Homoclinic orbits for a perturbed nonlinear Schrödinger equation

Abstract

In recent years, there have been extensive studies on the existence of homoclinic orbits for nearly integrable Hamiltonian PDEs, which are closely related to chaos. In this work, we consider a perturbed nonlinear Schrödinger equation for u even and periodic in x. The diffusion iεuxx is an unbounded perturbation term. When the diffusion is replaced by its bounded Fourier truncation, Li, McLaughlin, Shatah, and Wiggins [26] proved the existence of homoclinic orbits for the perturbed equation. The method was based on invariant manifolds, foliations, and Melnikov analysis. The unboundedness of the diffusion prevents the equation from being solved for t < 0 for general initial values and destroys some geometric structures, however. We overcome these difficulties and prove the existence of homoclinic orbits for the diffusively perturbed NLS. © 2000 John Wiley & Sons, Inc.