Dynamics of Semi-Discretizations of the Defocusing Nonlinear Schrodinger Equation

It has been demonstrated that the nonlinear Schrödinger(NLS) equation is sensitive to discretizations. In the focusingcase this is due to the homoclinic structure associated withthe NLS equation. In this paper we show that various numericalschemes for the defocusing case are also prone to instabilities,although not as severe as those of the focusing equation. Anintegrable discretization due to Ablowitz and Ladik does notsuffer from the same instabilities. However, it is shown thatit develops a focusing singularity if a threshold conditionis exceeded. Numerical examples illustrating the phenomena pertainingto the defocusing equation are given.