Breather continuation from infinity in nonlinear oscillator chains

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 ẍ + V ′ (x) = 0.

(

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:

Ass