Study On Exact Solutions Of Some Higher Dimensional Nonlinear Evolution Equations

The nonlinear evolution equations are closely related to the nonlinear phenomena in the fields of astronomy,biology,medicine,mechanics,physics and so on.Therefore,study-ing the exact solutions of the nonlinear evolution equations plays an important role in the development of nonlinear science.At present,although there are many feasible methods for solving the exact solutions of nonlinear evolution equations,due to their complexity and uniqueness,so far there is no one method that is universally effective.In this paper,we mainly use Hirota bilinear method,positive definite quadratic function method and KP reduction method,for(3+1)-dimensional generalized BKP equation;(3+1)-dimensional Jimbo Miwa(JM)equation and(2+1)-dimensional Boiti Leon Pempinelli(BLP)equation to obtain some exact solutions,and then make a certain analysis of the nature of the solutions.The main contents are as follows:The first chapter introduces the general situation and research significance of the nonlinear evolution equations,and summarizes several main research methods for solving the exact solutions of the nonlinear evolution equations.In the second chapter,we use the simplified Hirota bilinear method to obtain the 1-soliton solution,2-soliton solution and 3-soliton solution of the(3+1)-dimensional gen-eralized BKP equation,and then summarize the general form of its N-soliton solutions.Next,the lump solution of the reduced generalized BKP equation is obtained by the pos-itive definite quadratic function method.Finally,using the generalization of the positive definite quadratic function method,the lump-kink solution of the reduced generalized BKP equation is obtained,and the dynamics of the solution is analyzed.In the third chapter,we mainly use the KP reduction method to successfully obtain the rational solution of the(3+1)-dimensional JM equation,and further obtain the lump solution of the equation by analyzing the parameters.The fourth chapter mainly constructs the N-soliton solutions of the(2+1)-dimensional BLP equation by using the Hirota bilinear method,and analyzes the dynamics of the solution.Then we will apply the KP reduction method to the BLP equation,and the Grammian-type determinant solution with different mathematical structures and rational solution are derived.