Hirota Bilinear Method And Study On Complex Solutions Of Nonlinear Evolution Equations

The research contents of Soliton theory are mainly divided into the following categories.To establish a more accurate mathematical model;to give the general solution mode;to give the integrability of differential systems;symmetries and conserved quantities of differential systems;Hamiltonian structure of differential systems;dynamic behavior of differential system;to study the algebraic geometry properties of the solution;the nature of stability and practical significance.Based on the function transformation method,hiroda bilinear method,trial function method,variable separation method and auxiliary equation method,this paper studies the solutions and properties of nonlinear evolution equations such as(3+1)—dimensional Jimbo—Miwa—like equation,(4+1)—dimensional BLMP equation and(3+1)— dimensional BKP equation.The specific research work is as follows.In the first chapter,the development history of soliton theory is briefly reviewed,including the discovery of soliton,several solving methods,the achievements and the main work of this paper.In the second chapter,based on the hiroda bilinear method,the(3+1)—dimensional Jimbo—Miwa—like equation and(3+1)—dimensional variable coefficient BKP equation are transformed into bilinear differential equations.On this basis,by using the trial function method,new compound solutions of nonlinear evolution equations are obtained,including the compound solutions of trigonometric function and exponential function,hyperbolic function and trigonometric function and exponential function.Then we use computer algebra system to analyze the properties of the solution.In Chapter 3,based on the method of separating variables,we study the construction of new solutions for the(4+1)—dimensional variable coefficient BLMP equation and(3+1)—dimensional soliton equation.Results two new high dimensional solutions are expressed by arbitrary functions.According to the arbitrariness of any function,many new solutions are obtained.Step 1,the nonlinear equation is can transformed into bilinear form by function transformation;step 2,a suitable function transformation is given,and the partial differential equations are obtained by combining the variable separation method;step 3,by solving the partial differential equations,the arbitrary function representation solutions of(4+1)— dimensional variable coefficient BLMP equation and(3+1)—dimensional high-dimensional soliton equation are obtained respectively.In addition,the symbolic computing system Mathematica is used to analyze the interaction of solutions.In Chapter 4,based on hiroda bilinear method,the(3+1)—dimensional KP equation with variable coefficients is transformed into bilinear form.Then,by using the method of combining function transformation with two first elliptic equations,a compound solution of(3+1)—dimensional KP equation with variable coefficients is constructed,which consists of trigonometric function,hyperbolic function,exponential function,Jacobi elliptic function and Riemann ? function.It includes double periodic solution,double soliton solution and the combination of soliton solution and periodic solution.Finally,the interaction of understanding is analyzed.