Abstract
Existence of large-amplitude time-periodic breathers localized neara single site is proved for the discrete Klein--Gordon equation, in the casewhen the derivative of the on-site potential has a compact support.Breathers are obtained at small couplingbetween oscillators and under nonresonance conditions.Our method is different from the classical anti-continuum limitdeveloped by MacKay and Aubry, and yields in generalbranches of breather solutions that cannotbe captured with this approach.When the coupling constant goes to zero,the amplitude and period of oscillationsat the excited site go to infinity.Our method is based on near-identity transformations, analysis ofsingular limits in nonlinear oscillator equations, andfixed-point arguments.
Highlights
Recent studies of spatially localized and time-periodic oscillations in lattice models of DNA [7, 16] call for systematic analysis of such excitations in the discrete Klein–Gordon equation xn + V ′(xn) = γ, n ∈ Z, (1)where γ > 0 is a coupling constant, V : R → R is a nonlinear potential, and x(t) = {xn(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 → ∞
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
Note that the contraction mapping theorem has been used by Treschev [19] to prove the existence of other types of localized solutions in Fermi-Pasta-Ulam lattices, in which nearest-neighbors are coupled by an anharmonic potential having a repulsive singularity at a short distance
Summary
To cite this version: Guillaume James, Dmitry Pelinovsky. Breather continuation from infinity in nonlinear oscillator chains. Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2012, 32 (5), pp.1775-1799. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés
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