Applications Of The Exp Function Method And The Symmetries For Two Nonlinear Equations

Nonlinear evolution equation plays a very important role in the field of mathematics and physics, Therefore, research on the method of solving nonlinear evolution equations becomes very important topic in the soliton theory. Although many people have done a lot of research in this area,but because of the complexity of nonlinear evolution equation, exact solution method has no general unified.Fortunately,there is still exists some exact solutions of proven construction methods in the soliton theory,such as Jacobi elliptic function method, homogeneous balance method, complete approximation, the trial function method, hyperbolic function method, the reductive perturbation method, the auxiliary equation method, Fexpansion method, Exp function method, bilinear transformation method and Darboux transformation method,etc.This paper mainly using the Exp function method and the auxiliary equation method for solving nonlinear equations, and studies the nonlinear equations of the symmetry and Lie algebra structure.The first chapter is the introduction part, It mainly introduces the development of soliton theory, the research status of nonlinear evolutionequation solving and symmetrical. And we introduced the main work of this article in the end.In the first section of the second chapter, the basic idea of Exp function method for solving was introduced; the second and the third section respectively on the literature [10] and [19] in the two nonlinear elastic rod wave equation using the Exp function method and with aid of Mathematica software to solve the exact solution of the wave equation,and finally we got the exact solutions of the two elastic rod.In the first section of the third chapter, It introduce the solving process of the auxiliary equation method. In the second,the third and the fourth section,respectively by the traveling wave solution assumption in reference[29] three five order nonlinear equations into ordinary differential equation and with the aid of the auxiliary equation method and Mathematica software, finally got three exact solutions of five order nonlinear equation is more abundant.In the first section of the fourth chapter, the method of determining the study of nonlinear equation symmetry undetermined coefficient method and the Lie algebra structure was introduced. In the second section and the third section,we gives literature [30] and composed of these symmetric Lie algebra structure, accurate reduction and ordinary differential equations using symmetry reduction method the corresponding. Through some simple symmetric two nonlinear equationsdirectly hypothesis method.In the fifth chapter, we made summarized about the work during the period of graduate, and we also made a plan for work in the future.