Research On The Exact Solutions Of Several Nonlinear Development Equations

The nonlinear problem is a common problem in modern science,so we can apply common nonlinear models in the mathematical research to describe a large number of objective problems.Because nonlinear evolution equations possess physical background of reality,so the gain of their exact solutions is not only the need to solve practical problems,but also promote the progress of other related fields,which has great application value no doubt.In this paper,Lie group method and auxiliary equation method are used to study several kinds of nonlinear evolution equations.These nonlinear evolution equations can be generalized into high-dimensional ordinary coefficient partial differential equations or equations set and variable coefficient partial differential equations.They are successively the generalized(2+1)-dimensional shallow water wave equation,(3+1)-dimensional potential-YTSF equation,(2+1)-dimensional dissipative long water wave equation set,variable coefficient GKP equation and variable coefficient Zhiber-Shabat equation.Finally,some new exact solutions and conservation law of these equations are obtained.In the first chapter,the generalized(2+1)-dimensional shallow water wave equation is transformed into ordinary differential equation by using traveling wave method,and then this ordinary differential equation is solved by using auxiliary equation.In this way,the initial problem is simplified,and more abundant new exact solutions are obtained.In the second chapter,the symmetry of the(3+1)-dimensional potential-YTSF equation is obtained by direct reduction method,and the corresponding reduction equation can be gotten.Then the exact solutions of the target equation can be promoted which have played a certain role in promoting the results of the equation in the existing literature.Finally,the conservation law of the equation is obtained by using symmetry.In the third chapter,the symmetry of(2+1)-dimensional dissipative long wave equation is obtained by Lie group method.The original equation can be reduced by symmetry and then solve the reduction equation,so the exact solutions of the target equation can be promoted.Finally,the conservation laws of(2+1)-dimensional dissipative long wave equation is gotten by symmetry.The obtained exact solutions not only improvethe structure of the solutions of(2+1)-dimensional dissipative long wave equation,but also provide new possibilities for the study of this equation.The fourth chapter constructs the form of solution of variable coefficient GKP equation combining the new solutions of different auxiliary equations and B?cklund transform.Then,Maple software is used to determine the formal solutions and link some conclusions of auxiliary equations to obtain new infinite sequence exact solutions of variable coefficient GKP equation.In the fifth chapter,it's be found that variable coefficient Zhiber-Shabat equation possess Painlevé integrability by using Painlevé analysis,and naturally a transformation formula that reflects the relationship between the solutions of the variable coefficient ZS equation is given.Then Jacobi elliptic function is selected as an auxiliary function according to the homogeneous balance principle,finally some new different types of the equation are obtained.