KdV waves in atomic chains with nonlocal interactions

We consider atomic chains with nonlocal particle interactions and prove the existence of near-sonic solitary waves. Both our result and the general proof strategy are reminiscent of the seminal paper by Friesecke and Pego on the KdV limit of chains with nearest neighbor interactions but differ in the following two aspects: First, we allow for a wider class of atomic systems and must hence replace the distance profile by the velocity profile. Second, in the asymptotic analysis we avoid a detailed Fourier pole characterization of the nonlocal integral operators and employ the contraction mapping principle to solve the final fixed point problem.


Introduction
Since the pioneering paper [ZK65], the so-called KdV limit of atomic chains with nearest neighbor interactions -often called Fermi-Pasta-Ulam or FPU-type chains -has attracted a lot of interest in both the physics and the mathematics community, see [FML14] for a recent overview. The key observation is that in the limiting case of long-wave-length data with small amplitudes the dynamics of the nonlinear lattice system is governed by the Korteweg-de Vries (KdV) equation, which is a completely integrable PDE and hence well understood. For rigorous results concerning initial value problems we refer to [SW00] and to [CBCPS12,GMWZ14] for similar result in chains with periodically varying masses.
Of particular interest are the existence of KdV-like solitary waves and their stability with respect to the FPU dynamics. Both problems have been investigated by Friesecke and Pego in the seminal four-paper series [FP99,FP02,FP04a,FP04b], see also [HW13] for simplifications in the stability proof and [FML14] concerning the existence of periodic KdV-type waves. The more general cases of two or finitely many solitary waves have been studied in [HW08,HW09] and [Miz11,Miz13], respectively. In this paper we generalize the existence result from [FP99] and prove that chains with interactions between further than nearest-neighbors also admit KdV-type solitary waves. The corresponding stability problem is beyond the scope and left for future research.

Setting of the problem
We consider an infinite chain of identical particles which interact with up to M neighbors on both sides. Assuming unit mass, the equations of motion are therefore given bÿ where u j (t) denotes the position of particle j at time t. Moreover, the potential Φ 1 describes the interactions between nearest-neighbors, Φ 2 between the next-to-nearest-neighbors, and so on. A traveling wave is an exact solution to (1) which satisfies where the parameters r * and v * denote the prescribed background strain and background velocity, respectively. Moreover, ε > 0 is an additional scaling parameter which will be identified below and becomes small in the KdV limit. A direct computation reveals that the wave speed c ε as well as the rescaled wave profile U ε must solve the rescaled traveling wave equation where the discrete differential operators are defined by (3) Note that v * does not appear in (2) due to the Galilean invariance of the problem and that the solution set is invariant under the addition of constants to U ε . It is therefore natural to interpret (2) as an equation for the rescaled velocity profile W ε := U ε ; the corresponding distance or strain profile ∇ +ε U ε can then be computed by convoluting W ε with the rescaled indicator function of an interval, see formula (9) below. For M = 1 and fixed ε > 0 there exist -depending on the properties of Φ 1 -many different types of traveling waves with periodic, homoclinic, heteroclinic, or even more complex shape of the profile W ε , see for instance [Her10,HR10,HMSZ13] and references therein. In the limit ε → 0, however, the most fundamental waves are periodic and solitary waves, for which W ε is either periodic or decays to 0 as x → ±∞.
In this paper we suppose r * = 0 -this condition can always be ensured by elementary transformations -and split off both the linear and the quadratic terms from the force functions Φ m . This reads or, equivalently, Φ m (r) = 1 2 α m r 2 + 1 3 β m r 3 + Ψ m (r) with Ψ m (r) = O r 4 . In order to keep the presentation as simple as possible, we restrict our considerations to solitary waves -the case of periodic profiles can be studied along the same lines -and rely on the following standing assumption.
x W " (x) Figure 1: Sketch of the rescaled velocity profile W ε for ε > 0 (black) and ε = 0 (gray) as function of the rescaled phase variable x. The grid with spacing ε describes the rescaled particle index εj while the dashed arrows indicate the height and the width of the pulse W ε . The rescaled distance profile A ε W ε has a similar shape.
Note that the usual requirements for M = 1 are α 1 > 0 and β 1 = 0 but the case β 1 < 0 can be traced back to the case β 1 > 0 by a simple reflection argument with respect to the strain variable r. Below we discuss possible generalizations of Assumption 1 including cases in which the coefficients come with different signs.

Overview on the main result and the proof strategy
The overall strategy for proving the existence of KdV-type solitary waves in the lattice system (1) is similar to the approach in [FP99] but many aspects are different due to the nonlocal coupling. In particular, we base our analysis on the velocity profile W ε = U ε and not on the distance profile ∇ ε U ε , deviate in the justification of the key asymptotic estimates, and solve the final nonlinear corrector problem by Banach's fixed point theorem. A more detailed comparison is given throughout the paper.
As for the classical case M = 1, we prescribe a wave speed c ε that is slightly larger than the sound speed c 0 and construct profile functions that satisfy (2) and decay for x → ±∞. More precisely, we set i.e., the small parameter ε quantifies the supersonicity of the wave. Note that the subsonic case c ε < c 0 is also interesting but not related to solitary waves, see discussions at the end of §2 and the end of §3. The asymptotic analysis from §2 reveals that the limiting problem as ε → 0 is the nonlinear ODE where the positive constants d 1 and d 2 depend explicitly on the coefficient α m and β m , see formula (24) below. This equation admits a homoclinic solution, which is unique up to shifts (see §3.1) and provides via w(t, x) = W 0 (x − t) a solitary wave to the KdV equation For ε > 0 we start with the ansatz and derive in §3 a fixed point equation for the corrector V ε , where the operator F ε is introduced in (37). The definition of F ε requires to invert a linear operator L ε , which is defined in (26) and admits a singular limit as ε → 0. The linear leading order operator L 0 stems from the linearization of (5) around the KDV wave W 0 and can be inverted on the space L 2 even (R) but not on L 2 (R) due to the shift invariance of the problem. The first technical issue in our perturbative existence proof is to show that these invertibility properties persist for small ε > 0, see Theorem 12. The second one is to guarantee that F ε is contractive on some ball in L 2 even (R), see Theorem 13. Our main findings are illustrated in Figure 1 and can be summarized as follows, see also Corollary 14.
Main result. For any sufficiently small ε there exists a unique even and nonnegative solution W ε to the rescaled traveling wave equation (2) with (4) such that holds for some constant C independent of ε, where W 0 is the unique even solution to (5).
The asymptotic analysis presented below can -for the price of more notational and technical effort -be applied to a wider class of chains. Specifically, we expect that the following generalization are feasible: 1. We can allow for M = ∞ provided that the coefficients α m , β m and γ m decay sufficiently fast with respect to m (say, exponentially).
2. Some of the coefficients α m and β m might even be negative. In this case, however, one has to ensure that the contributions from the negative coefficients are compensated by those from the positive ones. A first natural condition is 3. The non-quadratic contributions Ψ m might be less regular in the sense of The paper is organized as follows. In §2 we introduce a family of convolution operators and reformulate (2) as an eigenvalue problem for W ε . Afterwards we provide singular asymptotic expansions for a linear auxiliary operator B ε , which is defined in (17) and plays a prominent role in our method. §3 is devoted to the proof of the existence theorem. We first study the leading order problem in §3.1 and show afterwards in §3.2 that the linear operator L ε is invertible. In §3.3 we finally employ Banach's contraction principle to construct solutions V ε to the nonlinear fixed problem (6) and conclude with a brief outlook.

Preliminaries and linear operators
In this section we reformulate the nonlinear advance-delay-differential equation (2) as an integral equation and provide asymptotic estimates for the arising linear operators.

Reformulation in terms of integral operators
For any η > 0, we define the convolution operator A η by and regard (2) as an equation for the rescaled velocity profile W := U .
Lemma 2 (reformulation as nonlinear eigenvalue problem). Suppose that W ε = U ε belongs to L 2 (R). Then, the nonlinear eigenproblem is equivalent to the traveling wave equation (2).
Proof. The operators defined in (3) and (7) satisfy so (2) follows from (8) after differentiation with respect to x and defining U ε as the primitive of W ε . In order to derive (8) from (2), we first notice that W ε ∈ L 2 (R) implies A mε W ε ∈ W 1,2 (R) (cf. Corollary 4 below) and hence (A mε W ε )(x) to 0 as x → ±∞. Afterwards we integrate (2) with respect to x and eliminate the constant of integration by means of the decay condition at infinity.
In the case M = 1, we can derive from (8) the identity which is the equation for the distance profile A ε W ε and has been studied in [FP99] (see equation (2.7) there for the function φ = A ε W ε ). For M > 1, however, we have to work with the velocity profile W ε since for a general function W it is not possible to express A mε W for m > 1 in terms of A ε W .
We next summarize important properties of the convolution operators defined in (7).
Lemma 3 (properties of A η ). For each η > 0, the integral operator A η has the following properties: 3. A η respects the even-odd parity, the nonnegativity, and the unimodality of functions. The latter means monotonicity for both negative and positive arguments.

5.
A η is self-adjoint in the L 2 -sense.
Proof. All assertions follow immediately from the definition of A η ; see [Her10] for the details.

Asymptotic analysis for the convolution operators A η
The symbol function a η from (11) is analytic with respect to z = ηk and in view of The integral operator (7) therefore admits the formal expansion which reveals that A mε should be regarded as a singular perturbation of the identity operator id. This singular nature complicates the analysis because the error terms in (12) can only be bounded in terms of higher derivatives.
One key observation for dealing with the limit ε → 0 is -roughly speaking -that the resolventtype operator is well-defined and almost compact as long as κ > 0. It thus exhibits nice regularizing properties which allows us to compensate bad terms steming from the expansion (12). The same idea has been employed in [FP99] in the context of the distance profile A ε W , showing that the Yosida-type regularization is well-defined and bounded by C/ 1 + ε 2 k 2 , cf. [FP99, Corollary 3.4.]. Before we establish a related but weaker result in next subsection, we derive explicit error bounds for the singular expansion of A mε .
Lemma 5 (small-parameter asymptotics of A η ). There exists a constant C, which does not depend on η, such that the estimates hold for any sufficiently regular W . In particular, we have for any W ∈ L 2 (R).
Final argument: Let W ∈ L 2 (R) be arbitrary but fixed. Since A η is self-adjoint, see Lemma 3, and in view of (13) we readily demonstrate and this implies W 2 ≤ lim inf η→0 A η W 2 . On the other hand, the estimate (10) 2 ensures that lim sup η→0 A η W 2 ≤ W 2 . We therefore have W 2 = lim η→0 A η W 2 and combining this with the weak convergence (16) we arrive at (15) since L 2 (R) is a Hilbert space.
2.3 Asymptotic properties of the auxiliary operator B ε As already outlined above, we introduce for any given ε > 0 the operator which appears in (8) if we collect all linear terms on the left hand side, insert the wave-speed scaling (4), and divide the equation by ε 4 . We further define the operator which can -thanks to Lemma 5 -be regarded as the formal limit of B ε as ε → 0. In Fourier space, these operators correspond to the symbol functions which are illustrated in Figure3 and satisfy for any fixed k. However, this convergence does not hold uniformly in k since B ε is a singular perturbation of B 0 . Using the positivity of these symbol functions, we easily demonstrate the existence of the inverse operators where B −1 ε maps into the Sobolev space W 1,2 (R) and is hence compact since 1/b 0 (k) decays quadratically at infinity. The inverse of B ε , however, is only continuous because b ε (k) remains bounded as k → ±∞. In order to obtain asymptotic estimates for B −1 ε , we introduce the cut-off operator Π ε : L 2 (R) → L 2 (R) by defining its symbol function π ε as follows π ε (k) := 1 for |k| ≤ 4 ε , 0 else .
One of our key technical results is the following characterization of B −1 ε , which reveals that B ε admits an almost compact inverse. For m = 1, a similar but slightly stronger results has been given in [FP99,Corollary 3.5] using a careful Fourier pole analysis of the involved integral operators. For m > 1, however, the symbol functions possess more poles in the complex plane and hence we argue differently.
Proof. In view of (17), (19) and Lemma 3, it remains to show (20). Using the properties of the sinc function, see Figure 2, we readily verify that for all z ∈ R and m ∈ N .
There exists another useful characterization of B −1 ε , which relies on the non-expansive estimate A mε W ∞ ≤ W ∞ , see Lemma 3.
Lemma 7 (von Neumann representation). We have where the series on right hand converges for any W ∈ L 2 (R).
Proof. In the first step we regard all operators as defined on and taking values in L ∞ (R). We also use the abbreviations Since the operator norm of I ε -computed with respect to the ∞-norm -satisfies the von Neumann formula provides in the sense of an absolutely convergent series of L ∞ -operators. In the second step we generalize this result using the estimates from Lemma 3. In particular, the right-hand side in (21) is well-defined for any W ∈ L 2 (R) since Lemma 3 ensures I ε W ∈ L ∞ (R).
Proof. For ε > 0, all assertions follow from the representation formula in Lemma 7 and the corresponding properties of the operators A mε , see Lemma 3. For ε = 0 we additionally employ the approximation result from Lemma 6.
Note that all results concerning B −1 ε are intimately related to the supersonicity condition c 2 ε > c 2 0 . In a subsonic setting, one can still establish partial inversion formulas, see for instance [HMSZ13], but the analysis is completely different.

Proof of the main result
In view of the wave-speed scaling (4) and the fixed point formulation (8), the rescaled traveling wave problem consists in finding solutions W ε ∈ L 2 (R) to the operator equation where the linear operator B ε has been introduced in (17). Moreover, the nonlinear operators encode the quadratic and cubic nonlinearities, respectively, and are scaled such that the respective formal ε-expansions involve nontrivial leading order terms. In particular, we have for any fixed W ∈ L 2 (R), see (15). Note also that (22) always admits the trivial solution W ε ≡ 0.
In what follows we solve the leading order problem to obtain the KdV wave W 0 , transform (22) via the ansatz W ε = W 0 + ε 2 V ε into another fixed point equation, and employ Banach's contraction principle to prove the existence of a corrector V ε for all sufficiently small ε. In [FP99], the last step has been solved using a operator-valued variant of the implicit function theorem.

The leading order problem and the KdV wave
Passing formally to limit ε → 0 in (22), we obtain the leading order equation which is the ODE (5) with parameters In particular, the leading order problem is a planar Hamiltonian ODE with conserved quantity E = 1 2 (W ) 2 + 1 3 d 2 W 3 − 1 2 d 1 W 2 and admits precisely one homoclinic solution, see Figure 4.
Lemma 9 (linear and nonlinear leading-order problem). There exists a unique solution W 0 ∈ L 2 even (R) to (23), which is moreover smooth, pointwise positive, and exponentially decaying. Moreover, the L 2 -kernel of the linear operator L 0 with is simple and spanned by the odd function W 0 .
Proof. The existence and uniqueness of W 0 follow from standard ODE arguments and the identity L 0 W 0 = 0 holds by construction. Moreover, the simplicity of the L 2 -kernel of the differential operator L 0 can be proven by the following Wronski-type argument. Suppose for contradiction that V 1 , V 2 ∈ L 2 (R) are two linearly independent kernel functions of L 0 such that The ODE L 0 V i = 0 combined with V i ∈ L 2 (R) implies that V i and V i are continuous functions with and we conclude that ω(x) → 0 as |x| → ∞. On the other hand, we easily compute ω (x) = 0 and obtain the desired contradiction.
Since W 0 is smooth, it satisfies (22) up to small error terms. In particular, the corresponding linear and the quadratic terms almost cancel due to (23).
Lemma 10 (ε-residual of W 0 ). There exists a constant C such that holds for all ε ≤ 1.
Proof. Since W 0 is smooth, Lemma 5 provides a constant C such that holds for j ∈ {1, 2}, and this implies Since W 0 solves (23), we get where the second inequality stems from the definitions of B ε and B 0 , see (17) and (18). Lemma 5 also yields and combining this with (10) 2 and we arrive at The desired result is now a direct consequence of (25).
For completeness we mention that

The linearized traveling wave equation for ε > 0
For any ε > 0, we define the linear operator L ε on L 2 (R) by where W 0 ∈ L 2 even (R) is the unique even KdV wave provided by Lemma 9. This linear operator appears naturally in the linearization (22) around W 0 since holds due to the linearity of B ε and the quadraticity of Q ε .
Lemma 11 (elementary properties of L ε ). For any ε > 0, the operator L ε is self-adjoint in L 2 (R) and respects the even-odd parity. Moreover, we have for any W ∈ L 2 (R) with W ∈ L 2 (R).
Proof. All assertions follow immediately from the properties of A mε and B ε , see (17) and Lemma 5, and the smoothness of W 0 .
Our perturbative approach requires to invert the operator L ε on the space L 2 even (R), see the fixed point problem in Theorem 13 below. In view of Lemma 6 we conclude that the operator L ε is invertible if and only if has this property. On the other hand, the formal limit operator id −B −1 0 M 0 can be inverted as it is a Fredholm operator on L 2 even with index 0 and trivial kernel thanks to the properties of B −1 0 and L 0 , see Lemma 9. Due to these observations we are now able to derive our main asymptotic result, which ensures the ε-uniform invertibility of L ε on the space of even L 2 -functions.
The proof is actually at the core of our method and does not employ standard results since B −1 ε is not compact and because A mε is not a regular but a singular perturbation of id. Note that the analogue for M = 1 is not stated explicitly in [FP99] but could be derived from the asymptotic formulas therein.
Theorem 12 (uniform invertibility of L ε ). There exists ε * < 0 such that for any 0 < ε ≤ ε * the operator L ε is continuously invertible on L 2 even (R). More precisely, there exists a constant C which depends on ε * but not on ε such that holds for all 0 < ε ≤ ε * and any G ∈ L 2 even (R).

Proof. Preliminaries:
Our strategy is to show the existence of a constant c * > 0 such that holds for all V ∈ L 2 even (R) and all sufficiently small ε, because this implies the desired result. In fact, (27) ensures that the operator L ε : L 2 even (R) → L 2 even (R) has both trivial kernel and closed image. The symmetry of L ε gives ker L ε = coker L ε and due to the closed image we conclude that L ε is not only injective but also surjective. Moreover, the ε-uniform continuity of the inverse is a further consequence of (27). Now suppose for contradiction that such a constant c * does not exist. Then there exist a sequence (ε n ) n∈N ⊂ (0, 1] with ε n → 0 as well as sequences (V n ) n∈N ⊂ L 2 even (R) and (G n ) n∈N ⊂ L 2 even (R) such that Weak convergence to 0: By weak compactness we can assume that there exists V ∞ ∈ L 2 even (R) such that and using Lemma 11 we find for any sufficiently smooth test function φ. In other words, the even function V ∞ belongs to the kernel of L 0 , so Lemma 9 provides Further notations: For the remaining considerations we abbreviate the constant from Lemma 6 by D and denote by C any generic constant (whose value may change from line to line) that is independent of n. We further choose K > 2M sufficiently large such that and denote by χ K the characteristic function of the interval I K := [−K, +K]. We also write V n = V (1) and observe that these definitions imply We finally set and notice that the estimates from Lemma 3 combined with the smoothness of W 0 provide for some constant C.

Strong convergence of V
(1) n and V (3) n : By definition, we have and Lemma 6 ensures that where the second inequality follows from (32) and (31). From this we infer as well as Since the functions V (1) n are supported in the interval I K and since W 1,2 (I K ) is compactly embedded into L 2 (I K ) we conclude that the sequence V (1) n n∈N is precompact in L 2 (I K ). On other hand, the weak convergence (29) implies and in summary we find where we used that V for any given ξ ∈ R. We therefore estimate and this implies

Nonlinear fixed point argument
Setting W ε = W 0 + ε 2 V ε , the nonlocal traveling wave equation (22) is equivalent to Since L ε can be inverted for all sufficiently small ε > 0, we finally arrive at the following result. and hence after integration. A particular consequence is the estimate Concluding arguments: Combining all estimates derived so far with the definition of F ε and the bounds for L −1 ε -see Lemma 12 -we verify To complete the proof we first set D := 2 C and choose afterwards ε sufficiently small.
Corollary 14 (main result from §1). For any sufficiently small ε > 0, the reformulated traveling wave equation (8) admits a unique even solution W ε with speed c 2 0 + ε 2 such that holds for some constant C independent of ε. Moreover, W ε is nonnegative and smooth.
Proof. The existence and local uniqueness of W ε = W 0 + ε 2 V ε along with the L 2 -estimate is a direct consequence of Theorem 13. Moreover, re-inspecting the proof of Theorem 13 we easily derive an uniform L ∞ -bound for the corrector V ε . By (22) we further get where the right hand side is -at least for small ε -nonnegative due to the properties of the KdV wave W 0 and the potential Φ, see Lemma 9 and Assumption 1. The nonnegativity of W ε is hence granted by Corollary 8.
The constants from the proof of Theorem 13 are, of course, far from being optimal. In general, a solution branch ε → W ε ∈ L 2 even (R) on an interval [0, ε * ] can be continued as long as the linearization of the traveling wave equation around W ε * provides an operator L * that can be inverted on the space L 2 even (R). Since the shift symmetry always implies that W ε * is an odd kernel function of L * , the unique continuation can hence only fail if the eigenvalue c 2 ε of the linearized traveling wave operator is not simple anymore. Unfortunately, almost nothing is known about the spectral properties of the operator (38) for moderate values ε * . It remains a challenging task to close this gap, especially since any result in this direction should have implications concerning the orbital stability of W ε * . For M = 1 it has also been shown in [FP99, Propositions 5.5 and 7.1] that the distance profile A ε W ε is unimodal ('monotonic falloff') and decays exponentially for x → ±∞. For M > 1, it should be possible to apply a similar analysis to the velocity profile W ε but the technical details are much more involved. It remains open to identify alternative and more robust proof strategies. For instance, if one could show that the waves from Corollary 14 can be constructed by some variant of the abstract iteration scheme the unimodality of W ε would be implied by the invariance properties of A εm and B −1 ε , see Lemma 3 and Corollary 8. A similar argument could be used for the exponential decay because A mε maps a function with decay rate λ to a function that decays with ratē λ = sinh 1 2 εmλ 1 2 εmλ and since the von Neumann formula from Lemma 7 provides corresponding expressions for B −1 ε ; see [HR10] for a similar argument to identify the decay rates of front-like traveling waves. In this context we further emphasize that only supersonic waves can be expected to decay exponentially. For subsonic waves with speed c 2 ε < c 2 0 , the linearization of the traveling wave equation (2) predicts tails oscillations and hence non-decaying waves, see [HMSZ13] for a similar analysis with non-convex interaction potentials.