The complex multi-symplectic scheme for the generalized sinh-Gordon equation

In this paper, the complex multi-symplectic method and the implementation of the generalized sinh- Gordon equation are investigated in detail. The multi-symplectic formulations of the generalized sinh-Gordon equation in Hamiltonian space are presented firstly. The complex method is introduced and a complex semi-implicit scheme with several discrete conservation laws (including a multi-symplectic conservation law (CLS), a local energy conservation law (ECL) as well as a local momentum conservation law (MCL)) is constructed to solve the partial differential equations (PDEs) that are derived from the generalized sinh- Gordon equation numerically. The results of the numerical experiments show that the multi-symplectic scheme has excellent long-time numerical behavior and high accuracy.