Nonlinear Waves, Symbolic Integration And Its Application

In this dissertation, under the guidance of mathematical mechanization and the AC=BD theory put forward by Prof. Zhang Hongqing, and by means of symbolic computation software Maple, some topics on symbolic integration and differential equations are studied, including exact solutions, rational integration, differential extension, elementary integration, Liouville Theorem, Order function and its applications to differential equations.Chapter 1 is to introduce the related development of mathematical physics mechanization and symbolic integration, emphasizing on the relation between differential equations and computer algebra.Chapter 2 and 3 are devoted to investigating exact solutions of nonlinear partial differential equations. Firstly, the basic theories of AC=BD model and C-D pairs are introduced. Then, we illustrate them in Chapter 3. We introduce two independent sub-equations with different variables, and study extended rational expansion method. Finally, illustrative examples of complexiton solutions are exhibited.Chapter 4 is to introduce the relative theory of the symbolic integration, including squarefree factorization, Hermite reduction, Rothstein-Trager algorithm and differential extensions.In the last chapter, we study basic theory of Order function and its applications. We apply Order function to nonlinear partial differential equations for obtaining many exact solutions, and study the solutions of ordinary differential equations in terms of special functions by Order function. Finally, we introduce the Liouville Theorem and its proof in the paper.