The Study Of The Exact Solutions And Conservation Laws For Two Classes Nonlinear Evolution Equations

Nonlinear evolution equations are widely used in the fields of mathematics,physics,chemistry,fluid mechanics,vibration mechanics,celestial mechanics,biol-ogy,ecology,finance and other natural science and social science.A large number of researchers in mathematics and other fields had chronically devoted to the study of non-linear evolution equations.On the basis of fruitful results,they promoted the sustain-able development of nonlinear problems continually and provided an important guiding significance and application value for the production,life and scientific research of hu-man society.In this paper,we study the analytical solutions and conservation laws of two nonlinear evolution equations by using the classical Lie symmetry method.Lie symmetry analysis method is one of the most important theories for solving nonlinear partial differential equations and computing conservation laws.This paper firstly introduces the background and basic theory of the method,and gives a series of key theorems and formulas which are used in the follow-up study.The second part of this paper focuses on studying the Qiao equation.The in-finitesimal generators of the Qiao equation are obtained by using the Lie symmetry analysis.According to several linear combinations of infinitesimal generators,we ob-tain the corresponding reduced equations and group-invariant solutions.Based on the group-invariant solutions,the non-group-invariant solutions of the original equation are obtained by using a new construction method.In order to show the difference between group-invariant solutions and non-group-invariant solutions,a lot of figures are given by numerical simulation.According to the special symmetry,the iterative solutions of the original equation are also obtained.Finally,we get the nonlocal conservation laws of the Qiao equation corresponding to each generator.In the third part of this paper,the Gardner-KP equation is studied.Based on the Lie symmetry analysis,we obtain some group-invariant solutions and iterative solu-tions of the equation.Since the Gardner-KP equation has three independent variables,the reduced equations are still partial differential forms.It is also difficult to find the solutions of the original equation.Those reduced equations are transformed into a variety of ordinary differential equations by symmetry analysis secondly.The power series solutions of the ordinary differential equations are obtained by using the power series method,and the convergences of those solutions are proved.Corresponding to each generator of the first reduction,the nonlocal conservation laws are obtained in this chapter.At the end of this paper,the research methods,processes and results are summa-rized,and the goals and directions of the next research work are put forward.