Breather Solutions of the Discrete p-Schrödinger Equation

Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 7)

Abstract

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 α. In the limit of weak nonlinearity (α → 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 α is close to unity, whereas pinning becomes predominant for larger values of α.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Laboratoire Jean KuntzmannUniversité de Grenoble and CNRSGrenoble Cedex 9France
  2. 2.INRIA, Bipop Team-Project, ZIRST MontbonnotSaint IsmierFrance
  3. 3.Faculty of Mechanical EngineeringTechnion – Israel Institute of TechnologyTechnion City, HaifaIsrael

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