Multi-symplectic scheme and simulation of solitary wave solution for generalized KdV-mKdV equation

Aim. Many practical problems are nonlinear. Linearization often brings poor long-time numerical behavior. In order to keep long-time numerical behavior satisfactory, we consider the multi-symplectic formulations of the generalized KdV-mKdV equation with initial value condition in the Hamilton space. In the full paper, we explain our multi-symplectic scheme in some detail; in this abstract, we just add some pertinent remarks to listing the two topics of explanation. The first topic is: the multi-symplectic formulation of the generalized KdV-mKdV equation and its conservation laws. In this topic, we derive eq.(6) as the multi-symplectic formulation and eqs.(7), (8) and (9) as the conservation laws. The second topic is: the multi-symplectic Preissmann scheme and its equivalent form. In this topic, we construct the equivalent scheme of the Preissmann integrator, which is given as eq.(14). To verify the validity of eq.(14), we simulate the solitary wave solution of the generalized KdV-mKdV equation. The computer simulation results, shown in Figs.1 and 2 in the full paper, indicate preliminarily that the multi-symplectic scheme can keep unchanged the wave form of the solitary wave solution and preserve well the local energy and local momentum in the Hamilton space.