Breather solutions of the discrete pSchrödinger equation
Abstract
We consider the discrete pSchrödinger (DpS) equation, which approximates small amplitude oscillations in chains of oscillators with fullynonlinear nearestneighbors interactions of order alpha = p1 >1. Using a mapping approach, we prove the existence of breather solutions of the DpS equation with even or oddparity 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 Schrödinger 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 oddparity 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.
 Publication:

arXiv eprints
 Pub Date:
 July 2013
 DOI:
 10.48550/arXiv.1307.8324
 arXiv:
 arXiv:1307.8324
 Bibcode:
 2013arXiv1307.8324J
 Keywords:

 Nonlinear Sciences  Pattern Formation and Solitons;
 Mathematics  Dynamical Systems
 EPrint:
 To appear in Springer Series on Wave Phenomena