Some Investigation On Solving Nonlinear Evolution Equations Under The Model Of AC=BD

In this dissertation, some topics are studied with the idea of mathematical mechaniza-tion and AC=BD model, including the exact solutions of nonlinear evolution equations, and developing the local discontinuous Galerkin methods to solve the numerical solutions of nonlinear evolution equations.In Chapter 1, we introduce the history and development of soliton theory, mathemat-ical mechanization, solutions of the mathematical and physical equations, discontinuous Galerkin method and the local discontinuous Galerkin method, as well as the main work of this dissertation.In Chapter 2, intoduces the main content and idea of AC= BD model theory and C-D pair, as well as the mechanized construction of transformation of differential equations under the instruction of the AC= BD theory are presented. Furthermore, we give some applications on solving numerical solutions under BD=AC model.In Chapter 3, based on the theories in Chapter 2, and the idea of solving nonlinear evolution equations, algebraic method, algorithm reality, mechanization, we improve the generalized compound Riccati equations rational expansion method and extend the gen-eralized sub-equation rational expansion method. Furthermore, more exact solutions of nonlinear evolution equations are obtained.In Chapter 4, the LDG schemes for three classes of BBM-Burgers equations is de-signed. We prove L2 stability and a cell entropy inequality for the BBM-Burgers equation, generalized BBM-Burgers equation with dispersive term and Rosenau-Burgers equation, and we give an error estimate for the nonlinear cases of the BBM-Burgers equation. Nu-merical examples are shown to illustrate the capability of this method.In Chapter 5, we design the LDG methods for a class of the higher order nonlinear Schrodinger equations. Our schemes extend the previous work of Xu and Shu solving NLS equations on LDG method. The energy stability of the LDG methods is proved for any orders of accuracy for the general nonlinear cases. Numerical examples for the three order Schrodinger equation and four order Schrodinger equation are presented. The numerical results illustrate the accuracy and capability of the methods.