Research On Exact Solutions,Integrability And Symmetry-preserving Discrete Scheme To Nonlinear Differential Equations

The research on the explicit solution and integrability of nonlinear differential e-quation play an important role in scientifically explaining of the corresponding physical phenomenon and engineering application.Chapter 2 and 3 focus on the binary Bell polynomials and Riemann theta functions,and extend them to GKdV equation and(2+1)-dimensional Boussinesq equation.Their bilinear representation,Backlund transformation,Lax pair,infinite conservation laws are obtained.Then the Riemann theta function periodic wave solutions and soliton solutions of the equations are constructed.In addition,the asymptotic properties of the periodic wave solutions are analyzed in detail,and which strictly shows that the solutions can be reduced form the soliton solutions under the small amplitude limit.In chapter 4,by extending the Lie symmetry analysis method to coupled Burger equation and high-order Beam equation,their symmetry reductions and group invariant solutions are obtained.Then exact power series solutions are derived,and its convergence is also proved.The investigation of integrability for a equation can be viewed as a pre-test and the first step of its exact solvability,so this dissertation investigate the Painleve test for nonlinear differential equations in detail.The WTC approach is systematically introduced and promoted,then the necessary conditions which coupled Burger equa-tion and variable coefficients Bogoyavlenskii's breaking solition equation possess the Painleve property is obtained.Then the auto-Backlund transformation is also success-fully constructed.Finally,a method to obtain the discrete scheme which preserves the symmetries of differential equation is introduced,and then apply it to high order Beam equation,(2+1)-dimensional diffusion-convection equation and generalized KP-MKdV equation.Corresponding symmetry-preserving discrete scheme which inherit Lie point symmetries of continuous equations are obtained.