Research On Numerical Algorithms To Some Nonlinear Solitary Wave Equations

All the physical courses whose dissipative effects are negligible can be expressed as Hamiltonian systems which preserve energy conservation and symplectic geometric structure. The Hamiltonian system is universal in the nature, in other words, most soliton equations can be written into Hamiltonian formalism. The basic principle of modern numerical computation is to preserve the intrincal character of the original problems. Therefore, it is necessary to study numerical methods which preserve the energy conservation or the symplectic structure of the Hamiltonian system. However, most known conservative schemes for many nonlinear solitary wave equations are implicit and coupled. There even exist some nonlinearly coupled equations which have no conservative finite difference schemes, or have very few conservative schemes without essential numerical analysis. Up to now, there is very few rigorous numerical analysis for the stability and convergence of the symplectic and multisymplectic difference schemes for nonlinear partial differential equations, i.e., it is an open problem.In the dissertation, some nonlinear solitary wave equations and some nonlinear coupled systems of solitary wave equations are numerically studied, some new conservative schemes including some explicit ones for these equations are constructed. Some new inequalities are proposed which and discrete energy analysis method are used to prove the convergence and stability of all the schemes. For the convergence of those explicit schemes, it is known that the prior estimates of their numerical solutions are difficult to obtain. In order to avoid obtaining the prior estimates, mathematical induction method and discrete energy analysis method are used to prove the convergence. Taking the symplectic and multi-symplectic difference method for solving the nonlinearly coupled Schr(o|¨)dinger equations as an example, the rigorous proof of the convergence of symplectic and multi-symplectic difference algorithm are given. The method of proof can be extended for symplectic and multi-symplectic schemes of other equations. A nonlinear Schr(o|¨)dinger equation with a weakly damped term which possesses a global attractor is also numerically analyzed. The difficult proof of long-time estimate is overcomed, the approximate attractor is obtained and the long-time stability and convergence of the two numerical schemes are proved.