Multisymplectic Schemes For The 2+1 Dimensional Soliton Equations

The soliton theory is an important branch of applied mathematics and mathematical physics. It has many important applications in fluid mechanics, nonlinear optics, classical and quantum fields theories etc. Though many important progress of soliton theory, such as the inverse scattering method, have been made in the recent years, the numerical methods are still very important and absolutely necessary in studying the soliton equations.In this thesis, higher-dimensional extensions of the ubiquitous Korteweg-de Vries (KdV) equation are investigated. One is the Zakharov-Kuznetsov (ZK) equation, and the other is the generalized Kadomtsev-Petviashvili (KP) equation, they are both the 2+1 dimensional soliton equations. Recently, Marsden, Patrik and Shkoller and Bridges and Reich proposed the concept of multisymplectic partial differential equations (PDEs) and multisymplectic schemes that can be viewed as the generalization of symplectic schemes. As we know, there are two basic multisymplectic schemes: the Preissman scheme and the Euler box scheme. The Preissman scheme has been hot in the last years, however, little work has been done with the Euler box scheme. We know that the Preissman scheme is compact while the Euler box scheme is not. So, we want to know whether the non-compact Euler box scheme has as good a numerical performance as has the Preissman scheme? The Euler box scheme is also an important concept of the multisymplectic theory. So, we investigate the noncompact Euler box for the two equations. A 16-point scheme for the KP equation, and a 10-point scheme for the ZK equation are derived. They are both explicit schemes in sense that they do not need iteration in each time integration. Furthermore the new schemes can be easily carried out, so they may be useful tools for engineering applications to study the KP and the ZK equations. The results of back error analysis for the Euler box scheme show that the modified equations associated with the new schemes are not multisymplectic but of high order approximations to the multisymplectic PDEs with the modified structures. At last, numerical results on some waves are reported to illustrate the efficiency of the multisymplectic schemes.