Several Nonlinear Evolution Equations Of The Exact Solution

In recent decades, the research on the nonlinear evolution equations has becomean important branch in nonlinear science. And it is playing an increasingly importantrole in modern science. Nonlinear evolution equations can not only describe manynatural phenomenon, dynamic process and the change rule of things, but also canprovide important enlightenment in many applications. It is very important to relevantscience and technology. Along with the development of computer symbol calculation,it opened a new chapter of solving nonlinear evolution equations, many more directand effective algebraic methods are emerged, such as: the homogeneous balancemethod, F-expansion method, tanh-function expansion method, Jacobi elliptic functionexpansion method, G′/G-expansion method and the high-order subsidery equationmethod etc. Among them, the G′/G-expansion method with its direct, concise, basic,the characteristics of the effective cause many scholars’ attention, it not only establishsrelation between the exact solutions of the nonlinear evolution equations with thesolutions of two order linear differential equation G′′+λ G′+μG=0, and the exactsolutions of the nonlinear evolution equations obtained by using this method containmore free parameters. Therefore, the G′/G-expansion method and the high-ordersubsidery equation method are hired to study the exact solution of the variablecoefficient soliton equation, time-delay differential equation and the generalized thefractional power equation, and the equations above have been widely used inmathematics, physics, control theory, and other fields.In this paper, the extended G ’/G-expansion method is used to derive the exactsolutions of the variable coefficient KdV equation, mKdV equations and theapproximate solution of a few time-delay partial differential equations, the influenceabout the time-delay parameters on the approximate time-delay equation for the exactsolution is discused; and then the high-order subsidery equation is used toinvestigation the exact solutions of the generalized time-delayed Burgers-Fisher equation with positive fractional power, and analyzed the state of these solutions.Chapters and the content of this article:In the first chapter, firstly, I summarize the theoretical and development oflooking for the exact solution of the nonlinear evolution equationsand then G ’/G-expansion method and the extended G ’/G-expansion method are introduced, andfinally the main content of this paper is presented.In the second chapter, G ’/G-expansion method is used to solve the exactsolutions of variable coefficient KdV equation and variable coefficient mKdVequations. It proves that when we want to solve the exact solutions of some complexnonlinear evolution equations, we can use a few auxiliary equation to find out theexact solution of the nonlinear evolution equations through the appropriatetransformation.In the third chapter, G ’/G-expansion method is applied to solve theapproximate solution of delay partial differential equations. Firstly, time-delay B-BBMequation, time-delay KdV equation and time-delay KPP equation are approximated,and then find out the solitary wave solutions of these approximated time-delayequation, including periodic wave solution and the rational function forms of travellingwave solutions. Finally the effect of the time-delay parameters on approximate time-delay equation is discussed.In the fourth chapter, the high-order subsidery equation is used to derive theexact solutions of the generalized time-delayed Burgers-Fisher equation with positivefractional power and its special types. And through the expression of the solution, thequestion of time-delay effect on the soliton speed and width is researched.In the fifth chapter, the article is summarized and discussed.