Research On The Algebraic Soliton Solutions And Symmetric Reduction Of Nonlinear Equations

In recent years,the study of soliton phenomenon has become an important research direction of non-linear science.Soliton theory has also been widely applied to magnetohydrodynamics,biology,quantum mechanics,optics and many other fields.Solution of soliton equation has also become the research focus of scientists at home and abroad.Among them,algebraic geometry method and symmetric reduction and solution of non-linear equation can describe many physical phenomena deeply,and have important application value.In this paper,the algebraic and geometric solutions of soliton equations and the symmetric reduction of nonlinear equations are mainly studied.The specific contents are as follows:In section 1,this paper briefly describes the development of soliton theory,the integrability of integrable systems,and the construction and application of several solutions of solitons.In section 2,it mainly introduces the basic knowledge applied in the process of solving algebraic and geometric solutions,such as Riemann surface,function and Abel differential.In section 3,through the given lie algebra,a set of discrete spectral operators is constructed.The equation clusters of integrable systems can be obtained by using the zero curvature equation.Then,according to the hyperelliptic curve,the Riemann surface is introduced,so that a set of Abel-Jacobi coordinates can be constructed,and the corresponding flow is straightened on the Riemann surface.Finally,the algebraic soliton solution of the equation is obtained by using the inversion theorem and the θ funnction.In section 4,starting from a set of new lie algebras and given spectral problems,three solvable ordinary differential equations can be derived by using zero curvature equation and Lenard sequence.Next,elliptic coordinates are introduced,and the corresponding flow can be straightened in Abel-Jacobi coordinates.Finally,the algebraic and geometric solutions of the equation can be obtained by using Riemann surface and the θ function.In section 5,Clarkson-Kruskal direct method is an algebraic method that does not involve group operation to solve nonlinear partial differential equations and does not need to solve complex initial value problems.By applying Clarkson-Kruskal direct method and using the corresponding rules,the symmetric reduction of nonlinear coupled long water wave equation is obtained.