New Solutions Of WBK Equation?AKNS Equation And Fractional Advection-Dispersion Equation

With the development of soliton theory and science and technology,in recent years,constructing the new exact solutions of soliton equations has become a priority among priorities of the soliton theory.In many methods for solving soliton equations,Hirota bilinear method plays an important role,is the focus of many scholars' attention,Hirota bilinear method belongs to constructive solving,the advantage of the constructive method compared with other methods is that it does not depend on the corresponding Lax pair or spectral problem.It is because that this constructive method is simple and intuitive,the research upsurge of scholars has been inspired.In recent years,the fractional problem has aroused wide spread concern and the fractional order nonlinear partial differential equations has become a hot research topic.In this dissertation,on the one hand,Hirota bilinear method is used to construct single-soliton solutions,double-soliton solutions,three soliton solutons and expressions of N-soliton solutions of generalized WBK equations and generalized AKNS equations.On the other hand,tightly around the related knowledge of fractional calculus,fractional derivative,this dissertation constructs the new exact solution with initial and boundary conditions of variable coefficient of time fractional advection-dispersion equation.The main results of this dissertation are summarized as follows:Firstly,in the third and fourth chapters,Hirota bilinear method is improved and extended to construct single-soliton solutions,double-soliton solutions and three soliton solutions of the generalized WBK equations and the generalized AKNS equations,from which the expressions of N-soliton solutions are summarized.This benefits from the successful transformation of the generalized WBK equation and the generalized AKNS equation.In solving the generalized WBK equations and the generalized AKNS equations,the key is to find their bilinear forms by a series of effective transformations.And then new soliton solutions are obtained.Secondly,in the fifth chapter,by using the separation variable method and the properties of Mittag-Leffler function,we obtain a uniform expression for the exact solution,which satisfies a certain initial and boundary conditions,of a class of time-fractional advection-dispersion equations with variable coefficients.Furthermore,by considering the special examples of the advection-dispersion equations and the initial and boundary conditions,a new solution is obtained,This provides an important reference value for solving fractional order nonlinear partial differential equations.