A Galerkin energy-preserving method for two dimensional nonlinear Schrödinger equation

Abstract

In this paper, a Galerkin energy-preserving scheme is proposed for solving nonlinear Schrödinger equation in two dimensions. The nonlinear Schrödinger equation is first rewritten as an infinite-dimensional Hamiltonian system. Following the method of lines, the spatial derivatives of the nonlinear Schrödinger equation are approximated with the aid of the Galerkin methods. The resulting ordinary differential equations can be cast into a canonical Hamiltonian system. A fully-discretized scheme is then devised by considering an average vector field method in time. Moreover, based on the fast Fourier transform and the matrix diagonalization method, a fast solver is developed to solving the resulting algebraic equations. Finally, the proposed scheme is employed to capture the blow-up phenomena of the nonlinear Schrödinger equation.