Multi-symplectic Preserving Integrator for the Schr\"{o}dinger Equation with Wave Operator

In the article, we discuss the conservation laws for the nonlinear Schr\"{o}dinger equation with wave operator under multisymplectic integrator (MI). First, the conservation laws of the continuous equation are presented and one of them is new. The multisymplectic 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 multisymplectic Hamiltonian system excellent 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 over long-term.