Numerical and analytical studies of the discrete nonlinear Schroedinger equation.

Abstract

Certain conservative discretizations of the Nonlinear Schroedinger (NLS) Equation 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 for the two unstable mode regime and, in this case, many and continual homoclinic crossings occur throughout the irregular time series. We undertake an analysis to determine the mechanism that causes the "chaotic" behavior to appear in this conservatively perturbed NLS equation. This analysis involves the construction of explicit formulas for the homoclinic orbit, a description of the relevant finite dimensional phase space and a Melnikov analysis for the various regimes studied.