Trial Equation Method Of Research, The Exact Solution Of Certain Types Of Equations

With the development of modern science and technology, physicists and math-ematicians have been paying more and more attention to explore the exact solutionsof nonlinear evolution equations. Since the exact solutions of nonlinear evolutionequations could o?er good explanations for so much physical phenomenon, theyhave been widely applied in the physical fields of chemistry, biology, fiber opticalcommunications, ?uid mechanics, plasma physics, quantum field theory and so forth.This paper aims at turning nonlinear evolution equations into elementary in-tegral forms by using trial equation method and finding the exact travelling wavesolutions of the equations by using polynomial total discriminant system. The exacttravelling wave solutions include rational function solutions, solitary wave solutions,polynomial-type Jacobi elliptic function periodic solutions and so on.This paper include three parts:Chapter 1 simply introduces the ideas of the mathematics mechanization, soli-ton theory and the achievements of some scholars at home and abroad in this field.Chapter 2 mainly introduces several methods for solving nonlinear evolutionequations which are Ba¨cklund transformation, homogeneous balance method, ellip-tic function expansion method, first integral method, index expansion method andWu-di?erential elimination method.Chapter 3 introduces the main ideas and procedures of trial equation methodand the specific application of solving and correcting kawachara equation!Brethertonequation and finding exact solutions of these equations by using trial equationmethod.