Heteroclinic travelling waves for the lattice sine-Gordon equation with linear pair interaction

The existence of travelling heteroclinic waves for the sine-Gordon 
lattice is proved for a linear interaction of neighbouring atoms. The 
asymptotic states are chosen such that the action functional is 
finite. The proof relies on a suitable concentration-compactness 
argument, which can be shown to hold even though the associated 
functional has no sub-additive structure.

1. Introduction. We consider the lattice sine-Gordon equation q k (t) = V � (q k+1 (t) − q k (t)) − V � (q k (t) − q k−1 (t)) − K sin (q k (t)) , k ∈ Z, (1) with a constant K > 0. Equation (1) describes the evolution of an infinite chain of atoms with elastic nearest neighbour interaction and an on-site potential, according to Newton's law. The interaction potential V : R R takes as argument the discrete → strain, which is given by the difference of the positions of the atoms q k+1 (t) − q k (t).
R 2 (u � (τ )) 2 dτ minus the potential The action functional is the kinetic energy c R 2 2 0 (u(τ + 1) − u(τ )) 2 dτ , and on-site part, This specific choice of the on-site potential is made for c energy, consisting of interaction part, � � 2

CARL-FRIEDRICH KREINER AND JOHANNES ZIMMER
We are interested in heteroclinic waves (that is, waves that connect two different asymptotic states at ±∞) for supersonic velocities c > c 0 . Before stating the precise results, we give a brief overview of some related work.
(4) 0 The on-site potential energy can here be taken to be [K cos(u(τ )) − 1] dτ . Bates R and Zhang [2] consider homoclinic waves that have their asymptotic states in the maximum of the on-site potential. We study the analogous situation for heteroclinic waves. That is, we consider waves with asymptotic states in two different maxima of the on-site potential. For the choice −K(1 + cos(u(τ ))) made above for the on-site potential, this leads to the boundary conditions lim u(τ ) = −π and lim u(τ ) = +π.
The existence proof will rely on minimisation and a novel type of concentrationcompactness. The main difficulties are: Firstly, the action functional, which is to be minimised, is highly nonconvex due to the periodicity of the on-site potential. The second challenge is a lack of compactness due to the infinite domain R. We show in Section 4 that these difficulties can be overcome with a suitable variant of concentration-compactness [9]. This is not obvious, since the functional (3) is not subadditive. We show that a concentration-compactness result holds nevertheless. This argument relies on the fact that the lattice action functional (3) can be related to the Mortola-Modica functional [10], so that a crucial L ∞ -a-priori bound can be inferred. This connection to the Mortola-Modica functional is made explicit in Section 3. Concentration-compactness arguments for lattice models were introduced by Friesecke and Wattis [6] (see also, e.g., [1]).
These two difficulties, namely a highly nonconvex functional and lack of com pactness also persist for other boundary conditions, in particular lim u(τ ) = 0 and lim u(τ ) = 2π, τ →−∞ τ →+∞ that is, asymptotic states in the minima of the on-site potential (possibly to be understood in an averaged sense). These boundary conditions correspond to a moving dislocation in the Frenkel-Kontorova model [5]. The existence of periodic solutions and sliding solutions for the two-dimensional generalisation of the Frenkel-Kontorova model can be shown with topological and variational methods [4]. A survey over some related results can be found in the book by Pankov [11]. For the one-dimensional Frenkel-Kontorova model, there are existence results for hetero clinic waves with asymptotic states (6) for the special case of a piecewise quadratic on-site potential in the physics literature [8]. There, it is assumed that the solution satisfies the sign condition of the kind u(τ ) < π for τ < 0 and u(τ ) > π for τ > 0.
Under this assumption, the analogue of the Euler-Lagrange equation (2) for piece wise on-site potential simplifies to an equation with a nonlinearity that depends only on τ , rather than u(τ ). This simplified system is then solved by Fourier methods, where the solution is represented as a sum of Fourier components. The difficulty

TRAVELLING WAVES FOR THE LATTICE SINE-GORDON EQUATION
is to show that the solution satisfies the sign condition (7). Kresse and Truski novsky [8] observe that this condition probably does not hold for a specific interval of subsonic velocities. A rigorous proof that the sign condition holds in some regime seems, at the time of writing, only to be available for the Fermi-Pasta-Ulam chain with piecewise quadratic pair interaction [12]. The extension of this result to more general potentials is an open problem.
2. Main result. We set X := u ∈ H 1 (R) : u � ∈ L 2 (R) and remark that X is a loc Hilbert space when equipped with the inner product We are now in a position to formalise the connection of Equation (2) and the action functional J : X R ∪ {∞} given in (3).

Proof. Every critical point
This means that u is a weak solution of (2). Applying a classical bootstrap argu ment, we find that u ∈ C 2 (R) is a strong solution of (2).
Lemma 2.1 shows in particular that a minimiser of the variational problem minimise J, as defined in (3), on M −π,π ⊂ X is a solution to (2) with boundary conditions (5). We now formulate the existence result for (2), for sufficiently large supersonic wave speed and heteroclinic boundary conditions. Theorem 2.2. Let c 2 > 9 c 0 2 . Then there exists a minimiser u 0 of J on M −π,π ⊂ 8 X, that is, the variational problem (9) possesses a solution. This minimiser u 0 is a C 2 -function which satisfies (2) and the asymptotic boundary condition (5).
The proof of this Theorem will follow easily from Lemma 2.1 and the statements in Sections 3 and 4; it is given in Section 5.

3.
A-priori bound. For more a compact notation, we introduce on X a difference operator A as Au(z) this follows with Jensen's inequality and Fubini's theorem, [13]). This implies R for all u ∈ X. Modica and Mortola [10] have studied a very similar functional to those in this inequality. We quote a relevant result on the minimal values of such functionals from [3, Section 6.2].
Proof. The proof relies on the fact that (12) and (13) provide, loosely speaking, an estimate for the "cost" for u 0 to traverse a height of 2π from one minimum of cos( ) · to the next. More precisely, we have

TRAVELLING WAVES FOR THE LATTICE SINE-GORDON EQUATION
Hence, (13) Let T 1 , T 2 ∈ R ∪ {±∞} with T 1 < T 2 be such that u 0 (T 1 ) = νπ and u 0 (T 2 ) = (ν + 2)π for some odd integer ν. Then the contribution of the interval [T 1 , T 2 ] to the value I (c −c 0 ) (u 0 ) is, from (12), The boundary conditions (5) imply that the height 2π needs to be covered; any further increase in height of 2π has to be compensated by a decrease in height of 2π and vice versa. Hence there is an odd number of such increases or decreases. We write this odd number as 2κ+1 with κ ∈ N, so that (κ+1) 2π ≤ � {u 0 (z) : z ∈ R} � < · (κ + 2) 2π; thus κ can be understood as a lower bound on the number of times that · u 0 grows by full 2π in excess to the one time required by u 0 ∈ M −π,π . Then (14) and (16) show that On the other hand, J(u 0 ) is bounded by the Modica-Mortola bound (13). Therefore, using (15), note in particular that the inequality is strict.

4.
Concentration-compactness. The next step is to prove a variant of the con centration-compactness lemma of P.-L. Lions [9, Lemma I.1] that is adapted to our situation. The setting in this classical paper [9] (see also [6]) is as follows. The general problem is to minimise a functional E : U R on a Banach space U subject to a → constraint L(u) = λ > 0. It is shown that, for fixed λ, that any minimising sequence is, up to a subsequence, either relatively compact, or vanishes, or splits into two or more parts which drift away arbitrarily distant from each other. Vanishing can usually be excluded quite easily. Setting In comparison to the classical setting, a major difference in the present paper is that the constraint u(±∞) = ±π cannot be varied continuously. Hence it is impossible to consider the minimum value of the functional on level sets of the constraint as a function of a continuous parameter in the constraint. As a consequence of this, no meaningful analog to the above subadditivity inequality can be formulated. Instead, we will exclude splitting by means of the a priori bound from Lemma 3.2. The most important difference in contrast to other variants of the concentrationcompactness lemma is therefore in the alternative of splitting. The value of the functional J is split up between sequences (f n ) n∈N , (g n ) n∈N (whose sum is essen tially the original sequence (u n ) n∈N )-not the value of the constraint, as usual. On the other hand, the present lemma holds not just for minimising sequences (u n ) n∈N ⊂ M −π,π , but for all sequences for which the values of the functional converge.
� � The following proof will be formulated using symmetrised differences u τ + 2 1 � � − u τ − 2 1 , rather than u(τ + 1) − u(τ ), in order to exploit the symmetry of the in tegration domains. It is clear that J, and hence the minimisation problem, remains We point out that all integrals are taken over symmetric intervals around η which simplifies some estimates later in this proof. For use in Section 5, we mention that the second summand equals � shows that, roughly speaking, the second term could be interpreted as an integration over the same domain. This idea has already been suggested by (10). n→∞ possesses a subsequence, not relabelled and still denoted by (u n ) n∈N , which satisfies one of the following three alternatives: (i) Tightness: There is a sequence (η n ) n∈N ⊂ R such that, for every ε > 0, there exists T 0 > 0 such that for all T > T 0 J(u n ) − J T (u n ; η n ) < ε for every n ∈ N.
(ii) Vanishing: For all T > 0, n→∞ η∈R (iii) Splitting: There exists ε 1 > 0 such that for every 0 < ε < ε 1 , there are f n , g n ∈ X such that n→∞ n→∞ for some 0 < α, β < θ. (π is needed in the first inequality to ensure J (f n ) < ∞ and J (g n ) < ∞.) The condition (19) is in particular satisfied if (u n ) n∈N is a minimising sequence for J. We actually need Lemma 4.1 only for that case, in which θ = inf J(u)| .

M−π,π
Proof. The proof is given in four steps. First we introduce a concentration func tional, discuss its properties (Step 1). The rest is concerned with the proof that the only alternative to cases (i) and (ii) is case (iii).
Step 2 identifies the intervals which will become the support of f n and g n , respectively. Further estimates show the statements about the sequences f n (Step 3) and g n (Step 4).
Step 1. As in Lions' proof [9], a concentration function is introduced. Namely, given a sequence (u n ) n∈N ⊂ M −π,π with (19) and a parameter η ∈ R, define a sequence of functions P n ( · ; η) : (0, ∞) → R, the shift by 1 2 from the definition of J T saves some summands 1 2 in subsequent estimates of this proof while the form of J T is more useful in Section 5 below.
Note that, for every fixed n ∈ N and η ∈ R, P n is nondecreasing in T . Namely, for all ε > 0 and all η ∈ R, (10) and c > c 0 show that the increment of the second η+T +s η+T − +s − and the very same estimate holds on (η − T − ε, η − T ). This implies P n (T + ε; η) ≥ P n (T ; η) for all T, ε > 0 because 1 + cos (u n ) is always non-negative. Now we can define for each n ∈ N the concentration function Q n (T ) := sup P n (T ; η).
η∈R As supremum of monotone and nondecreasing functions, Q n enjoys the same prop erties. It is clear that Q n is bounded on (0, ∞) because, for each n ∈ N, lim Q n (T ) = J (u n ) .
Step 2. Let ε > 0. By definition of l in (24), there exists T 0 ∈ R such that Q (T 0 ) ≥ l − 1 ε. Since Q n (T ) Q(T ) as n → ∞ for almost every T , we may 3 → 2 assume, possibly after increasing T 0 , that Q n (T 0 ) → Q(T 0 ). Thus, Q n (T 0 ) ≥ l− 3 ε, if we consider only large enough n. The definition (23) of Q n implies that we can find η n ∈ R such that for all large enough n P n (T 0 ; η n ) ≥ l − ε.
We can also find a sequence (T n ) n∈N with T n → ∞ as n → ∞ (and in particular T n � T 0 for all n ∈ N) such that Q n (T n ) ≤ l + ε; this follows from the facts that Q n (T ) → Q(T ) as n → ∞ for almost every T , and that Q(T ) → l as T → ∞, see (24). Since Q n has been defined as supremum over P n in (23), the sequence (T n ) n∈N satisfies P n (T n ; η n ) ≤ l + ε. As P n is monotone and nondecreasing in T for each n ∈ N, Now we are going to analyse the behaviour of u n (τ ) for |τ − η n | ∈ [T 0 , T n ]; the goal is to show that there exist k ± ∈ Z such that n � � � � � � � u n (τ ) − 2k n + + 1 π � ≤ δ(ε) for τ ∈ η n + T 0 + 1 2 , η n + T n − 1 2 and To prevent a complicated labelling of axes, the situation has been sketched for η n = 0.

2
Step 3. Define The goal is now to show that, up to a subsequence, J (f n ) α ∈ (0, θ) for n → ∞. To do so, we are going to estimate, with l from (24), → P n (T ) = P � n T 0 + 5 2 (∞) = J (f n ) for all T > T 0 + 5 .