Some soliton-type analytical solutions and numerical simulation of nonlinear Schrödinger equation

In this article, we study some soliton-type analytical solutions of Schrödinger equation, with their numerical treatment by Galerkin finite element method. First of all, some analytical solutions to the equation are obtained for different values of parameters; thereafter, the problem of truncating infinite domain to finite interval is taken up and truncation approximations are worked out for finding out appropriate intervals so that information is not lost while reducing the domain. The benefit of domain truncation is that we do not need to introduce artificial boundary conditions to find out numerical approximations. To verify theoretical results, numerical simulations are performed by Galerkin finite element method. Crank–Nicolson method is used for the time discretization, and non-linearity is resolved using predictor corrector method, which is second order accurate and computationally efficient.