Homoclinic manifolds and numerical chaos in the nonlinear Schrödinger equation

Crossings of homoclinic manifolds is a well known mechanism underlying observed chaos in low dimensional systems. We discuss an analogous situation as it pertains to the numerical simulation of a well known integrable partial differential equation, the nonlinear Schrödinger equation. In various parameter regimes, depending on the initial data, numerical chaos is observed due to either truncation effects or errors on the order of roundoff. The explanation of the underlying cause of the chaos being due to crossing of homoclinic manifolds induced by the numerical errors is elucidated. The nonlinear Schrödinger equation is prototypical of a much wider class of nonlinear systems in which computational chaos can be a significant factor.