Breather solutions of the discrete p-Schrödinger equation

Guillaume James (1, 2) , Yuli Starosvetsky (3)

1 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems
2 CASYS - Calculs Algébriques et Systèmes Dynamiques
3 Technion - Israel Institute of Technology [Haifa]

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.

Domaines: Formation de Structures et Solitons [nlin.PS]; Systèmes dynamiques [math.DS]

Guillaume James, Yuli Starosvetsky. Breather solutions of the discrete p-Schrödinger equation. Ricardo Carretero-González and Jesús Cuevas-Maraver and Dimitri Frantzeskakis and Nikos Karachalios and Panayotis Kevrekidis and Faustino Palmero-Acebedo. Localized Excitations in Nonlinear Complex Systems, Springer International Publishing, pp.77-115, 2014, Nonlinear Systems and Complexity, volume 7, 978-3-319-02056-3. ⟨10.1007/978-3-319-02057-0_4⟩. ⟨hal-00849582⟩