Homoclinic orbits for a perturbed nonlinear Schrodinger equation

In recent years, there have been extensive studies on the existence of homo clinic orbits for nearly integrable Hamiltonian PDEs, which ore closely rel ated to chaos. In this work, we consider a perturbed nonlinear Schrodinger equation iu(t) = u(xx) + 2(u (u) over bar - w(2))u + i epsilon(u(xx) - alpha u - bet a(-)) for u even and periodic in x. The diffusion i epsilon u(xx) is an unbounded perturbation term. When the diffusion is replaced by its bounded Fourier t runcation, Li, McLaughlin, Shatah, and Wiggins [26] proved the existence of homoclinic orbits for the perturbed equation. The method was based on inva riant manifolds, foliations, and Melnikov analysis. The unboundedness of th e diffusion prevents the equation from being solved for t < 0 for general i nitial values and destroys some geometric structures, however. We overcome these difficulties and prove the existence of homoclinic orbits for the dif fusively perturbed NLS. (C) 2000 John Wiley & Sons, Inc.