Symplectic methods for the nonlinear Schrödinger equation

Various symplectic discretizations of the nonlinear Schrödinger equation are compared, including one for the integrable discretization due to Ablowitz and Ladik. The numerical experiments are performed with initial values taken near a homoclinic orbit, i.e., in a situation where integrability is crucial. It is shown that symplectic discretizations can sometimes lead to remarkable improvements, and that in even more sensitive situations some of our best numerical schemes fail.