TRAVELING WAVES IN FERMI-PASTA-ULAM CHAINS WITH NONLOCAL INTERACTION

. The paper is devoted to traveling waves in FPU type particle chains assuming that each particle interacts with several neighbors on both sides. Making use of variational techniques, we prove that under natural assumptions there exist monotone traveling waves with periodic velocity proﬁle (periodic waves) as well as waves with localized velocity proﬁle (solitary waves). In fact, we obtain periodic waves by means of a suitable version of the Mountain Pass Theorem. Then we get solitary waves in the long wave length limit.


1.
Introduction. In the present paper we consider an infinite chain of identical particles on the line such that each particle interacts with M neighbors on both sides. The dynamics of such a chain is governed by equations where q j (t) is the position of jth particle at time t. The potentials U m , m = 1, 2, . . . , M , represent the interaction between a particle and its neighbors so that U 1 corresponds to the interaction with nearest neighbors, U 2 with second nearest neighbors, and so on. In the case M = 1 this is the famous Fermi-Pasta-Ulam (shortly, FPU) lattice introduced and studied numerically in the pioneering paper [6]. Since that time the FPU lattice constantly attracts a lot of interest in physics and mathematics communities. A representative overview of the subject can be found in the monograph [14]. Notice, however, that [14] does not contain any information about traveling waves on the FPU lattice.
A traveling wave solution of (1) is a solution of the form q j (t) = u(j − ct), where u(s) and c > 0 are the profile function and speed of the wave, respectively. For definiteness we consider the case of positive wave speed, the case c < 0 being similar. The relative displacement profile is defined by where the derivative u (s) is the velocity profile. Two types of traveling waves are of interest: solitary waves and periodic waves. A solitary wave is localized in space in the sense that its velocity profile vanishes at infinity, while the velocity profile of a periodic wave is periodic. Automatically, these waves have vanishing at infinity and periodic displacement profiles, respectively. Notice that the profile function of a periodic wave is not necessarily periodic.
The profile function of a traveling wave is a solution of the following forwardbackward differential-difference equation where D + m u(s) = u(s + m) − u(s) and D − m u(s) = u(s) − u(s − m) . We exclude trivial waves with linear profile functions u(s) = as + b. Notice that physically meaningful are waves with monotone, either nondecreasing or nonincreasing, profile functions. Only such waves are considered in this paper.
The first result on the existence of supersonic monotone solitary FPU traveling waves is obtained in [7]. This is done by minimizing the average kinetic energy subject to the constraint that the average potential energy is given. In this approach the wave speed appears as the Lagrange multiplier. Another constrained minimization is used in [16] to study both periodic and solitary waves in FPU with convex potentials.
Completely different approach to the existence of solitary waves is suggested in [32]. With the aid of the Mountain Path Theorem, it is obtained an existence result for supersonic waves with prescribed speed. The starting point of paper [28] is to study periodic waves, still supersonic. More precisely, it is shown that periodic waves with arbitrarily large periods do exist. Then a solitary wave is obtained as the limit of a sequence of periodic waves with periods tending to infinity (periodic approximation). The approach of [28] makes use of the Mountain Path Theorem and the Nehari manifold. A detailed presentation of these and other results on the FPU system, including the existence of subsonic periodic waves, can be found in [26]. Notice that the results mentioned so far concern the case of superquadratic interaction potential. The case of asymptotically quadratic (saturable) potential is studied in [29]. Other results on FPU traveling waves obtained with the help of variational methods can be found in [16,31]. Let us point out that periodic approximation approach is proven to be efficient in many other variational problems (see, e.g., [25,27,29]).
In remarkable paper [9] it is discovered that near-sonic, supersonic FPU solitary waves can be obtained as perturbations of KdV solitons. In subsequent papers [10,11,12] the stability of such waves is studied. Recently, with the help of this approach cnoidal FPU waves have been found [8]. In [3,4,13,20] generalized KdV equations is used to study other FPU type systems. Another perturbation approach is used in [18] where different types of traveling waves on FPU lattices are obtained by means of bifurcation theory.
More general lattice systems (1) of FPU type are not well-studied. To the best of our knowledge, there are only two papers devoted to this subject. In [34] certain approximations of solitary waves on lattices with second-neighbor interaction are suggested and numerical results are discussed. Recent paper [17] is devoted to near-sonic solitary waves for (1) of KdV type and follows the ideas developed in [9]. However, the techniques used there is different in many aspects due to the nonlocal character of the interaction. Let us point out paper [1] where the nonlinear Klein-Gordon lattice with non-local interaction is considered and solitary traveling waves are obtained as perturbations of continuum Klein-Gordon solitary waves. Notice that nonlocal discrete Klein-Gordon equations are extensively studied in physics literature (see, e.g., [30] and references therein). Also we mention other types of lattice systems with nonlocal interaction such as discrete nonlinear Schrödinger equations (see [5] and references therein).
The organization of the paper is as follows. In Section 2 we provide a variational setting of the problem and formulate the main result of the paper. Section 3 contains some technical preliminaries, while in Section 4 we discuss the Mountain Pass Geometry of the energy functionals associated with the problem considered and Nehari type characterization of mountain pass values of these functionals. In Section 5 we prove the existence of periodic traveling waves and obtain a uniform bound for their energy levels. In Section 6 we make the passage to the limit as the period of periodic waves tends to infinity and obtain the existence of solitary waves. At this point we make use of a rudimentary concentration-compactness based on inequality (13). Deep results from [22] are not needed here because the equation for traveling waves is set up on the real line. Finally, in Section 7, we show that, under additional assumptions, periodic waves converge to solitary ones in certain non-local sense and consider some examples. Considering T -periodic problem with integer T we assume, in addition, that m 0 = 1. (A 4 ) All functions |r| −1 V m (r), extended by 0 to r = 0, are nondecreasing and at least one of them is strictly increasing. Notice that, in general, not all of the functions V m are nonzero. are nondecreasing (respectively, nonincreasing) for r > 0 (respectively, r < 0). A simple proof in a more general setting can be found in [23,Lemma 2.3]. Furthermore, G m0 (r) > 0 for all r = 0.
We are looking for solutions of equation (2) subject to one of the following conditions where T > 0 is a given period (periodic waves), or (solitary waves). Given T > 0, we introduce the space . Since all functions in H 1 loc (R) are continuous, condition u(0) = 0 makes sense. The inner product onX T is given by Here and later on, I T = [−T /2, T /2]. Also we introduce the Hilbert space We denote by · T and · the norms on X T and X induced by the inner products (·, ·) T and (·, ·), respectively.
We denote by X + T and X + the cones of nondecreasing functions in X T and X, respectively. The cones of nonincreasing function are X − T = −X + T and X − = −X + . All these cones are closed.

Remark 2.
It is easily seen that if T ≥ M is an integer, all operator D ± m , m = 1, 2, . . . , M , have nontrivial kernels in the space X T and ker D ± 1 ⊂ ker D ± m , m = 2, . . . , M . Conversely, if one of those operators has a nonzero kernel in X T , T ≥ M , then T is an integer. In the space X all operators D ± m have trivial kernels. We associate to problems (2), (4) and (2), (5) the following energy functionals on X, respectively. Later on we shall show that these are well-defined C 1 functionals, and their critical points are solutions of the problems we consider. The quadratic and non-quadratic parts of J T (respectively, J) are denoted by 1 2 q T (u) and Ψ T (u) (respectively, 1 2 q(u) and Ψ(u)). The spaces X T and X, as well as the functionals J T and J, are invariant with respect to modified translations Let us point out that the only trivial solution of equation (2) in the space X is u = 0, while every space X T contains trivial solutions of the form u(s) = ks, k ∈ R.
Our main result is the following.
, then the previous statement holds for all integer T ≥ T c as well.
(b) Let T n ≥ T c be a sequence of noninteger numbers such that T n → ∞. Then there exist a sequence of nontrivial solutions u n ∈ X + Tn (respectively, u n ∈ X − T k ) to problem (2), (4), and a nontrivial solution u ∈ X + (respectively, u ∈ X − ) of problem (2), (5) such that, along a subsequence, J Tn (u n ) → J(u) and u n → u uniformly on compact intervals together with first and second derivatives. If m 0 = 1 in Assumption (A 3 ), then the same holds for all integer sequences T n ≥ T c .

Remark 3.
In [17] it is not assumed that all the coefficients a m in the quadratic parts of interaction potentials are nonnegative. We do not know whether variational methods may work if some of a m are negative. Therefore, in the last case the existence of traveling waves with arbitrary supersonic speed is an open problem.
Looking for monotone profile functions, me may add, without loss of generality, the following assumption The proof of main result shall be carried out in the case of nondecreasing waves, the other case being similar.

Preliminaries. We begin with simple estimates for difference operators
Proof. We prove part (a) in the case of D + m . The other one is similar. Since Interchanging the order of integration, we obtain that the inequality for the H 1 norm follows immediately. From Lemma 3.1 it follows immediately that the quadratic parts of functionals J T , T ≥ M , and J are bounded on X T and X, respectively. Furthermore, u ∈ X , that is, the quadratic parts are positive definite.
By assumption (A 2 ), for every R > 0 there exists a constant C R > 0 such that |V m (r)| ≤ C R |r| whenever |r| ≤ R. Then standard arguments together with estimates from Lemma 3.1 give rise to the following result. and Let u ∈ X T be a critical point of J T and ϕ be a T -periodic C ∞ function. Making use of (6), equation that is, u is a weak solution of (2) and, hence, a classical solution because the righthand side of (2) is a continuous function of s. Taking as ϕ a finitely supported C ∞ function and making use of (7), we obtain that critical points of J are solutions of equation (2).
Remark 4. Not all solutions of (2) in X T are critical points of J T . Indeed, taking h(s) = s as a test function in (6), we see that any critical point of J T satisfies The function u(s) = ks is a solution in X T for all T . By (8), it is a critical point of Under our assumptions, this equation has exactly two nonzero solutions k + > 0 and k − < 0. It would be interesting to find examples of solutions to (2) in X which are not critical points of J.

4.
Mountain Pass Geometry. Let Φ be a C 1 functional on a Banach space E.
(C) If u n ∈ E n is a Cerami sequence at a level l, that is, Φ(u n ) → l and as n → ∞, then u n contains a convergent subsequence.
Now we remind a version of the Mountain Pass Theorem for functionals satisfying the Cerami condition (see, e.g., [15,24]).
Theorem 4.1. Let Φ be a C 1 functional on a Banach space E that satisfies the Cerami condition, and let P : E → E be a continuous mapping such that Φ(P u) ≤ Φ(u) for all u ∈ E. Assume that Φ possesses the Mountain Pass geometry with extra condition P (e) = e. Then Φ has a nonzero critical point u ∈ P (E) such that Φ(u) = d ≥ α. The critical value d has the following minimax characterization where Notice that [15,24] contain "no P " version of this theorem. Under the Palais-Smale condition, Theorem 4.1 is obtained in [2], but the arguments used in that paper apply in the case of Cerami condition as well.
If the functional does not satisfy the Cerami condition, we shall still consider Mountain Pass values defined by (9). In general, such a value is not necessarily a critical value. Proof. We sketch the standard argument just to highlight the independence of α and ρ on T . Notice that Assumption (A 2 ) implies that all interaction potentials U m are subquadratic at 0. Making use of Lemma 3.1(a), we see that given ε > 0, there exists ρ > 0 independent of T ≥ M and such that Ψ Similarly, changing ρ, we have that Since J T (v)v = 0 for all critical points, we obtain the required after an appropriate choice of ε. Part (b) is similar.
Now we define the mapping P by It is easily seen that P : X T → X T is a continuous mapping. We have that Since the interaction potentials U m (r) are supposed to be even and increasing as |r| increases, we obtain that J T (P (v)) ≤ J T (v). Furthermore, if 0 = e ∈ X + T , then D + m0 > 0 on certain interval and, by (A 3 ), J T (te) < 0 for large enough t > 0. Hence, J T possesses the Mountain Pass geometry.
The operator P acts continuously in the space X, J(P (v)) ≤ J(v) for all v ∈ X, and J possesses Mountain Pass geometry as well. As we will see below, J T satisfies the Cerami condition (C), while (C) does not hold for J.
Notice that P (X T ) = X + T and P (X) = X + are closed. It is easy that, defining Mountain Pass values d T and d for the functionals J T and J, we may assume that Γ J T and Γ J consist of paths with values in X + T and X + , respectively.
Mountain Pass values d T and d have another minimax characterization. We introduce partial Nehari manifolds Notice that, in general, these sets are not smooth manifolds, but we follow the traditional terminology.
Denote by d * the common value of the two right-hand sides in (12). If v ∈ X + , then a part of the row {tv, t > 0}, after an appropriate rescaling, is a member of Γ J . Hence, d * ≥ d. On the other hand, let γ ∈ Γ J . By Remark 5, in a neighborhood of 0 there exists t > 0 such that J (γ(t))γ(t) > 0. Since J(γ(1)) < 0, making use of Remark 1 we obtain that Therefore, γ(t 0 ) ∈ N + for some t 0 ∈ (0, 1). Since J(γ(t)) , and γ ∈ Γ J is an arbitrary path, we get d * ≤ d.
The proof of (12) is similar.
Remark 6. The Nehari manifold approach developed in [33] does not apply in our situation because the whole of the Nehari manifold for J T is not homeomorphic to the unit sphere in X T if T is an integer.
5. Periodic waves. In this section we prove part (a) of Theorem 2.1 making use of Theorem 4.1.
In the proof of the next lemma, as well as in the proof of Lemma 6.1, we employ a normalization trick that goes back to [19] and is now used in many paper (see, e.g., [21,23,33]). Proof. Let u n ∈ X T be a Cerami sequence for the functional J T at a level l. First, we prove that the sequence u n is bounded. Assuming the contrary and passing to a subsequence, we have that u n T → ∞. Let v n = u n / u n T . Then v n T = 1, and, passing to a subsequence again, we may assume that v n → v weakly in X T . Lemma 3.1 and the compactness of embedding Observing that q T (u) ≤ c 2 u 2 and making use of Assumption (A 3 ), we obtain This is a contradiction. Let w n = kv n , where k > 0. For all n large enough, 0 < k/ u n T < 1 and, hence, Notice that D + m v n → 0, uniformly on I T for all m = 1, 2, . . . , M and, hence, Ψ T (w n ) → 0. Since k > 0 is an arbitrary number, we obtain that J T (r n u n ) → ∞. Observe that 0 < r n < 1 for sufficiently large n because J T (0) = 0 and J T (u n ) → l. As consequence, r n is an interior maximum point of the function J T (ru n ) on [0, 1] and, therefore, J T (r n u n )(r n u n ) = r n (J T (ru n )) | r=rn = 0 .
As consequence, J T (r n u n ) = 1 2 Ψ T (r n u n )(r n u n ) − Ψ T (r n u n ) .
By Remark 1, we obtain that On the other hand, and we obtain a contradiction. Thus, u n is bounded, and then u n → u 0 weakly in X T along a subsequence. Hence, D + m u n → D + m u 0 weakly in H 1 (I T ) and, by the compactness of embedding, uniformly on I T for all m = 1, 2, . . . , m. A straightforward calculation shows that The first term in the right-hand side converges to 0 because u n is a Cerami sequence and u n → u 0 weakly in X T . The second term converges to 0 due to uniform convergence D + m u n → D + m u 0 , m = 1, 2, . . . , M . Therefore, q T (u n − u 0 ) → 0. Since the quadratic form q T is positive definite, u n → u 0 strongly in X T .
Part (a) of Theorem 2.1 is a consequence of the following proposition. It is easily seen that J T (te T ), t ≥ 0, is independent of T ≥ 2M and, by Assumption (A 3 ), is negative for sufficiently large t > 0.
By Theorem 4.1 with P defined in (10), the functional J T , T ≥ 2M , possesses a Mountain Pass critical point u T ∈ X + T with critical value d T = J T (u T ). Obviously, By Remark 4, the only trivial non-zero critical point in X T is the function u(s) = k + s. The value of J T at this point is const · T and, hence, can not be the Mountain Pass critical value d T if T is large. 6. Infinite wave length limit. To obtain solutions in X we pass to the limit as T → ∞. With this aim first we need a uniform bound for the norms u T T , where u T ∈ X + T stands for nontrivial Mountain Pass critical points.
Lemma 6.1. If T n → ∞, u n ∈ N + Tn and the sequence J Tn (u n ) is bounded, then the sequence u n Tn is bounded.
Proof. Assume the contrary. Then, passing to a subsequence, we may assume that u Tn Tn → ∞ and J Tn (u n ) → l. Notice that l ≥ ρ > 0 by Lemma 4.1. Let v n = u n / u n Tn . Making use of suitable modified translations S a , we may assume that |D + m0 v n | attains its maximum value at 0. Since v n Tn = 1, also we may assume that u n → v 0 weakly in H 1 loc (R) and uniformly on compact intervals. It is easily seen that v 0 ∈ X and v 0 X ≤ 1.
Consider two cases.
Case 1. Suppose that v 0 = 0. Then, by Remark 2, D + m0 v 0 = 0 and the maximum value of |D + m0 v 0 | is attained at 0. As consequence, there is an interval I 0 centered at 0 on which |D + m0 v 0 | ≥ ε 0 for some ε 0 > 0. Now the same reasoning as in Case 1 of the proof of Lemma 5.1 leads to a contradiction. In the proof of the next lemma we use the following elementary inequality. If f ∈ L ∞ (T ) ∩ L 2 (I), then f ∈ L p (I), p > 2, and Here I ⊂ R is an interval (not necessarily finite).
Lemma 6.2. Let T n → ∞. Assume that either T n is a noninteger sequence, or it is integer and m 0 = 1. Then for every sufficiently large T n there exist a nontrivial Mountain Pass critical point u n ∈ X + Tn for J Tn and a Mountain Pass critical point u ∈ X + such that, along a subsequence, u n → u uniformly on compact intervals together with first and second derivatives.
Proof. Proposition 5.1 yields the existence of nontrivial Mountain Pass critical points u n ∈ X Tn for all sufficiently large T n and the sequence d Tn = J Tn (u n ) is bounded. Due to the invariance of J T with respect to translations S a , we may assume that D + m0 u n ≥ 0 attains its (positive) maximum value at 0. By Lemmas 4.1 and 6.1, the norms u n Tn are bounded below and above by positive constants independent of n and, therefore, u n → u weakly in H 1 loc (R) along a subsequence. Obviously, u is nondecreasing. The uniform boundedness of u n Tn implies that u ∈ X + .
By the compactness of Sobolev embedding, D + m u n → D + m u for all m = 1, 2, . . . , M , uniformly on compact intervals. Due to the standard approximation argument, to show that u is a critical point of J it is enough to verify that J (u)h = 0 for all h ∈ X with compactly supported h . For any such h, all D + m h are compactly supported as well. Let Then A ⊂ I Tn for all sufficiently large k, and there is a unique h n ∈ X Tn such that h n = h on A. Then Since u n → u weakly in L 2 loc (R) and D + m u n → D + m u uniformly on compact intervals, we see that J(u)h = 0.
Finally, we show that u is a nontrivial critical point of J. Notice that in this case nontrivial means nonzero. Assume the contrary. Then D + m u = 0 and, therefore, for all |r| ≤ R. As consequence, renaming ε and C ε , and making use of the boundedness of D + m u n L 2 (In) , we obtain that Since ε > 0 is arbitrary and all L p -norms in the right-hand side tend to 0, we have that Ψ Tn (u n )u n → 0. Then equation J Tn (u n )u n = 0 implies that Tn ≤ q Tn (u n ) = Ψ Tn (u n )u n → 0 .
This contradicts to the fact that the norms u n 2 Tn are bounded below by a positive constants. Thus, u = 0 and the proof is complete.
The next lemma finalize the proof of Theorem 2.1(b). Lemma 6.3. Let T n → ∞ be a sequence as in Lemma 6.2. Assume that u n ∈ X + Tn is a sequence of nontrivial Mountain Pass critical points for J Tn such that u n → u weakly in H 1 loc (R) and, hence, uniformly on compact intervals, where u ∈ X + is a nontrivial critical point of J. Then u is a Mountain Pass critical point and , it vanishes at infinity, and we can assume that its maximum value, say, r 0 > 0 is attained at 0. Choose v n ∈ X + such that v n is supported on [−T n /2 + M, T n /2 − M ] and v n → v in X. Then D + m v n → D + m v, m = 1, 2, . . . , M , uniformly on compact intervals. Let w n ∈ X + Tn be a unique function such that w n = v n on I Tn . Obviously, J Tn (w n ) = J(v n ) → J(v). Let τ n > 0 be a unique number such that τ n w n ∈ N + Tn . In fact, we have also that τ n v n ∈ N + .
Let us prove that τ n → 1. For, it is enough to show that the sequence τ n is bounded. Assuming the contrary, we have that τ n → ∞ and J(τ n v n ) = J Tn (τ n w n ) > 0. On the other hand, D + m0 v n ≥ r 0 /2 for all n large enough. Due to Assumption (A 3 ), this implies that τ −2 n Ψ(τ n v n ) → ∞ and for all sufficiently large n, a contradiction. Thus, τ n → 1, τ n v n → v and, therefore, J Tn (τ n w n ) → J(v). Since d Tn ≤ J Tn (τ n w n ) and v ∈ N + is arbitrary, we obtain that lim sup d Tn ≤ d . As I is arbitrary, u is a critical point of J and G m ≥ 0, we obtain that Therefore, lim d Tn = d = J(u).
for all m = 1, 2, . . . , M . Then, along a subsequence, the solutions obtained in Theorem 2.1 satisfy Proof. Let v n ∈ X + be a function such that v n → u in X and v n is supported on the interval [−T n /2 + M, T n /2 − M ]. Obviously, D + m v n → D + m u, m = 1, 2, . . . , M , uniformly on compact intervals and in L 2 (R). If w n ∈ X + Tn is a unique function such that w n = v n on I Tn , then it is easily seen that u − w n L 2 (I Tn ) → 0 and J Tn (w n ) → d = J(u) . It is enough to prove that J Tn (u n − w n ) → 0 (16) and J Tn (u n − w n ) → 0 .
Since u n − w n L 2 (Tn) is bounded, we obtain that q Tn (u n − w n ) → 0, and the result follows.
To prove (16) we use the following elementary identity J Tn (u n − w n ) = J Tn (u n ) − J Tn (w n ) + The difference of J Tn 's in the right-hand side is o(1). Let I be any finite interval and n is large enough so that I ⊂ I Tn . We split the second integral in (18) into the sum of integrals over I and I Tn \I. Since u n −w n → 0 in H 1 loc (R) and w n Tn is bounded, while D + m (u n − w n ) → 0 uniformly on compact intervals and D + m w n is bounded in L ∞ (R) for every m = 1, 2, . . . , M , the integral over I tends to 0. As the norm u n − w n Tn is bounded, w n = v n on I Tn and v n → u in X, the integral over I Tn \ I does not exceed a constant multiple of u n − w n Tn v n L 2 (I Tn \I) ≤ C u L 2 (R\I) which can be made arbitrarily small by choosing sufficiently large interval I. Here and thereafter C stands for a generic positive constant.
Splitting the second integral in (18) similarly, we see that its part over I tends to 0 because u n → u and w n → u uniformly on compact intervals. Now we estimate the Following the same lines as in the proof of [26,Proposition 3.7] one can show that, under additional C 2 smoothness of the interaction potentials, any monotone nontrivial solution of (2) in the space X is strictly monotone provided its velocity profile decays exponentially fast at infinity.
Finally, we consider two examples to illustrate the results obtained above. where all β m ≥ 0 and at least one is positive. Then our results provide us the existence and nonlocal "periodic-to-solitary" convergence for both types of waves -nondecreasing and increasing.