Multi-symplectic preserving integrator for the Schrödinger equation with wave operator

In this article, we discuss the conservation laws for the nonlinear Schrödinger equation with wave operator under multi-symplectic integrator (MI). First, the conservation laws of the continuous equation are presented and one of them is new. The multi-symplectic structure and MI are constructed for the equation. The discrete conservation laws of the numerical method are analyzed. It is verified that the proposed MI can stably simulate the Hamiltonian PDEs excellently over long-term. It is more accurate than some energy-preserving schemes though they are of the same accuracy. Moreover, the residual of mass is less than energy-preserving schemes under the same mesh partition in a long time.