Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator

We consider the U(1)-invariant nonlinear Klein-Gordon equation in discrete space and discrete time, which is the discretization of the nonlinear continuous Klein-Gordon equation. To obtain this equation, we use the energy-conserving finite-difference scheme of Strauss-Vazquez. We prove that each finite energy solution converges as $t\to\pm\infty$ to the finite-dimensional set of all multifrequency solitary wave solutions with one, two, and four frequencies. The components of the solitary manifold corresponding to the solitary waves of the first two types are generically two-dimensional, while the component corresponding to the last type is generically four-dimensional. The attraction to the set of solitary waves is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent radiation. For the proof, we develop the well-posedness for the nonlinear wave equation in discrete space-time, apply the technique of quasimeasures, and also obtain the version of the Titchmarsh convolution theorem for distributions on the circle.


Introduction
In this paper we study the long-time asymptotics for dispersive Hamiltonian systems. The first results in this direction were obtained by Segal [Seg63a,Seg63b], Strauss [Str68], and Morawetz and Strauss [MS72], who considered the nonlinear scattering and local convergence to zero for finite energy solutions to nonlinear wave equations. Apparently, there can be no such convergence to zero when there are localized standing wave solutions; in the case of U(1)-invariant systems, these solutions are solitary waves of the form φ(x)e −iωt , with ω ∈ R and φ decaying at infinity (one could say, "nonlinear Schrödinger eigenstates"). In this case, one expects that generically any finite energy solution breaks into a superposition of outgoing solitary waves and radiation; the statement known as the Soliton Resolution Conjecture (see [Sof06,Tao07]). The Soliton Resolution Conjecture implies that any finite energy solution locally converges either to zero or to a solitary wave. Thus, for a U(1)-invariant dispersive Hamiltonian system, one expects that the weak attractor is formed by the set of all solitary waves. For a translation invariant system, this implies that the convergence to solitary waves is to take place -locally -in any inertial reference frame.
Existence of finite-dimensional attractors (formed by static stationary states) is extensively studied for dissipative systems, such as the Ginzburg-Landau, the Kuramoto-Sivashinsky, and the 2D forced Navier-Stokes equations, where the diffusive part of the equation damps higher frequencies and in some cases leads to existence of a finite-dimensional attractor [BV92,Tem97,CV02]. Existence of attractors for finite difference approximations of such dissipative systems, as well as the relation between the attractors of continuous systems and their approximations, was considered in [KK90,FT91,FJKT91].
We are interested in extending these results to the Hamiltonian systems, where the convergence to a certain attracting set (for both large positive and negative times) takes place not because of the dissipation, but instead due to the dispersion, and thus takes place "weakly", in the weighted norms, with perturbations dispersing because of the local energy decay. In [KK07], we considered a weak attractor of the U(1)-invariant nonlinear Klein-Gordon equation in one dimension, coupled to a nonlinear oscillator located at x = 0: t ψ(x, t) = ∆ψ(x, t) − m 2 ψ(x, t) − δ(x)p(|ψ(x, t)| 2 )ψ(x, t), x ∈ R, (1.1) where ψ(x, t) ∈ C and p(·) is a potential with real coefficients, with positive coefficient at the leading order term. We proved in [KK07] that the attractor of all finite energy solutions is formed by the set of all solitary waves, φ ω (x)e −iωt , with ω ∈ R and φ ω ∈ H 1 (R). The general strategy of the proof has been to consider the omega-limit trajectories and then to prove that each omega-limit trajectory has a point spectrum, and thus is a solitary wave.
In this paper, we extend this result to the finite difference approximation of the n-dimensional Klein-Gordon equation interacting with a nonlinear oscillator. Our intention was to show that in the discrete case, just as in the continuous one, the attractor is formed by the set of solitary waves. This turned out to be true, except that in the discrete case, besides usual one-frequency solitary waves, the set of solitary wave solutions may contain the two-and four-frequency components. This is in agreement with our version of the Titchmarsh convolution theorem for distributions on the circle, which we needed to develop to complete the argument. These multifrequency solitary waves disappear in the continuous limit. To our knowledge, this is the first result on the weak attraction for the Hamiltonian model on discrete space-time.
The discretized models are widely studied in applied mathematics and in theoretical physics, in part due to atoms in a crystal forming a lattice, in part due to some of these models (such as the Ising model) being exactly solvable. Moreover, it is the discretized model that is used in numerical simulations of the continuous Klein-Gordon equation. The ground for considering the energy-conserving difference schemes for the nonlinear Klein-Gordon equations and nonlinear wave equations was set by Strauss and Vazquez in [SV78]. The importance of having conserved quantities in the numerical scheme was illustrated by noticing that instability occurs for the finite-difference schemes which do not conserve the energy [JV90]. Let us mention that our approach is also applicable to other energy-conserving finitedifference schemes so long as there are a priori bounds on the norm of the solution. Such schemes have been constructed in [LVQ95,Fur01,CJ10].
Our approach relies on the well-posedness results and the a priori estimates for the Strauss-Vazquez finite-difference scheme which we developed in [CK11]. While the discrete energy for the Strauss-Vazquez scheme given in [SV78] contained quadratic terms which in general are not positive-definite, we have shown [CK11] that the conserved discrete energy is positive-definite under the condition on the grid ratio, where ε is the space step with respect to each component of x ∈ R n and τ is the time step; the Strauss-Vazquez finite-difference scheme with the grid ratio τ /ε = 1/ √ n also preserves the discrete charge. The positivedefiniteness of the conserved energy provides one with the a priori energy estimates and results in the stability of the finite-difference scheme. (The relation (1.2) agrees with the stability criterion in [Vir86].) While the charge conservation does not seem to be particularly important on its own, it could be considered as an indication that the U(1)-invariance of the continuous equation is in a certain sense compatible with the chosen discretization procedure. See the discussion in [LVQ95, Section 1]. We reproduce our results on the well-posedness for the Strauss-Vazquez finite-difference scheme in Appendix A.
There is another important feature of our approach to the finite difference equation, compared to the approach which we developed in [KK07,KK08,KK10] for the continuous case. In the discrete case, the spectral gap, where the frequencies of the solitons are located and where, as it turns out, the spectrum of the omega-limit trajectory could be located, consists of two open neighborhoods of the circle. This does not allow us to apply the Titchmarsh convolution theorem in a direct form as in [KK07,KK08,KK10]. To circumvent this problem, we derive a version of the Titchmarsh convolution theorem for distributions on the circle; see Appendix B. This version of the Titchmarsh convolution theorem does not allow one to reduce the spectrum of omega-limit trajectories to a single point; we end up with the spectrum consisting of one, two, and four frequencies. Indeed such omega-limit trajectories exist; we explicitly construct solitary waves with one, two, and four frequencies.
Here is the plan of the paper. In Section 2, we describe the model and state the main results. In Section 3, we introduce the omega-limit trajectories and describe the proof of the main result: the convergence of any finite energy solution to the set of solitary waves. The main idea is that such a convergence is equivalent to showing that each omega-limit trajectory itself is a solitary wave. In Section 4, we separate the dispersive part of the solution, and consider the regularity of the remaining part in Section 5. In Section 6 we obtain the spectral relation satisfied by the omega-limit trajectory. For this, we use the technique of quasimeasures, which we borrow from [KK07]. In Section 7 we apply to the spectral relation our version of the Titchmarsh convolution theorem on the circle, proving that the spectrum of any omega-limit trajectory consists of finitely many frequencies. This completes the proof that each omega-limit trajectory is a (multifrequency) solitary wave. We give an explicit construction of multifrequency solitary waves in Section 8. Appendix A gives the well-posedness for the finite difference scheme approximation. The versions of the Titchmarsh convolution theorem for distributions on the circle are stated and proved in Appendix B.
ACKNOWLEDGMENTS. The author is grateful to Alexander Komech for the suggestion to consider the discrete analog of the Klein-Gordon equation, to Evgeny Gorin for interesting discussions, to Juliette Chabassier and Patrick Joly for the references and the preprint of their paper [CJ10]. Special thanks to the anonymous referee for pointing out the misprints.

Definitions and main results
In [KK07], we considered the weak attractor of the U(1)-invariant nonlinear Klein-Gordon equation in one dimension, coupled to a nonlinear oscillator located at x = 0: where ψ(x, t) ∈ C and W (·) is a real-valued polynomial which represents the potential energy of the oscillator: Equation (2.1) is a Hamiltonian system with the Hamiltonian In this paper, we will consider the discrete version of equation (2.1). We pick ε > 0 and τ > 0 and substitute the continuous variables (x, t) ∈ R n × R by where the nonlinear term is given by In (2.3), we used the notations with e 1 = (1, 0, 0, 0, . . . ) ∈ Z n , e 2 = (0, 1, 0, 0, . . . ) ∈ Z n , etc. (2.6) Remark 2.2. By the little Bézout theorem, We introduce the phase space We will denote by l(Z, X ) the space of functions of T ∈ Z with values in X .
Definition 2.5 (Discrete energy). The energy of the function ψ ∈ l(Z, X ) at the moment T ∈ Z is (2.8) Remark 2.6. In the case n = 1 the continuous limit of the energy E in (2.8) coincides with the classical energy functional of the Klein-Gordon equation interacting with an oscillator described by the potential W ; see (2.2).
We consider the Cauchy problem where f T is defined by (2.4).
(ii) The value of the energy functional is conserved: (iii) ψ T X satisfies the a priori estimate The main part of this theorem (all the statements but the last one) is proved in [CK11]; we reproduce this proof in Appendix A (the existence, uniqueness, and continuous dependence on continuous data are proved in Theorems A.1, A.2, and A.4, and the a priori estimates are proved in Theorem A.8). Let us mention that the bound (2.10) follows from the energy conservation since the first term in (2.8) is nonnegative, while inf λ≥0 W (λ) > −∞ due to Assumption 2.4.
Let us sketch the proof of the weak continuity of U (T ), which is the last statement of the theorem. Let Ψ j ∈ X , j ∈ N, be a sequence in X weakly convergent to some Ψ ∈ X . By the Banach-Steinhaus theorem, Ψ j , j ∈ N, are uniformly bounded in X ; so are U (T )(Ψ j ). Now one can see that the weak convergence of U (T )(Ψ j ) to U (T )(Ψ) in X follows from the continuity of U (T ) in X , from the finite speed of propagation (the value ψ T X only depends on the initial data (u 0 Y , u 1 Y ) in the ball |Y | ≤ |X| + |T |), and from the convergence Ψ j → Ψ in the topology of l 2 (B) for any bounded set B ⊂ Z n .
We will use the standard notation T := R mod 2π.
(2.11) (ii) The solitary manifold is the set where ψ T are solitary wave solutions to (2.3) of the form (2.11).
This assumption is needed so that the Titchmarsh convolution theorem for distributions on the circle (see Theorem 7.4 below) will be applicable for the analysis of omega-limit trajectories. One can see from (2.13) that for any fixed m > 0 Assumption 2.9 is satisfied as long as the time step τ > 0 is sufficiently small.
Our main result is that the weak attractor of all finite energy solutions to (2.3) coincides with the solitary manifold S. Theorem 2.10 (Solitary manifold as the weak attractor). Assume that τ ∈ (0, τ 0 ), with τ 0 > 0 as in Theorem 2.7. Let Assumptions 2.4 and 2.9 be satisfied. Then: (i) For any initial data (u 0 , u 1 ) ∈ X the solution to the Cauchy problem (2.9) weakly converges to S as T → ±∞.
with φ ∈ l 2 (Z n ). The corresponding component of the solitary manifold is generically two-dimensional.
Definition 2.8 and Theorem 2.10 show that the set S satisfies the following two properties: (i) It is invariant under the evolution described by equation (2.3).
(ii) It is the smallest set to which all finite energy solutions converge weakly.
It follows that S is the weak attractor of all finite energy solutions to (2.3).
Remark 2.11. The convergence of any finite energy solution to S, stated in Theorem 2.10, also holds in certain normed spaces. For example, let s > 0; for u ∈ l 2 (Z n ), denote u l 2 −s = X∈Z n |u X | 2 (1 + X 2 ) −s 1/2 , and for (u, v) ∈ X , denote (u, v) X−s = ( u l 2 −s + v l 2 −s ) 1/2 . Then, for any finite energy solution, one has with the initial data at an omega-limit point of ψ T X : Lemma 3.2. If (z 0 , z 1 ) = w-lim Tj →∞ (ψ Tj , ψ Tj +1 ) (weakly in X ) is an omega-limit point of ψ ∈ l(Z, l 2 (Z n )) and if ζ ∈ l(Z, l 2 (Z n )) is the omega-limit trajectory with then ψ Tj +T → ζ T , weakly in l 2 (Z n ), and in particular there is the convergence Proof. This immediately follows from the weak continuity of U (T ) stated in Theorem 2.7.
We will deduce Theorem 2.10 from the following proposition.
Proof of Theorem 2.10. Let T j ∈ N, j ∈ N, be a sequence such that T j → +∞. By the Banach-Alaoglu theorem, a priori bounds on (ψ T , ψ T +1 ) X stated in Theorem 2.7 allow us to choose a subsequence {T jr : r ∈ N} such that weakly in X . (3.4) Let ζ ∈ l(Z, X ) be the corresponding omega-limit trajectory, that is, the solution to the Cauchy problem (2.9) with the initial data (ζ T , ζ T +1 )| T =0 = (z 0 , z 1 ). By Proposition 3.3, ζ T is a solitary wave. It follows that (z 0 , z 1 ) = (ζ T , ζ T +1 )| T =0 ∈ S. Thus, the first two statements of Theorem 2.10 follow from Proposition 3.3.
Let us prove the last statement of Theorem 2.10, namely, that the set of all solitary waves only consists of one-, two-, and four-frequency solitary waves. It will follow from Proposition 3.3 if we can show that each solitary wave solution is itself (its own) omega-limit trajectory, has to be of the form specified by Proposition 3.3.
so that ψ T is the omega-limit trajectory of itself.
Proof. Pick any sequence T j ∈ N, T j → ∞ such that ω 1 T j → 0 ∈ T as j → ∞. Then either {ω 2 T j : j ∈ N} is dense in T, or it is a subset of { kπ q ∈ T : k ∈ Z 2q }, for some q ∈ N. In the former case, we take a subsequence T ′ j of T j such that ω 2 T ′ j → 0 in T; In the latter case, we consider a new sequence, T ′ j = qT j , so that ω 2 T ′ j = 0 (and we still have Repeating this process, we end up with a sequence such that Hence, ψ T itself is an omega-limit trajectory of a finite energy solution, and has to be of one of the three types mentioned in Proposition 3.3.
The dimension of the components of the solitary manifold corresponding to one-, two-, and four-frequency solitary waves are computed in Lemma 8.1, Lemma 8.2, and Lemma 8.3 below.
This finishes the proof of Theorem 2.10.
It remains to prove Proposition 3.3, which is the contents of in the remaining part of the paper. We will prove it analyzing the spectrum of omega-limit trajectories. Everywhere below, we suppose that the conditions of Proposition 3.3 (that is, conditions of Theorem 2.10) hold.

Separation of the dispersive component
We rewrite (2.9) as a linear nonhomogeneous equation where f T is given by (2.4). Let a(ξ, ω) be the symbol of the operator A in the left-hand side of (4.1): For a fixed value of ω ∈ T, the dispersion relation admits a solution ξ ∈ T n if and only if | cos ω| ≤ 1 + τ 2 m 2 2 −1 , or, equivalently, when ω belongs to the continuous spectrum Ω c of the linear discrete equation (4.1), which is given by the spectral gaps Ω 0 and Ω π have been defined in (2.14).
Note that due to the factor 1 n in (4.1) the continuous spectrum does not depend on the dimension n.

Regularity on the continuous spectrum
Now we consider the equation on ϕ T ; see (4.6). Let us recall that f T is defined by (2.4) for T ∈ Z, but is only considered in (4.6) for T ≥ 1. The function ϕ T X is defined by (4.6) for T ≥ 0 (with ϕ 0 X = ϕ 1 X = 0). We extend f T and ϕ T by zeros for T ≤ 0, so that Then equation (4.6) is satisfied for all T ∈ Z: We introduce the Fourier transforms Since in the above relations the summation is over T ∈ N, we can extend (5.3)-(5.5) to the upper half-plane as analytic functions of ω ∈ C + . Then equation (4.6) yields where a(ξ, ω) is defined by extending (4.2) to ω ∈ C: a(ξ, ω) := (2 + τ 2 m 2 ) cos ω − 2 n n j=1 cos ξ j , ξ ∈ T n , ω ∈ C. (5.7) Lemma 5.1. For ξ ∈ T n , ω ∈ C \ Ω c , one has a(ξ, ω) = 0.
By Lemma 5.1, for ω away from Ω c , equation (5.6) yieldŝ For ω ∈ (C + mod 2π) \ Ω c , since inf ξ∈T n |a(ξ, ω)| = min ξ∈T n |a(ξ, ω)| > 0, the operator of multiplication by 1/a(ξ, ω) is a bounded linear operator from L 2 (T n ) to L 2 (T n ). In the coordinate representation, (5.8) can be written as where R(ω) is a bounded linear operator with the Fourier transform F : l 2 (Z n ) → L 2 (T n ) and its inverse given by The expression in the right-hand side of (5.9) can be written as where X ∈ Z n , ω ∈ (C + mod 2π) \ Ω c , and G X (ω) stands for the fundamental solution, which is the inverse Fourier transform of 1/a(ξ, ω): where ω ∈ (C + mod 2π) \ Ω c . Let us study properties of G X (ω). We start by introducing the set of the singular points in the continuous spectrum Ω c : These frequencies correspond to the critical points of the symbol a(ξ, ω), that is, for ω ∈ Σ, there is ξ ∈ T n such that both a(ξ, ω) = 0 and ∇ ξ a(ξ, ω) = 0. (The relation ∇ ξ a(ξ, ω) = 0 implies that cos ξ j = ±1 for all 1 ≤ j ≤ n; the value of l in (5.13) is the number of cosines in the denominator of (5.12) which are equal to −1.) Note that in the one-dimensional case one has Σ = ∂Ω c = {±ω m ; π ± ω m }, with ω m = arccos 1 + τ 2 m 2 2 −1 .

Lemma 5.2.
(i) For each fixed value of X ∈ Z n , the function G X (ω) is analytic in ω ∈ (C + mod 2π) \ Ω c and admits the trace which is a smooth function of ω ∈ T \ Σ.
(iii) (a) If n ≤ 4, G 0 (ω) is a monotonically increasing function on the interval 0 ≤ ω < ω m , where it is strictly positive, and on the interval π − ω m < ω ≤ π where it is strictly negative.
for all ǫ ∈ (0, 1), ω ∈ I. 2. For X ∈ Z n , ω ∈ Ω 0 ∪ Ω π , we have: 3. The monotonicity of G 0 (ω) for ω ∈ T \ Ω c immediately follows from the definition (5.12). For n ≥ 5, one notices that G 0 (ω) remains finite at ω = ±ω m and ω = π ± ω m 4. To prove that G X (ω + i0) is a multiplier in the space of distributions, it suffices to notice that for each X ∈ Z n the trace G X (ω + i0) is a smooth function of ω ∈ T \ Σ. 5. Using (5.12) and the Plancherel theorem, we compute: Note that since ω ∈ I ⊂ T\ (Ω 0 ∪Ω π ), one has 1 + τ 2 m 2 2 | cos ω| < 1, therefore Γ ω is a nonempty submanifold of T n of codimension one, piecewise smooth away from the discrete set {0; π} n ⊂ T n (on this set, one could have simultaneously a(ξ, ω) = 0, ∇ ξ a(ξ, ω) = 0; in fact, Γ ω does not contain these points for ω ∈ T \ Σ). It follows that |U ǫ (Γ ω )| = O(ǫ). Moreover, since the hypersurface Γ ω has strictly positive area for ω ∈ I, there is v I > 0 (dependent on I but not on ω) so that One can see that for all ξ ∈ U ǫ (Γ ω ) and all ǫ ∈ (0, 1) the denominator in the integral in the right-hand side of (5.17) is bounded from above by k I ǫ 2 , for some k I > 0 which depends on I but not on ω ∈ I. Thus, there is c I > 0 such that Proof. We will prove this lemma consideringf (ω) for ω ∈ C + and then taking the limit Im ω → 0. Since f T = 0 for T ≤ 0 (Cf. (5.1)) and f T is bounded due to (2.10), the Fourier transform (5.5) extended to ω ∈ C + , with the convergence in the sense of distributions. We need to show that for any closed subset which is an analytic function valued in l 2 (Z n ). Its limit as Im ω → 0+ exists as an element of D ′ (T, l 2 (Z n )). Due to equation (5.2), the complex Fourier transforms of ϕ T X and of f T are related by Using (5.22) and the Plancherel theorem, we see that due to the a priori estimates on ψ and χ (see (2.10) and (4.7)). Combining the bound (5.23) with the bound on G(ω + iǫ) l 2 (Z n ) obtained in Lemma 5.2 (Cf. (5.16)), we conclude that there is C I < ∞ such that (5.20) holds. This completes the proof of the lemma.

Spectral relation
Let ζ ∈ l(Z, l 2 (Z n )) be the omega-limit trajectory of the solution ψ ∈ l(Z, l 2 (Z n )) to the Cauchy problem (2.9), in the sense of Definition 3.1. That is, we assume that ζ is a solution to (3.1) and that there is a sequence Let us expressζ X (ω) in terms ofg(ω); this representation will allow us to express ζ T X in equation (3.1) via ζ T 0 . By the definition of omega-limit trajectoryζ X (ω) (see Definition 3.1), its Fourier transform in T satisfies the stationary Helmholtz-type equation (6.1) Lemma 6.1. The solution to the stationary problem (6.1) satisfies where Σ is the set of singular points defined in (5.13).
Proof. By (6.1), the Fourier transform of ζ in X and T , where ξ ∈ T n and ω ∈ T. By (2.4), (3.2), and (3.3), Moreover, by Theorem 2.7, the solutions ψ T X , χ T X (Cf. (4.5)), and hence their difference ϕ T X = ψ T X − χ T X are bounded uniformly in T and X. Therefore, similarly to how we arrived at (6.5), Now the proof follows from taking the limit Im ω → 0+ in (5.22) and using (6.5) and (6.7), and also taking into account that for each X ∈ Z n , G X (ω) is a multiplier in D ′ (T \ Σ) by Lemma 5.2.
Let us show that the spectra of ζ T 0 and g T are located inside the closures of the spectral gaps.
(i) The space l ∞ F (Z) is the vector space l ∞ (Z) endowed with the following convergence: (ii) The space of quasimeasures is the vector space of distributions with bounded Fourier transform, endowed with the following convergence: For example, any function from L 1 (T) is a quasimeasure, and so is any finite Borel measure on T. (ii) LetM ǫ ∈ l 1 (Z) be bounded uniformly for ǫ > 0. If Proof. We define M (ω)q(ω) := F [ M * q (T )](ω) that agrees with the case q ∈ C(T). The statement (1) follows from (2) with M ǫ = M and q ǫ ∈ C(T). To prove (2), by Definition 6.5, we need to show that Define the functions By Definition 6.5, to prove the convergence (6.11), we need to show that and that for any T 1 ∈ N one has lim To prove (6.12), we write: which is bounded uniformly in ǫ > 0. It remains to prove (6.13). We need to show that, given T 1 ∈ N, for any δ > 0 there is ǫ δ > 0 such that for any ǫ ∈ (0, ǫ δ ) one has sup T |f ǫ (T ) − f (T )| < δ. We have: (6.14) The first term in the right-hand side of (6.14) converges to zero uniformly in T ∈ Z sinceM ǫ −M → 0 in l 1 (Z) whilě q ǫ ∈ l ∞ (Z) are bounded uniformly for ǫ > 0. If ǫ δ > 0 is small enough and ǫ ∈ (0, ǫ δ ), then the first term in the right-hand side of (6.14) is smaller than δ/3. We break the second term in the right-hand side of (6.14) into where T δ ∈ N is chosen as follows: SinceM ∈ l 1 (Z), whileq ǫ −q is bounded in l ∞ (Z) uniformly in ǫ > 0, there exists T δ ∈ N so that On the other hand, since q ǫ → q in Q(T), one has so that, choosing ǫ δ > 0 smaller than necessary, we make sure that the second term in (6.15) is also smaller than δ/3 for ǫ ∈ (0, ǫ δ ), and therefore (6.14) is bounded by δ. Thus, as ǫ → 0+, (6.14) converges to zero uniformly in |T | ≤ T 1 , proving (6.13). The convergence (6.11) follows.
Lemma 6.7. For n ≥ 1, the function , ω ∈ T, (6.18) is continuous and real-valued for ω ∈ Ω 0 ∪ Ω π and satisfies It is a multiplier in the space of quasimeasures Q(T \ Σ ′ ), and moreover for any ρ ∈ C ∞ (T) with support away from Σ ′ one has (6.20) Remark 6.8. Due to Lemma 6.6, one concludes that Lemma 6.7 implies that, as ω → 0+, the ratio 1 G0(ω+iǫ) converges to r(ω) in the space of multipliers which act on quasimeasures with support in T \ Σ ′ .
Since G 0 (ω) is a smooth function of ω ∈ (C + mod 2π) \ Σ ′ , it is enough to check the convergence (6.20) for the Fourier transform We have: In the case n ≥ 3, the convergence of (6.21) in l 1 as ǫ → 0 is straightforward since at the points ω ∈ ∂Ω c the function G 0 (ω + iǫ) has a nonzero limit as ǫ → 0.
We define the "sharp" operation ♯ on D ′ (T) by (6.28) Remark 6.10. For F ∈ l(Z), one has (F ) ♯ =F . Now we will use the spectral representation stated in Lemma 6.9 to obtain the following fundamental relation satisfied by ζ 0 , which we call the spectral relation. In the next section we will apply our version of the Titchmarsh convolution theorem to this relation, proving that the ω-support ofζ consists of one, two, or four frequencies.

Nonlinear spectral analysis of omega-limit trajectories
We are going to prove our main result, reducing the spectrum of ζ T 0 to at most four points. Denote We use the following version of the Titchmarsh convolution theorem on the circle.
Remark 7.9. The above lemma is similar to Theorem B.4 in Appendix B.
If, instead, (7.12) is satisfied, we are led to the conclusion suppζ 0 ⊂ {a; b; π + a; π + b}, (7.16) and moreoverζ This again puts us in the framework of the third possibility stated in the lemma.
Lemma 7.10. ζ T is a multifrequency solitary wave with one, two, or four frequencies.
Note that in the case n ≥ 5, when one could have ω j ∈ ∂Ω c for some 1 ≤ j ≤ 4, the derivatives of δ(ω − ω j ) do not appear in (7.20) due to the uniform l 2 (Z n )-bounds on ζ T . By Lemma 7.10, we know that the set of all omega-limit trajectories consists of multifrequency waves. In Section 8, we will check that the two-and four-frequency solitary waves have the form specified in Proposition 3.3; see Lemma 8.2 and Lemma 8.3 below. This will finish the proof of Proposition 3.3.
Let us complete this section with a simple derivation of the form of four-frequency solitary waves in dimensions n ≤ 4.
Lemma 7.11. Let n ≤ 4. Each four-frequency solitary wave can be represented in the form with φ, θ ∈ l 2 (Z n ).
Proof. Recall that, by Lemma 3.4, each multifrequency solitary wave is its own omega-limit trajectory; therefore, it is enough to prove that each four-frequency omega-limit trajectory has the form (7.21).
Since suppζ X ⊂ Ω 0 ∪ Ω π by Lemma 7.10, the relation (7.19) could be extended to ω ∈ T. To have a four-frequency omega-limit trajectory,ζ 0 is to be given by (7.15). Then (7.19) yields where we took into account that, by Lemma 5.2 (Cf. (5.15)), It follows that finishing the proof.

Analysis of solitary wave solutions
Here we discuss in more detail one-, two-, and four-frequency solitary waves, prove that they have the form specified in Proposition 3.3, and construct particular examples.
(ii) For a particular value ω ∈ Ω 0 ∪ Ω π (ω ∈ Ω 0 ∪ Ω π if n ≥ 5), there is a nonzero solitary wave if and only if (iv) In the case n = 1, the necessary and sufficient criterion for the existence of nonzero solitary waves is Proof. Let us substitute the Ansatz ψ T X = φ X e −iωT , ω ∈ T, into (2.3). Using the relations (2.4)), we see that φ X satisfies Equivalently, the Fourier transformφ(ξ) = X∈Z n φ X e −iξ·X is to satisfy with C ∈ C. By Lemma 4.1, φ is of finite l 2 -norm if and only if ω ∈ Ω 0 ∪ Ω π for n ≤ 4, and ω ∈ Ω 0 ∪ Ω π for n ≥ 5. This proves the first statement of the lemma.
Substituting (8.4) into (8.3), we see that C = 0 is to satisfy the equation For each ω ∈ Ω 0 ∪ Ω π , the set of solutions C to equation (8.5) (if it is nonempty) admits the representation C = ae is with a > 0 and s ∈ T; for each particular ω, the set of values a is discrete (under the assumption that W (λ) is a polynomial of degree larger than 1). The solitary manifold can be locally parametrized by two parameters, a > 0 and s ∈ T, proving the third statement of the lemma.

Two-frequency solitary waves
Let us study two-frequency solitary wave solutions. By Lemma 7.8, the two frequencies of a two-frequency solitary wave differ by π, hence we need to consider solitary wave solutions of the form with p, q ∈ l 2 (Z n ) not identically zero. We have: We can write Taking into account that we see that the Fourier transform of ψ T X (with respect to both time and space variables) satisfies a(ξ, ω) pδ ω1 +qδ ω1+π = −2πτ 2 M δ 0 + N δ π * (p 0 δ ω1 − q 0 δ ω1+π ) cos ω 1 .
To prove the second statement of the lemma, we notice that, by (8.9), In the last equality, we took into account Lemma 5.2 (Cf. (5.15)). This finishes the proof.
Proof. As follows from the above discussion, once we have one solitary wave of this type, we can vary ω 1 and ω 2 . Then The second statement of the lemma follows from (8.13) after noticing that, by (8.21), one has q X = (−1) Λ·X p X , s X = −(−1) Λ·X r X .
We finished studying the structure of multifrequency solitary wave solutions. Now the proof of Proposition 3.3 is complete.

A.1 Continuous case
Let us first consider the U(1)-invariant nonlinear wave equation where ψ(x, t) ∈ C and v(x, λ) is such that v ∈ C(R n × R) and v(x, ·) ∈ C 2 (R) for each x ∈ R n . Equation (A.1) can be written in the Hamiltonian form, with the Hamiltonian The value of the Hamiltonian functional E and the value of the charge functional are formally conserved for solutions to (A.1). A particular case of (A.1) is the nonlinear Klein-Gordon equation, with v(x, λ) = m 2 2 λ + z(x, λ), with m > 0: If z(x, λ) ≥ 0 for all x ∈ R n , λ ≥ 0, then the conservation of the energy yields an a priori estimate on the norm of the solution:

A.2 Finite difference approximation
Let us now describe the discretized equation. Let (X, T ) ∈ Z n × Z denote a point of the space-time lattice. We will always indicate the temporal dependence by superscripts and the spatial dependence by subscripts. Fix ε > 0, and let V X (λ) = v(εX, λ) be a function on Z n × R, so that V X ∈ C 2 (R) for each X ∈ Z n . For λ, µ ∈ R and X ∈ Z n , we introduce We consider the Vazquez-Strauss finite-difference scheme for (A.1) [SV78]: where ψ T X ∈ C is defined on the lattice (X, T ) ∈ Z n × Z. Above, e 1 = (1, 0, 0, 0, . . . ) ∈ Z n , e 2 = (0, 1, 0, 0, . . . ) ∈ Z n , etc. (A.8) The continuous limit of (A.7) is given by (A.1), with εX corresponding to x ∈ R n and τ T corresponding to t ∈ R. Since ∂ λ V X (λ) = B X (λ, λ), the continuous limit of the last term in the right-hand side of (A.7) coincides with the right-hand side in (A.1).
An advantage of the Strauss-Vazquez finite-difference scheme (A.7) over other energy-preserving schemes discussed in [LVQ95,Fur01] is that it is explicit: at the moment T + 1 the relation (A.7) only involves the function ψ at the point X, allowing for a simple realization of the solution algorithm even in higher dimensional case.

A.3 Well-posedness
We will denote by ψ T the function ψ defined on the lattice (X, T ) ∈ Z n × Z at the moment T ∈ Z.
Note that we do not claim in this theorem that ψ T l 2 (Z n ) is uniformly bounded for all T ∈ Z. For the a priori estimates on ψ T l 2 (Z n ) , see Theorem A.8 below. One can readily check that any X-independent polynomial potential of the form satisfies (A.9). Note that since lim λ→+∞ V (λ) = +∞, this potential is confining.
Theorem A.2 (Uniqueness and continuous dependence on the initial data). Assume that the functions are bounded from below: Define Let τ ∈ (0, τ 2 ) and ε > 0.
(i) There exists a solution to the Cauchy problem for equation (A.7) with arbitrary initial data (ψ 0 , ψ 1 ), and this solution is unique.
(ii) For any T > 0, the map , the values of k 1 and k 2 from Theorem A.1 and Theorem A.2, whether k 2 > −∞, are related by k 2 ≤ k 1 , and then the values of τ 1 and τ 2 from these theorems are related by τ 2 ≤ τ 1 .
Theorem A.4 (Existence and uniqueness for polynomial nonlinearities).
(ii) Assume that where C X,q ≥ 0 for X ∈ Z n and 1 ≤ q ≤ 4, and C X,0 are uniformly bounded from below: Then for any τ ∈ (0, τ 3 ) and any ε > 0 there exists a solution to the Cauchy problem for equation (A.7) with arbitrary initial data (ψ 0 , ψ 1 ), and this solution is unique.
Thus, even though the potential (A.10) satisfies conditions (A.9) and (A.11) in Theorem A.1 and Theorem A.2, the corresponding values τ 1 and τ 2 could be hard to specify explicitly. Yet, the second part of Theorem A.4 gives a simple description of a class of X-dependent polynomials V X (λ) for which the range of admissible τ > 0 can be readily specified.
We will prove existence and uniqueness results stated in Theorems A.1, A.2, and A.4 in Appendix A.7.

A.4 Energy conservation
Theorem A.5 (Energy conservation). Let ψ be a solution to equation (A.7) such that ψ T ∈ l 2 (Z n ) for all T ∈ Z. Then the discrete energy is conserved.
Remark A.6. The discrete energy is positive-definite if the grid ratio satisfies Remark A.7. If ψ 0 and ψ 1 ∈ l 2 (Z n ), then, by Theorem A.1, one also has ψ T ∈ l 2 (Z n ) for all T ∈ Z as long as Proof. For any u, v ∈ C, there is the identity The presence of the second term which is not positive-definite deprives one of the a priori l 2 bound on ψ, such as the one stated in Theorem A.8. In view of this, the Strauss-Vazquez finite-difference scheme for the nonlinear Klein-Gordon equation is not unconditionally stable. Other schemes (conditionally and unconditionally stable) were proposed in [LVQ95,Fur01]. Now, due to the a priori bound (A.21), we deduce that, as the matter of fact, the Strauss-Vazquez scheme is stable in n dimensions under the condition that the grid ratio is τ /ε ≤ 1/ √ n. Note that in the case ψ ∈ R, the Strauss-Vazquez energy (A.22) agrees with the energy defined in (A.14).
The left-hand side takes the form which is clearly strictly positive for all 0 ≤ z < 1 and p ≥ 0, proving (A.41).
This finishes the proof of the first part of Theorem A.4; now we turn to the second part.
By (A.46) and Lemma A.16, condition (A.42) is satisfied. Since C X,q ≥ 0 for 1 ≤ q ≤ 4, each term C X,q b q (λ, µ) satisfies condition (A.43). Therefore, by Lemma A.15, there is a unique solution ψ T X to the Cauchy problem for equation (A.7). This finishes the proof of Theorem A.4.

B Titchmarsh convolution theorem for distributions on the circle
The Titchmarsh convolution theorem [Tit26] states that for any two compactly supported distributions f, g ∈ E ′ (R), inf supp f * g = inf supp f + inf supp g, sup supp f * g = sup supp f + sup supp g. (B.1) The higher-dimensional reformulation by Lions [Lio51] states that for f, g ∈ E ′ (R n ), the convex hull of the support of f * g is equal to the sum of convex hulls of supports of f and g.  ). Here we give a version of the Titchmarsh Theorem which is valid for distributions supported in n > 1 small intervals of the circle T = R mod 2π. For brevity, we only give the result for distributions supported in two small intervals, which suffices for the applications in this paper; a more general version is proved in [KK12]. First, we note that there are zero divisors with respect to the convolution on the circle. Indeed, for any two distributions f , g ∈ E ′ (T) one has (f + S π f ) * (g − S π g) = f * g + S π (f * g) − S π (f * g) − f * g = 0. (B.2) Above, S y , y ∈ T, is the shift operator, defined on E ′ (T) by where the above relation is understood in the sense of distributions. Yet, the cases when the Titchmarsh convolution theorem "does not hold" (in a certain naïve form) could be specified. This leads to a version of the Titchmarsh convolution theorem for distributions on the circle (Theorem B.1 below).
Let us start with the following problem which illustrates our methods.
Theorem B.1 (Titchmarsh theorem for distributions on the circle). Let f, g ∈ E ′ (T). Let I, J ⊂ T be two closed intervals such that supp f ⊂ R 2 (I), supp g ⊂ R 2 (J), and assume that there is no closed interval I ′ I such that supp f ⊂ R 2 (I ′ ) and no closed interval J ′ J such that supp g ⊂ R 2 (J ′ ).
Assume that |I| + |J| < π. Remark B.2. While E ′ (T) = D ′ (T), we use the notation E ′ (T) for the consistency with the requirements of the standard Titchmarsh convolution theorem (B.1).