Breather continuation from infinity in nonlinear oscillator chains

Existence of large-amplitude time-periodic breathers localized near a single site is proved for the discrete Klein--Gordon equation, in the case when the derivative of the on-site potential has a compact support. Breathers are obtained at small coupling between oscillators and under nonresonance conditions. Our method is different from the classical anti-continuum limit developed by MacKay and Aubry, and yields in general branches of breather solutions that cannot be captured with this approach. When the coupling constant goes to zero, the amplitude and period of oscillations at the excited site go to infinity. Our method is based on near-identity transformations, analysis of singular limits in nonlinear oscillator equations, and fixed-point arguments.


Introduction
Recent studies of spatially localized and time-periodic oscillations (breathers) in lattice models of DNA [16,7] call for systematic analysis of such excitations in the discrete Klein-Gordon equation where γ > 0 is a coupling constant, V : R → R is a nonlinear potential, and x(t) = {x n (t)} n∈Z is a sequence of real-valued amplitudes at time t ∈ R.
In the classical Peyrard-Bishop model for DNA [17], V is a Morse potential having a global minimum at x = 0, confining as x → −∞ and saturating at a constant level as x → ∞. However, recent studies [20,15,16] argued that the Morse potential should be replaced by a potential with a local maximum at x = a 0 > 0, which induces a double-well structure, where one of the wells extends to infinity (both kinds of potentials are depicted in Figure 1). The existence of breathers residing in the potential well near x = 0 can be proved with classical methods such as the center manifold reduction for maps [6,8], variational methods [3,14], and the continuation from the anticontinuum limit γ → 0 [2,10,18]. A more delicate problem is the existence of large-amplitude breathers residing in the other potential well which extends to infinity. Large-amplitude stationary solutions bifurcating from infinity as γ → 0 have been obtained in [16]. These solutions are localized near a single site, say n = 0, and their amplitude diverges as γ → 0. Large-amplitude breathers in a finite-size neighborhood of these stationary solutions have been constructed in [7] for small coupling γ, using the contraction mapping theorem and scaling techniques. These large-amplitude breathers oscillate beyond the potential barrier of V at x = a 0 , and their amplitude goes to infinity as γ → 0. Existence of large-amplitude breathers oscillating everywhere above the potential barrier of V was left open in [7].
Our goal is to show the existence of large-amplitude breathers oscillating in several potential wells, setting-up a continuation of these solutions from infinity as γ → 0. To illustrate some key points of our analysis, let us consider the example Here V has a global minimum at x = 0, a pair of symmetric global maxima at x = ±a 0 with a 0 > 0, and lim x→±∞ V (x) = 1 4 . In the standard anti-continuum limit, one sets γ = 0 and x n = 0 for all n ∈ Z\{0}, and one considers a time-periodic solution x 0 (t) ≡ x(t) of the nonlinear oscillator equation Under a nonresonance condition, this compactly supported time-periodic solution can be continued for γ ≈ 0 into an exponentially localized time-periodic breather solution using the implicit function theorem [10].
The phase plane (x,ẋ) and the frequency-amplitude (ω, a) diagram of the nonlinear oscillator equation (3) with the potential (2) are shown on Figure 2. In this case, the periodic solution x(t) has a cut-off amplitude at a = a 0 . Only the family of periodic solutions with a ∈ (0, a 0 ) can be continued by the anti-continuum technique developed by MacKay and Aubry [10].
In addition, there are two families of unbounded solutions: one corresponds to oscillations beyond the potential barrier of V for |x| > a 0 and the other one corresponds to oscillations above the potential barrier. Roughly speaking, the new technique developed in [7] allows one to obtain large amplitude breathers "close" to unbounded solutions of the first family for γ ≈ 0.
The present paper considers large-amplitude breathers near the second family of unbounded solutions. These two families of breathers are obtained by "continuation from infinity" for arbitrarily small values of γ, but without reaching γ = 0. In this case, the potential V in the nonlinear oscillator equation (3) can be simply replaced by The potential V γ includes a restoring force originating from the nearest-neighbors coupling in the discrete Klein-Gordon equation (1). As γ → 0, the amplitudes and periods of the resulting breathers go to infinity. As a result, we need a careful control of nonresonance conditions in order to prove the existence of such breathers. Although a part of our continuation procedure involving the contraction mapping theorem is close to the one developed in [7], our mathematical analysis is quite different because our breather solutions scale differently in the different potential wells, which induces some singular perturbation analysis and more delicate estimates than in [7]. Note also that the contraction mapping theorem has been used by Treschev [19] to prove the existence of other types of localized solutions (solitary waves) in Fermi-Pasta-Ulam lattices, in which nearest-neighbors are coupled by an anharmonic potential having a repulsive singularity at a short distance. In this case, the existence problem yields an advance-delay differential equation with other kinds of mathematical difficulties.
To simplify our analysis, we assume that V is symmetric and bounded, whereas V ′ has a compact support. To be precise, the following properties on V are assumed: Assumption (P1) allows us to consider symmetric periodic oscillations, which can be studied on the quarter of the fundamental period. This assumption simplifies the presentation but is not essential, and our analysis could be extended e.g. to potentials confining at −∞ (as in Figure 1). Assumption (P2) allows us to develop a contraction mapping argument for the small-amplitude oscillations on the sites n = 0, a procedure which cannot be carried out if a quartic term is present in the expansion of V near the origin. It would be useful to relax this condition, which assumes a very weak anharmonicity of small amplitude oscillations. Note that the quartic term in V (x) near x = 0 is also excluded in the recent analysis of scattering of small initial data to zero equilibrium by Mielke & Patz [13].
Assumption (P3) allows for large-amplitude oscillations at the central site n = 0. Assumption (P4) allows us to consider linear oscillations of the central site outside the compact support of V ′ . This property is used in Lemma 3 below to solve the singularly perturbed oscillator equation for renormalized oscillations at n = 0. This compact support assumption is quite restrictive, and it would be interesting to relax it in a future work, by considering e.g. exponentially decaying potentials (as in example (2)) and treating exponential tails as perturbations of the present case.
We note that Fura & Rybicki [5] have proved the existence of periodic solutions bifurcating from infinity for a class of finite-dimensional Hamiltonian systems with asymptotically linear potentials using degree theory. Our analysis is different and consists in two steps. We first reduce the infinite-dimensional Hamiltonian system to a perturbed oscillator equation describing large amplitude oscillations at the breather center, using the contraction mapping theorem. Once this has been achieved, we solve the reduced problem using a topological method (Schauder's fixed point theorem).
Our main result is the existence of the large-amplitude breathers if the potential V satisfies assumptions (P1)-(P4) as well as the technical non-degeneracy condition in equation (11) below. As further problems, it would be interesting to analyze the existence of multibreather solutions bifurcating from infinity, as well as the stability of such solutions, as it was done previously for finite-amplitude breathers near the standard anti-continuum limit (see, e.g., [11,4,2,12,1,9]).
The article is organized as follows. Section 2 describes the main results. Large-amplitude oscillations near n = 0 are analyzed in Section 3. Small-amplitude oscillations for n = 0 are considered in Section 4. The proof of the main theorem is given in Section 5. Section 6 gives a proof that the large-amplitude breather decays exponentially in n ∈ Z.
Acknowledgement. This work was initiated during the visit of D.P. to Laboratoire Jean Kuntzmann supported in part by the Ambassade de France au Canada. D.P. thanks the members of the Laboratory for hospitality during his visit.

Main results
We shall consider the discrete Klein-Gordon equation (1) for small γ > 0 and assume that the breather is localized near the central site n = 0. We consider oscillations in the potential V γ (x) at the energy level E:ẍ Thanks to assumption (P3), the anti-continuum limit γ → 0 is singular for E > V L in the sense that a bounded trajectory of system (5) trapped by the quadratic potential γx 2 degenerates into an unbounded trajectory as γ → 0. We would like to select a unique T -periodic solution of (5) by fixing its energy E > V L and choosing γ small enough. For a fixed E > V L , we will be working for sufficiently small γ > 0 to ensure that V γ (a) = E admits a unique positive solution a(E, γ). More precisely, thanks to assumptions (P3) and (P4), we obtain a unique solution a = (E − V ∞ ) 1/2 γ −1/2 for γ < (E − V L )/a 2 0 . Fixingẋ(0) = 0, we can parameterize periodic solutions by x(0) = a(E, γ) > 0, and their period can be written T = T (E, γ). Thanks to assumption (P4), we shall prove (in Section 3) that for any E > V L Thanks to assumption (P1), the T -periodic solution x(t) of the nonlinear oscillator equation (5) with x(0) = a > 0 andẋ(0) = 0 is symmetric with respect to reflections about the points t = 0 and t = T 2 and anti-symmetric with respect to reflection about the points t = T 4 and t = 3T 4 . Therefore, the T -periodic solution satisfies The normalized frequency of oscillations is defined by such that ω 0 (E, γ) → √ 2 as γ → 0 for a fixed E > V L . To avoid resonances of large-amplitude oscillations at the central site n = 0 with small-amplitude oscillations at the other sites n ∈ Z\{0}, we will show (in Section 4) that the following non-resonance conditions For a fixed E > V L , we shall now consider the non-resonant set of parameters (breather frequency ω = 2π T and coupling constant γ). We plot on Figure 3 the prohibited regions between the boundaries of the non-resonant set, given by the curves ω = κ m and ω = √ κ 2 +4γ m , together with the curve ω = √ γ ω 0 (E, γ). Non-resonance conditions (9) are satisfied if γ belongs to the set C E = ∪ m≥m 0 (Γ m , γ m ), where γ = Γ m and γ = γ m correspond to the intersections of the above curves starting with some m 0 ≥ 1, i.e. Γ m and γ m satisfy implicit equations √ Equations (10) can be solved for m large enough thanks to expansion (6) and the implicit function arguments, yielding as m → ∞ In particular, we note that Γ m < γ m and |γ m − Γ m | = O(m −3 ) as m → ∞.
We can now state the main result of this article. Note that the existence of breathers is only obtained for a subsetC E,ν ⊂ C E of the non-resonant values of the coupling constant γ, because our method breaks down near the boundary of C E . Theorem 1 Assume (P1)-(P4) on V (x) and fix E > V L . Let x(t) be a T (E, γ)-periodic solution of the nonlinear oscillator equation (5) for small γ > 0 satisfying symmetries (7) and assume that λ ′ (E) = 0, i.e.
Fix ν ∈ (0, 1) and consider the set of coupling constantsC E,ν = ∪ m≥m 0 (Γ m ,γ m ) ⊂ C E , wherẽ Γ m ,γ m are defined by the implicit equations for m ≥ m 0 , and satisfy as m → +∞ For all sufficiently small γ inC E,ν , there exists a T -periodic spatially localized solution x(t) ∈ H 2 per ((0, T ); l 2 (Z)) of the Klein-Gordon lattice (1) such that and ∃C > 0 : sup (13) shows that the relative error Remark 2 If ν ∈ (0, 1), we still haveΓ m <γ m and |γ m −Γ m | = O(m −3 ) as m → ∞. Therefore, the rate of decrease of the interval widths in the setC E,ν corresponds to the rate of decrease of the widths of the non-resonant intervals in the set C E .
Remark 3 Although we do not attempt here to deal with non-compact potentials, we believe that assumption (P4) can be relaxed if V ′ (x) has a sufficiently fast decay to zero as |x| → ∞. In that case, we conjecture that the non-resonance condition (11) would be replaced by Figure 4 illustrates that this condition is satisfied for the particular potential (2), for any finite E > V L (we note that the value of Q approaches 0 as E → ∞).
Theorem 1 is proved in Section 5, using intermediate results established for the single oscillator equation (5) with a forcing term (in Section 3) and for the discrete KG equation (1) linearized at zero equilibrium (in Section 4). We finish the article with Section 6, where we prove that the amplitude of breather oscillations decays exponentially in n on Z in the following sense: for all sufficiently small γ ∈C E,ν .

Large-amplitude oscillations at a central site
We consider here solutions of the nonlinear oscillator equation (5) in the singular limit γ → 0. Assumptions (P1)-(P4) on the potential V are used everywhere, without further notes.
Since E = γa 2 + V (a), it follows from assumption (P4) that Asymptotic expansion of the period T of the periodic solution of (5) is found from the exact formula Thanks to assumption (P4), we know that where θ 0 is the smallest positive root of a sin(θ) = a 0 satisfying . As a result, the asymptotic expansion concludes the proof of the asymptotic expansion (14).
Let us represent the solution of (5) for E > V L in the form By Lemma 1, we have X L ∞ = O(1) and T 0 := γ 1/2 T = O(1) as γ → 0 with precise value X L ∞ = (E − V ∞ ) 1/2 and the asymptotic expansion We shall now derive a series of estimates that will be useful for the proof of Theorem 1.
Corollary 1 Let X(τ ) be the T 0 -periodic function defined by the solution of Lemma 1 in parametrization (16) for any fixed E > V L . Then, X ∈ C 3 per (0, T 0 ) and for sufficiently small γ > 0, there Proof. We recall that Let us rewrite the energy conservation (5) in parametrization (16): Since V ≥ 0 we have Ẋ L ∞ ≤ √ 2E, which gives the bound on X C 1 . This also gives the uniform bound on X H 1 per since T 0 = O(1) as γ → 0. Let us also rewrite the second-order equation (5) in parametrization (16): Thanks to assumption (P1), the solution X(τ ) is actually in C 3 per (0, T 0 ).
The potential term of the nonlinear equation (19) is a singular contribution to the linear equation as γ → 0. Because of the singular contribution, Ẍ L ∞ grows as γ → 0. Nevertheless, thanks to assumption (P4) of the compact support of V ′ (x), the solution X(τ ) stays in the domain |X| ≥ a 0 γ 1/2 , where V ′ (γ −1/2 X) = 0 for most of the times τ in the period [0, T 0 ]. The following lemma estimates the size of the time interval, for which the solution stays in the domain |X| ≤ a 0 γ 1/2 . Lemma 2 Let X(τ ) be the same as in Corollary 1. Let ∆T 0 be the measure of the subset of [0, T 0 ] in which |X(τ )| ≤ a 0 γ 1/2 . Then, ∆T 0 admits the asymptotic expansion Proof. Consider the splitting of [0, Thanks to the symmetries (7), we have In the first, third, and fifth intervals, the second-order equation (19) for sufficiently small γ > 0 becomes the linear oscillatorẌ + 2X = 0.
Corollary 2 Let X(τ ) be the same as in Lemma 2 and

Proof.
Using the bounds on ∆T 0 in Lemma 2, Corollary 1, and assumption (P1) on the potential V (x), we obtain for some constants C 1 , C 2 > 0. The bound on Y H 1 per follows from the above computation. We shall be working in the space of functions in H 2 per (0, T 0 ), H 1 per (0, T 0 ), and L 2 per (0, T 0 ) satisfying symmetry (7). Therefore, let us denote for all p ≥ 1 and use similar notations for L 2 e and L ∞ e . We are now prepared to deal with the singularly perturbed linear oscillator under the small source term: where ε > 0, F ∈ L 2 e , and F L 2 per = O(1) as γ → 0. It is necessary to consider the inhomogeneous problem (25) in order to control the effect of small coupling in the discrete Klein-Gordon equation (1) at the central site n = 0. Energy for the perturbed oscillator equation (25) can be written in the form Because the homogeneous equation with F ≡ 0 admits a T 0 -periodic solution X ∈ H 2 e with ∂ E T 0 (E, γ) = O(γ 1/2 ) (equation (25) is a singular perturbation of a linear oscillator), a source term of order one would generate a large output as γ → 0, i.e. the output Z − X H 1 per is going to be γ −1/2 larger than the source term, roughly speaking. This can be intuitively understood by linearizing equation (25) around X. Indeed, the linearized operator which generate large terms for γ → 0 because of the singular perturbation in the nonlinear potential. To avoid this difficulty, we use the fact that V ′ (γ −1/2 Z) has a compact support, and transform the search of periodic solutions to a root finding problem to which the Implicit Function Theorem can be applied. We shall prove the following.
Let E * := H(τ 0 ) = 1 2 W 2 0 + a 2 0 γ + V ∞ and assume that E * > V L ≥ V ∞ . Note that E * becomes now the parameter of the solution family in place of W 0 .
It remains to prove bounds (29) and (30). For this purpose, we first consider a time interval in which Z and X are both given by the explicit solution of a linear equation. Recall that By combining the explicit expressions of X and Z (see equation (33)), estimates (35), (47) and observing that Sinceτ 0 = T 0 4 + O(γ 1/2 ), it follows from equation (33) and (48) that As a result, we extend the first bound in (49) to To extend the second bound in (49), we write for all τ ∈ τ 0 , 1 4 T 0 , where we have used bounds (40) and (47). By bounds (41) and (50), hence we have As a conclusion, estimate (29) follows from estimates (49)  For the arguments in the proof of Theorem 1 based on the Schauder fixed-point theorem, we will need a continuity of the nonlinear map F → Z from L 2 e to H 2 e .
Lemma 4 Under Assumptions of Lemma 3, for all γ ∈ (0, γ 0 ) and all F ∈ B δ , the solution Z = G γ,ǫ (F ) ∈ H 2 e is continuous with respect to F ∈ L 2 e .

Small-amplitude oscillations on other sites
In our construction, the large-amplitude breather bifurcating from infinity as γ → 0 is localized at a single site n = 0, and close to the periodic solution of Lemma 1. The other amplitudes for n = 0 display the oscillatory motion guided by the central oscillation at n = 0 and powered by a small amplitude in γ. By symmetry of the discrete Laplacian, we may assume that x n = x −n , n ≥ 1.
Since the amplitudes of oscillations for n = 0 are small, we shall consider the linearized discrete Klein-Gordon equation (1) at the zero solution to study possible resonances with the oscillatory motion at n = 0. To this end, let us introduce the function spaces X := L 2 per ((0, T 0 ); l 2 (N)), D := H 2 per ((0, T 0 ); l 2 (N)), and denote ω 0 = 2π T 0 . Let us use assumption (P2) and consider the inhomogeneous linear problem where τ = γ 1/2 t denotes the rescaled time. We denote the sequence {F n (τ )} n≥1 ∈ X by F and look for solution U = {U n (τ )} n≥1 ∈ D of (56). We shall prove the following.
Lemma 5 Assume the non-resonance condition (9) to be satisfied for given values of κ, γ and ω 0 , i.e.
Then, for all F ∈ X problem (56) admits a unique solution U ∈ D. Moreover, one has where Moreover, if in addition F ∈ H 1 per ((0, T 0 ); l 2 (N)) then Proof. In order to solve (56), we expand F and U using Fourier series in the time variable t and band-limited Fourier transforms in the discrete spatial coordinate n. Defininĝ Note that the Fourier transform defines an isometric isomorphism between the Hilbert spaces X and X := ℓ 2 (Z; L 2 per (0, 2π)), and between D and D := ℓ 2 2 (Z; L 2 per (0, 2π)), where the latter denotes the usual Hilbert space consisting of sequences {û m } m∈Z in L 2 per (0, 2π) for which {m 2û m } ∈ ℓ 2 (Z; L 2 per (0, 2π)). In the Fourier space, differential equations (56) reduce to the set of algebraic equations wheref ∈ X and we look forû ∈ D. Thanks to condition (57), for all m ∈ Z, equation (61) has a unique solutionû m =Ĥ mfm ∈ L 2 per (0, 2π), wherê is 2π-periodic and analytic over R. To check thatû ∈ D, we use the estimate where .
For |m| ≥ ω −1 0 (4 + γ −1 κ 2 ) 1/2 one has Moreover, we have This proves estimate (58). Estimate (60) follows by differentiating (56) with respect to τ and using (58). Now we come back to the case in which ω 0 depends in fact on γ and on the fixed energy E of the excitation at site n = 0. More precisely, we recall that Non-resonance conditions (57) are satisfied if γ belongs to the disjoint set C E = ∪ m≥m 0 (Γ m , γ m ), where Γ m and γ m are roots of equations (10) for m large enough. For each γ ∈ C E , a unique solution of the inhomogeneous system (56) exists in the form (63). However, the norm U X diverges as γ approaches the boundary of C E , according to estimate (58)-(59). Following the approach introduced in [7], we quantify this divergence when γ tends towards 0 but remains in a well-chosen subsetC E,ν of C E far enough from resonances.
Proof. Equations (65) can be solved for m large enough (say m ≥ m 0 (E)) and smallΓ m ,γ m by combining expansion (17) and the implicit function arguments. In order to deduce estimates (67) and (68) from Lemma 5, we need a lower bound for Let us assume γ ∈ (Γ m ,γ m ) with m large enough. In what follows we will show that M (m, q) > 0 and M (m + 1, q) < 0. For n ≤ m − 1 it follows that M (n, q) > M (m, q) > 0, and for n ≥ m + 2 we have also M (n, q) < M (m + 1, q) < 0, hence the infimum of (69) will be reached for n = m or n = m + 1.
For q ∈ (0, 1), the interval (Γ m ,γ m ) converges to the interval (Γ m , γ m ) for the price of losing too much power of γ in the bound (68). Therefore, the estimate of Lemma 6 is sharp in this sense.

Proof of Theorem 1
Let us represent where {X n } n∈Z is a new set of unknowns in time τ = γ 1/2 t. From the discrete Klein-Gordon equation (1), we obtain where N (X) := V ′ (X) − κ 2 X. Let B δ ⊂ H 1 e be a ball of small radius δ > 0 centered at 0 ∈ H 1 e . By assumption (P2), N (X) : B δ → H 1 e is a C 5 map. Moreover, expansion V ′ (x) = κ 2 x + O(x 5 ) near x = 0 implies the existence of C > 0 such that for all δ > 0 small enough we have From system (72) and (73), we can see that oscillations near the zero solution would involve inverting the linearized operator in the inhomogeneous system (56). By estimate (68), we are going to loose γ 1/2 , which is the size of the inhomogeneous term γ 1/2 X 0 . This would prevent us from using the contraction mapping theorem in the neighborhood of the zero solution. To overcome this obstacle, we introduce the near-identity transformation X 1 = Y 1 + γ 1/2 κ −2 X 0 , X n = Y n , n ≥ 2 and rewrite system (71)-(73) in the equivalent form ExtractingẌ 0 + 2X 0 from (76), we can rewrite (77) in the equivalent form We shall solve the above system in two steps, using the contraction mapping theorem to solve (78)-(80) at fixed X 0 , and then Schauder's fixed point theorem to solve (76). In the latter case, we shall consider equation (76) similar to equation (25) with X 0 ∈ H 1 e being close to the solution X ∈ H 1 e of Lemma 1 rescaled by (16). The source term depends on Y 1 ∈ B δ ⊂ H 1 e and X 0 , and it will be proved that δ = O(γ ε ) is small as γ → 0.
6 Exponential decay on Z Our last result is to show that the large-amplitude discrete breather constructed in Theorem 1 decays exponentially in n on Z. The arguments repeat those of reference [7], to which we shall refer for some standard steps of the proof.
for all sufficiently small γ ∈C E,ν .
A simple application of the discrete maximum principle yields then (see [7], Lemma 3.3) Using equation (78), estimates (92) and (95) with n = 3, the fact that Y 1 H 1 e = O(γ 1/4 ) and X 0 H 1 e = O(1) (direct consequence of (90) and Corollary 1), we get Then one completes the proof by putting estimates (95) and (96) together and using the continuous embedding of H 1 e in L ∞ e .