2013/07/31 by Guillaume James, Yuli Starosvetsky · 1 citation
Physics and Astronomy · Mathematics · #nlin.PS #math.DS
paper · pdf · doi:10.1007/978-3-319-02057-0_4
To appear in Springer Series on Wave Phenomena
arxiv created 2013/07/31 · arxiv updated 2013/12/18
We consider the discrete p-Schrödinger (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 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 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.