Chapter

Localized Excitations in Nonlinear Complex Systems

Volume 7 of the series Nonlinear Systems and Complexity pp 77-115

Date:

Breather Solutions of the Discrete p-Schrödinger Equation

  • Guillaume JamesAffiliated withLaboratoire Jean Kuntzmann, Université de Grenoble and CNRSINRIA, Bipop Team-Project, ZIRST Montbonnot Email author 
  • , Yuli StarosvetskyAffiliated withFaculty of Mechanical Engineering, Technion – Israel Institute of Technology

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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.