Algebro-geometric Solutions Of A Kind Of Lattice Equations And N-soliton Solutions Of Kundu Equations

In recent years,algebro-geometric method and Riemann-Hilbert method have attracted more and more attention in the study of integrable equations.Algebro-geometric method can be used to study the algebro-geometric solutions of the integrable equations by using Riemann surface,hyperlliptic cures,Abel-Jacobi coordinates,Riemann theta function,inverse problem and so on.Riemann-Hilbert problem can be used to study the integrable equation based on the Riemann-Hilbert method.The main idea of Riemann-Hilbert method is construct the corresponding Riemann-Hilbert problem by using the Lax pair of integrable equation.According to the above method,the N-soliton solutions,the initial-boundary value problem and the long-term behavior of the solutions can be studied.In this thesis,the algebro-geometric solutions of a lattice equation can be constructed based on the algebro-geometric method.N-soliton solutions of Kundu equations are obtained by the Riemann-Hilbert method.In addtion,the single-soliton solutions and two-soliton solutions are discussed.The main structure of this paper is as follows:In Section 1,the generation and development of soliton theory are presented,the idea and application of Riemann-Hilbert method are briefly introduced.In Section 2,the algebro-geometric solutions of the lattice integrable equations are researched.Firstly,a family of lattice integrable equations can be constructed by using the Tu scheme and zero curvature equation.According to the Lenard's gradient sequences,an equivalent expression for the lattice integrable equations can be obtained.Then,the differential-difference equation is decomposed into the solveble ordinary differential equations by using the elliptic coordinates.The continuous and discrete flows of the lattice integrable equations are straightened with the help of Abel-Jacobi coordinates.Thus,the algebro-geometric solutions of the lattice integrable equations are obtained by using Riemann theta function and the Riemann theorem.In Section 3,the Riemann-Hilbert problem of Kundu equations is constructed.The Kundu equations are derived based on the zero curvature equation.According to the Lax pair of Kundu's equations,the spectral problem of Kundu's equations can be analyzed and the Riemann-Hilbert problem of the Kundu's equations can be constructed.Then,the solutions of Kundu equations can be transformed into the solutions of Riemann-Hilbert problem.In Section 4.the N-soliton solutions of Kundu equations are studied.Based on the Riemann-Hilbert problem of Kundu equations,the long-term behavior of the solutions and the N-soliton solutions of Kundu equations are discussed.When the jump matrix is the identity matrix,in the case of reflectionless,the N-soliton solutions of Kundu equations can be obtained.In addition,the single-soliton solutions and two-soliton solutions can be discussed.In Section 5,the whole thesis is summarized and the future research work is prospected.