Symplectic And Multisymplectic Scheme Of The Zakharov System

Any physical processes which dissipation can be neglected can be expressed as theform of Hamilton system which can maintain constant symplectic structure. This is ofgeneral applicability in the nature. That means most of soliton equations can be expressedas Hamilton system.Symplectic method which can maintain the symplectic structure of finite-dimensionalHamilton system has many advantages. There is a natural way to extent the theory of thesymplectic to the infinite-dimensional Hamilton system. First, discrete in the space; Secondapply the symplectic method to the exported finite-dimensional Hamilton system. Usingthis method to solve Hamilton PDEs has limitation. Symplectic conservation which is theaverage in the space, cannot represent the structural conservation in the strict sense. Toovercome this limitation, Bridges and Reich introduced the concept of multi-symplecticHamilton system. Multi-symplectic scheme has a strict local conservation law, that is themulti-symplectic conservation. Besides local energy and momentum conservation laws canbe derived from this scheme. These are the good features of multi-symplectic scheme.One example of infinite dimensional Hamilton system Zakharov equations which thispaper studied is an important model. In section one of the first chapter, I will introduce theZakharov system. To make this paper to be a complete system, I will give a comprehensiveoverview about the Hamilton system, symplectic method and multi-symplectic method inthe second chapter—Hamilton system. This helped naturally study into the third chapterand the fourth chapter.In the third chapter, I will give the related symplectic scheme, conservation laws forthe Zakharov system, and properties of the numerical method with numerical experiments.Similarly, in the fourth chapter, I will construct multi-symplectic Preissman scheme andgive the conservation laws for the Zakharov system with numerical experiments.