The Study Of The Integrability And Symmetry-preserving Discrete Scheme For The Nonlinear Differential Equations

In this paper, the integrability problems of the Boussinesq equation, which in-clude:Backlund transformations, Lax pairs, infinite conservation laws, are systemat-ically investigated using Bell polynomials method and Riemann theta function peri-odic wave solutions method. Meanwhile, the exact solutions of KdV-Sawada-Kotera-Ramani equation are obtained by using Lie symmetry analysis method. Finally, we introduce a method to construct the discrete schemes of the nonlinear differential equa-tion which preserve Lie point symmetries of the original equation.In the first chapter, the research background and research significance of soliton theory and Lie group are introduced, then the main works of this paper are also illus-trated.In the second chapter, we systematically construct the bilinear representation, Backlund transformation, Lax pair of the Boussinesq equation by using the proper-ties of the binary Bell polynomials. Moreover, the infinite conservation laws of the equation are also constructed by virtue of its Lax equation.In the third chapter, we obtain the Riemann theta function periodic wave solu-tions of the Boussinesq equation by using multidimensional Riemann theta functions. We analyze the relations between the periodic wave solutions and soliton solutions by making a limiting procedure, which strictly show that the periodic solutions tend to the known soliton solutions under the small amplitude limit.In the fourth chapter, by using Lie symmetry analysis method, the exact analytic solutions of the KdV-Sawada-Kotera-Ramani equation are derived. Then, the conver-gence of power series solution is also analyzed.In the fifth chapter, taking generalized Benjamin equation and third-order Burg-ers equation as examples, we introduce a method to obtain the discrete scheme which preserve the symmetries of nonlinear differential equation. The difference of this way to others is that we construct the symmetry-preserving discrete scheme of the potential system instead of the original equation.In the sixth chapter, we give some conclusions and prospects of this paper.