Chaotic and homoclinic behavior for numerical discretizations of the nonlinear Schrödinger equation

Certain conservative discretizations of the NLS can produce irregular behavior. We consider the diagonal discretization as a conservative perturbation of the integrable discretization and study the homoclinic crossings in its nonlinear spectrum. We find that irregularity sets in when two homoclinic structures are present and, in this case, many and continual homoclinic crossings occur throughout the irregular time series. We indicate a Melnikov analysis to study the consequences of this homoclinic behavior.