Mel'nikov analysis of numerically induced chaos in the nonlinear Schrödinger equation

Certain Hamiltonian discretizations of the periodic focusing Nonlinear Schrödinger Equation (NLS) have been shown to be responsible for the generation of numerical instabilities and chaos. In this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the NLS. Using heuristic Mel'nikov methods, the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system. The structure of the persistent homoclinic orbits that are predicted by the Mel'nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos. The Mel'nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments. In a neighborhood of the persistent homoclinic orbits, the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study.