k_{m}$ are obtained by projecting the propagation vector to the axes of symmetry. These are $(0,1)$, $(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$ and $(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})$ and provide

[TABLE]

as well as the effective potentials

[TABLE]

where we supposed for simplicity that all springs can be described by the same potential $V$. Finally, notice that the number of symmetries is considerably reduced as compared with the square lattice: we still have the two reflection symmetries and but no more invariance under rotations by 45 degrees.

Test of Assumption 5: In Figure 10 we see that the graph of $\sigma_{0}$ is always positive but exhibits considerable changes instead of the smooth and oscillatory behavior in the square lattice. Due to the axial symmetry of the lattice it suffices to consider $\alpha\in[0,\frac{\pi}{2}]$, and in the graph of KdV-shape parameters it is easily seen that $p_{1}\neq 0$ and $p_{2}\neq 0$ can be defined for all $\alpha\in(0,\frac{\pi}{4}]$ except for isolated critical values.

Test of Assumption 7: Figure 11 shows the numerical graphs of $T$ for $\alpha\in\{\frac{\pi}{12},\ \frac{\pi}{6},\ \frac{\pi}{3},\ \frac{\pi}{2}\}$, where we omitted the plot for $\alpha=0$ since the KdV-shape parameters explode for this value. Similarly as before, Assumption 5 is fulfilled locally around $z=0$ and Figure 11 gives numerical evidence for its global validity.

Velocity profiles: We observe in Figure 12 a clear flip of the velocity profiles, when $\alpha$ turns from $\frac{\pi}{12}$ to $\frac{\pi}{6}$. This means that the wave is compressive for small values of $\alpha$ and above a certain value it becomes expansive. The threshold for this change is exactly the singularity between [math] and $\frac{\pi}{2}$ as seen in the $(p_{1},\ p_{2})$–$\alpha$ graph of Figure 10. For $\alpha=\frac{\pi}{2}$ we observe the vanishing of the first component of $W_{0}$, which hints at a similar result as proven in [5], namely the existence of a unidirectional and longitudinal KdV-like solitary wave, whose first component completely vanishes.

3.3 Triangle lattice

In this example we have $M=3$ with

[TABLE]

so the effective potentials are given by

[TABLE]

Test of Assumption 5 and 7: As shown in Figure 13, the sound speed $\sqrt{\sigma}_{0}$ is constant for all $\alpha$. By comparing Figures 7, 10 we might guess that the $\alpha$-dependence of $\sigma_{0}$ is more dramatic in lattices with few symmetries, but we have no rigorous explanation for this observation. Moreover, the plots in Figure 14 indicate that Assumption 7 is satisfied for all values of $\alpha$.

Velocity profiles: Figure 15 illustrates the existence of expansion waves in all directions and for $\alpha\in 2\pi\{0,1/3,2/3\}$ we find again unidirectional waves which propagate longitudinal with respect to one of the principal axes in the lattice.

Acknowledgments

Both authors authors are grateful for the financial support by the Deutsche Forschungsgemeinschaft (DFG individual grant HE 6853/2-1). They also sincerely thank the referee for careful reading and for the substantial list of very helpful comments concerning the exposition of the material and the presentation of the technical details.