Method For Solving Nonlinear Evolution Equations And Researching On The Exact Solutions

With the development of modern science and technology,there are many nonlinear problems in social and natural areas,which arouses much concern. Many nonlinear problems are usually characterized by nonlinear evolution partial differential equations.So how to construct exact solutions of the associated nonlinear equations plays an important role in understanding the nonlinear problems.This paper mainly studies the following three aspects: firstly, the nonlinear evolution equations of soliton solutions are introducted, the new Jacobi elliptic function method, deformation mapping method and the improved truncated expansion method are given, and their application in nonlinear equations. Secondly, the introduction of the Darboux transformation (Darboux) is introduced, and its applications in nonlinear evolution equations. Finally, we study the soliton equation of MKdV-Burgurs saddle-node .This paper consists of four chapter composition: the first chapter describes the general form of nonlinear evolution equations, the historical background of the soliton, soliton theory of nonlinear evolution equations solving method influence ,and it also introduces Lie Group to impact the display solution of nonlinear evolution equations.The second chapter describes several nonlinear evolution equations solving method and its application. Among them, firstly introduces the new Jacobi elliptic function method, and with Zakharov equations for example is used to illustrate it's application , also obtains the 12 elliptic equations solution of Zakharov Equations.Second, the application of deformation mapping method gives exact solutions of a class of MKdV equation,which is introduced a class of MKdV equations solitary wave solutions, periodic wave solutions, Jacobi elliptic function solutions and the power function solutions.Again, introduces the deformation mapping method in solving variable coefficient MKdV new exact solutions .Also simply introduces the variable coefficient equation MKdV solitary wave solutions, periodic wave solutions, the power function and the Jacobi elliptic function solutions.Finally, to improve the truncated expansion method, and apply the obtained equation with variable coefficients MKdV exactsolutions.The third chapter describes the Darboux transformation (Darboux) and solves the auto-Backlund transformation of JM equations .It also obtains new solutions of equation JM.The fourth chapter, through the traveling wave transformation of MKdV-Burgurs,and obtained the saddle-nodal point of MKdV-Burgurs equation.