Excitation Thresholds for Nonlinear Localized Modes on Lattices

Abstract

Breathers are spatially localized and time periodic solutions of extended Hamiltonian dynamical systems. In this paper we study excitation thresholds for (nonlinearly dynamically stable) ground state breather or standing wave solutions for networks of coupled nonlinear oscillators and wave equations of nonlinear Schrödinger (NLS) type. Excitation thresholds are rigorously characterized by variational methods. The excitation threshold is related to the optimal (best) constant in a class of discr ete interpolation inequalities related to the Hamiltonian energy. We establish a precise connection among $d$, the dimensionality of the lattice, $2σ+1$, the degree of the nonlinearity and the existence of an excitation threshold for discrete nonlinear Schrödinger systems (DNLS). We prove that if $σ\ge 2/d$, then ground state standing waves exist if and only if the total power is larger than some strictly positive threshold, $ν_{thresh}(σ, d)$. This proves a conjecture of Flach, Kaldko& MacKay in the context of DNLS. We also discuss upper and lower bounds for excitation thresholds for ground states of coupled systems of NLS equations, which arise in the modeling of pulse propagation in coupled arrays of optical fibers.