Breather solutions of the discrete p-Schr\"odinger equation

TL;DR

This paper proves the existence of breather solutions in the discrete p-Schrödinger equation, derives analytical approximations, explores a continuum limit, and analyzes stability and dynamics of these localized oscillations.

Contribution

It introduces a mapping approach to establish breather solutions with symmetry, derives analytical profiles, and connects the discrete model to nonlinear Schrödinger equations in the weak nonlinearity limit.

Findings

Breather solutions exist with specific symmetries.

Analytical approximations for breather profiles are provided.

Stability analysis shows transition from pinned to traveling breathers.

Abstract

We consider the discrete p-Schr\"odinger (DpS) equation, which approximates small amplitude oscillations in chains of oscillators with fully-nonlinear nearest-neighbors interactions of order alpha = p-1 >1. Using a mapping approach, we prove the existence of breather solutions of the DpS equation with even- or odd-parity reflectional symmetries. We derive in addition analytical approximations for the breather profiles and the corresponding intersecting stable and unstable manifolds, valid on a whole range of nonlinearity orders alpha. In the limit of weak nonlinearity (alpha --> 1^+), we introduce a continuum limit connecting the stationary DpS and logarithmic nonlinear Schr\"odinger equations. In this limit, breathers correspond asymptotically to Gaussian homoclinic solutions. We numerically analyze the stability properties of breather solutions depending on their even- or odd-parity symmetry. A perturbation of an unstable breather generally results in a translational motion (traveling breather) when alpha is close to unity, whereas pinning becomes predominant for larger values of alpha.