Several Constructive Methods For Exact Solutions Of Nonlinear Evolution Equations

With the rapid development of science and technology, nonlinear science has beenwidely used in all fields of natural sciences. Because of the nonlinear partial differentialequations and other disciplines are closely linked, therefore, solving nonlinear partialdifferential equations and studying the properties of their solutions have attached muchattention.One of the primary tasks in discussing nonlinear partial differential equations is toobtain exact solutions of the equations, but it is not easy to solve nonlinear partialdifferential equations due to the equations theirself are more complex. Although someexact solutions of a lot of nonlinear partial differential equations are obtained, thesolving methods also have their own skills. At present, the methods for solving partialdifferential equations are Painlevé analysis method, Hirota bilinear method, Tanhfunction method, B cklund transform, Inverse scattering method, Auxiliary equationmethod, etc.Based on the theoretical basis of partial differential equation, this master’s thesiswith the helps of Maple and Mathematica softwares use Painlevé analysis method,Auxiliary equation method, Exp-function method to study the exact soltions ofnonlinear evolution equations. As a result, some new and meaningful results areobtained.The content of this paper mainly includes four aspects:1, Introducing the originand development of the soliton and several methods for solving nonlinear evolutionequaitons such as Painleve analysis method, Auxiliary equation method, and so on.2,using Painlevé analysis method to study the WZ equations, dispersive long-waveequations and variable-coefficient BK system. As a result, the conclusion that theconsidered equations possess Painlevé propeties under certain conditions is reached. Atthe same time, the Au-B cklund transform are drived and the Evolutionary behaviorcharacteristics are discribed.3, Some exact solutions of the KP equation is obtained bythe generalized auxiliary equation method and spatial structure of the obtained solutionis discribed.4, The Exp-function method is generalized through the design one genaralized ansatz solution. In the application of the generalized method, one solitarywave with arbitrary function of Burger equation is obtained. The results show that thegeneralized ansatz solution can be used to obtain exact solutins with new form.