Research On The B(?)cklund Transformation And Exact Solutions Of Some Nonlinear Evolution Equations

Nonlinear evolution equations are closely related to nonlinear phenolmena in physics,medicine,biology,finance,aerospace and other fields.Therefore,the research on the exact solution,integrability and B(?)cklund transformation of nonlinear evolution equations is of great significance to the development of science.In this paper,several high-dimensional nonlinear evolution equations are studied.Based on Hirota bilinear method,auxiliary equation method,Bell polynomial theory,separation of variables method and test function method,with the help of Mathematica system,the exact solutions of several equations are constructed.In addition,the B(?)cklund transformation,Lax pair and infinite conservation law of several equations are studied by using Bell polynomial theory.The first chapter briefly introduces the background,significance,content and methods of research in this field.In addition,the research methods and main research contents of this paper are described.In the second chapter,the multi-soliton solutions of the(2+1)-dimensional variable coefficients fifth-order Kd V model with forcing terms are constructed by using the method of function transformation and bilinear derivative.With the help of Mathematica system,the influence of forcing terms and model coefficients on the soliton solutions is analyzed through the three-dimensional diagrams and contour maps of single soliton solutions,double soliton solutions and triple soliton solutions.In addition,based on Hirota bilinear method,the(3+1)-dimensional K-S equation with variable coefficients is studied.Firstly,the homoclinic breathing wave solution of the equation is obtained by means of the extended homoclinic breathing test method.Based on the arbitrariness of the coefficient of the equation,the homoclinic breathing waves with different structures are obtained by selecting appropriate numerical values.Then,the rogue wave solution of the equation is derived by taking the limit of the periodic homoclinic breathing wave solution.Finally,the special high-order polynomial is selected as the test function,the first-order and second-order strange wave solutions of the equation are constructed,and the properties of the solutions are analyzed by drawing with mathematical software.In the third chapter,based on Hirota bilinear method and Cole-Hopf transformation,the following conclusions of(3+1)-dimensional HSIl equation are obtained.(1)Firstly,the N-soliton solution is obtained by using the small parameter expansion method.Then,the higher-order respiratory solution,higherorder Lump solution and mixed solution of the equation are constructed by appling the complex conjugate relation of parameters and the long-wave limit to the N-soliton solution.(2)By means of the test function method,the mixed solutions consisting of breathing waves and rogue waves are obtained.New interaction phenolmenas of these solutions are obtained by selecting different parameters.In the fourth chapter,the B(?)cklund transformation,Lax pairs and infinite sequence solutions of(3+1)-dimensional gKDKK equation and(4+1)-dimensional Fokas equation are studied.(1)Firstly,based on Bell polynomial theory,the bilinear form,B(?)cklund transformation,Lax pair and infinite conservation laws of the(3+1)-dimensional g KDKK equation are constructed.Several new exact solutions are given by means of test functions and separation of variables method,including Lump solution,breather solution,mixed solution,half periodic kink solution and compound solutions composed of exponential function,trigonometric function,hyperbolic function,rational function and Jacobi elliptic function in various forms.Then,with the aid of the combination method of Riccati equation and Bernoulli equation,the compound exact solutions composed of rational function,exponential function,trigonometric function and hyperbolic function are constructed.Finally,the interaction of solutions is analyzed by selecting appropriate parameters.(2)Firstly,the(4+1)-dimensional Fokas equation is transformed into ordinary differential equation by using two kinds of function transformations.Then,with the help of Riccati equation and the first kind of elliptic equation,the new solutions of infinite sequence composed of Jacobi elliptic function,Riemann ? function and exponential function of(4+1)-dimensional Fokas equation are constructed.Finally,by selecting appropriate parameters to plot,the properties of the solution are analyzed.In summary and prospect,this paper is briefly summarized,and the future contents worthy of deep thinking and research are prospected.