Breathers on a Background: Periodic and Quasiperiodic Solutions of Extended Discrete Nonlinear Wave Systems

TL;DR

This paper explores the emergence, stability, and metastability of time-periodic and quasiperiodic solutions in discrete nonlinear wave systems, revealing new localized solutions and their dependence on system parameters.

Contribution

It introduces a detailed analysis of internal modes leading to periodic and quasiperiodic solutions in discrete wave equations, including new localized solutions and stability insights.

Findings

Existence of time-periodic and quasiperiodic solutions on a breather background.

Identification of internal modes related to translational symmetry breaking.

Discovery of spatially localized, quasiperiodic solutions in discrete nonlinear Schrödinger equation.

Abstract

In this paper we investigate the emergence of time-periodic and and time-quasiperiodic (sometimes infinitely long lived and sometimes very long lived or metastable) solutions of discrete nonlinear wave equations: discrete sine Gordon, discrete $\phi^4$ and discrete nonlinear Schr\"odinger. The solutions we consider are periodic oscillations on a kink or standing wave breather background. The origin of these oscillations is the presence of internal modes, associated with the static ground state. Some of these modes are associated with the breaking of translational invariance, in going from a spatially continuous to a spatially discrete system. Others are associated with discrete modes which bifurcate from the continuous spectrum. It is also possible that such modes exist in the continuum limit and persist in the discrete case. The regimes of existence, stability and metastability of states as the lattice spacing is varied are investigated analytically and numerically. A consequence of our analysis is a class of spatially localized, time quasiperiodic solutions of the discrete nonlinear Schr\"odinger equation. We demonstrate, however, that this class of quasiperiodic solution is rather special and that its natural generalizations yield only metastable quasiperiodic solutions.