A linearized energy-conservative scheme for two-dimensional nonlinear Schrödinger equation with wave operator

Abstract

Based on the invariant energy quadratization approach, we propose a linear implicit and local energy preserving scheme for the nonlinear Schrödinger equation with wave operator, that describes the solitary waves in physics. In order to overcome the difficulty of designing an efficient scheme for the imaginary functions of the nonlinear Schrödinger equation with wave operator, we transform the original problem into its real form. By introducing some auxiliary variables, the real form of nonlinear Schrödinger equation with wave operator is reformulated into an equivalent system, which admits the modified local energy conservation law. Then the equivalent system is discretized by the finite difference method to yield a linear system at each time step, which can be efficiently solved. A numerical analysis of the proposed scheme is conducted to show its uniquely solvability and convergence. Our proposed method is validated by numerical simulations in terms of accuracy, energy conservation law and stability.