Research On The Solution Of Nonlinear Evolution Equations And The Property Of Solutions By Several Methods

With the development of science and technology,many mathematical models have been established in the fields of mechanics,thermology,rheology,oceanography and biology,including a large number of non-linear development equations.Based on the basic idea of applied mathematics,solving the solutions of the non-linear evolution equations(equations set)and studying the nature of the solutions have important reference value for explaining the practical significance of the mathematical model.Through the unremitting efforts of many researchers,many effective methods have been proposed in the field of solving nonlinear evolution equations,such as backscattering method,Hirota bilinear method,Darboux transform,homogeneous equilibrium method,hyperbolic function expansion method and auxiliary equation method.In this paper,based on several application steps of the auxiliary equation method and the research achievement,the improved auxiliary equation method is given,and the solution and nature of several non-line-ar evolution equations(equations set)such as Klein-Gordon equation,mBBM equation,mKdV equation,fractional order mBBM equation and fractional order WBK equation set are studied.Specific research work is as follows:In the first chapter,the history of soliton theory,the auxiliary equati-on method and its achievements,as well as the main work of this paper,are briefly introduced.In the second chapter,two improved auxiliary equation methods and new conclusions of Klein-Gordon equation.1.Linear traveling wave transformation in sincos method is improved to general function transformation.Based on the above improvements,the new solutions of Klein-Gordon equation are constructed.2.The linear traveling wave transformation in the projection Riccati equation method is improved to a general function transformation.On this basis,new trigonometric and hyperbolic solutions of Klein-Gordon equation are obtained.These solutions include that obtained under traveling wave transformation.In addition,the natures of the solution are analyzed.In the third chapter,two problems are studied based on the auxiliary equation method.1.By traveling wave transformation,the solution of several non-linear evolution equations are transformed into the solution of the first elliptic equation.On this basis,based on the relevant conclusions of the first elliptic equation,an infinite sequence exact solution of the mBBM equation and the nonlinear Schrdinger equation is constructed,which consists of Riemann theta function,Jacobi elliptic function and hyperbolic function.2.The third-order linear ordinary differential equation is selected in the third step of the auxiliary equation method,and the new solutions of mBBM,nonlinear Schr(?)dinger equation and Burgers equation,which are composed of exponential function,trigonometric function and rational function,are constructed by using the solutions of the auxiliary equation.In addition,the natures of solutions are studied.In the fourth chapter,based on Jumarie modified Riemann-Liouville fractional derivative definition and nonlinear traveling wave transformation,fractional order RLW equation,fractional order mBBM equation and fractional order WBK equation set are transformed into positive integer ordinary differential equation.On this basis,by using the solutions of the first elliptic equation and Backlund,a new infinite sequence solution of fractional order RLW equation,fractional order mBBM equation and fractional order WBK equation set are obtained.In addition,the natures of solutions are studied.