Numerical solution of nonlinear Schrodinger equation and investigation of singular solutions

The nonlinear Schrodinger equation (NLS) is one of the most important equations in quantum physics. It is frequently useful in describing the waveguide movement of minute particles, for example an electron in the atom. NLS can be divided into three different cases: critical, supercritical, and subcritical. In this paper we try to show numerical methods that solve critical case of NLS, for short CNLS, in two dimensions. We also study the effects of the discretization of the equation in the solutions. There are some initial conditions for which the solutions of CNLS become singular in the finite time interval, but by using the finite difference method for the discretization of the Laplacian term in the equation, it is shown that the resulting discrete NLS represents a more accurate discretized version of the modified CNLS equation which can have local solution as well. So, as such modified CNLS equations are simply obtained by inserting small perturbations in the original equation, evidently by using this method it is almost possible to avoid blowup in the numerical solutions of these equations.