Twigg said:Finite difference is the method I was thinking of. Specifically, a central-difference scheme for the Laplacian. The catch is that your problem is non-linear due to the |ψ|2ψ|\psi|^{2} \psi term. You will need to use Newton's method to get a solution, and that opens a new can of worms with convergence and validation. I've never tried this personally, but I know at least one piece of commercial software (COMSOL) that uses Newton's method to solve the matrix problems of finite element methods.
You're absolutely right, my mistake. I was thinking of a steady-state problem, which this is not.Strum said:Why would you think there is a problem with non-linear terms? I really can not see how this should pose a problem as long as he uses some explicit time integration scheme ( and even if he used an implicit I can not see why the difficulties would even be related to the finite difference method ).