Solutions To Several Soliton Equations Resorting To Hirota Bilinear Method

As an important theoretical branch of nonlinear science,soliton occupies an important position from the initial discovery to the exploration of the nature of soliton and to the jointness with our actual lives.For the basic and new subject of solving soliton equation,theoretically,it not only enriches the structure of understanding,but also promotes the common development of multiple disciplines.In practical applications,it can accurately describe the nature of scientific phenomena in nature,and also provides help for the exploration and application of new objective laws.There are many different ways to solve the soliton equation.Among them,the Hirota bilinear method has the advantages of simplicity and directness.The Wronskian solution and the Grammian solution link the soliton solution with the determinant.The Pfaff formula has more abundant properties than the determinant.The Pfaff equation can be used as a bridge to turn the Wronskian solution and the Grammian solution into a special case of the Pfaff identity.This paper mainly uses the Hirota bilinear method,Pfaff formula,Wronskian technique and other methods for research.The research content of this paper is as follows.In Chapter 3,the Boussinesq equation is considered.Using the properties of the D-operator,we can get the relationship satisfied by the coefficients when the equation is transformed into a bilinear equation.Taking the common Boussinesq equation as an example,the bilinear equation is expanded by perturbation,and the single,double and soliton solutions satisfying the Boussinesq equation can be obtained.With the help of Maple,the images of the single and double soliton solutions can be drawn.Finally,when the coefficients take different values,the change of the single soliton solution at different times is obtained.In Chapter 4,the bilinear form of the BLMP equation is obtained based on the Hirota bilinear method,and the Grammian solution of the BLMP equation is obtained with the aid of the Pfaff formula on the basis of the Wronskian solution.Through the two bilinear transformations of the BLMP equation,a(2+1)-dimensional BLMP equation with variable coefficients is obtained,and its Wronskian solution and Grammian solution are constructed using the balance method,and the obtained results are represented by Maya diagrams.In Chapter 5,we consider a new(2+1)-dimensional generalized KdV equation.Based on the bilinear form,its Wronskian solution and Grammian solution can be constructed using the balance method.In Chapters 4 and 5,when solving the Wronskian solution of the BLMP equation and the new(2+1)-dimensional generalized KdV equation,the result of the bilinear equation is transformed into the Pl cker relation.When solving the Grammian solution,bilinear equations are transformed into Jacobi identities.Through its Maya diagrams,it is not difficult to find that the identities of these two determinants are only special cases of Pfaff identities.