On homoclinic structure and numerically induced chaos for the nonlinear Schro¨dinger equation

It has recently been demonstrated that standard discretizations of the cubic nonlinear Schrodinger (NLS) equation may lead to spurious numerical behavior. In particular, the origins of numerically induced chaos and the loss of spatial symmetry are related to the homoclinic structure associated with the NLS equation. In this paper, an analytic description of the homoclinic structure via soliton type solutions is provided and some consequences for numerical computations are demonstrated. Differences between an integrable discretization and standard discretizations are highlighted.