A numerical solution and an exact explicit solution of the NLS equation

We consider traveling wave solutions of the nonlinear Schrödinger (NLS for short) equation. In this paper by considering the decomposition scheme, we first obtain the exact solutions of the NLS equation for the initial condition without using any classical transformations and then its numerical solutions are constructed without using any discretization technique. The numerical solutions are compared with the known analytical solutions. Its remarkable accuracy is finally demonstrated in the study of some initial values of the NLS equation.