New Solutions Of Generalized AKNS Systems And A Time-fractional Differential Equation

It is very important to solve nonlinear partial differential equations both in theory and in practical applications.After long-term exploration and research by many scholars,soliton theory has established and developed many effective methods for solving nonlinear partial differential equations.Inverse scattering transform method is a systematic method and has been continuously developed and widely used.The core work of the inverse scattering transform is to start with the linear problem associated with the given equation and reduce the required potential to a linear integral equation.One of the greatest advantages is that it can derive a hierarchy of isospectral or nonisospectral nonlinear equations based on the appropriate linear spectral problem.The variable separation method is a common method to solve the initial boundary value problem of wave equations.Its main idea is to separate the variables from the terms of the equations so that the original equation is splited into simple and easy equations.On the one hand,this dissertation studies how to use the variable separation method to solve a new time-fractional partial differential equation with forcing term.On the other hand,this dissertation studies how to use the inverse scattering transform to solve several new generalized AKNS systems.The main tasks of this dissertation are includes the following aspects:Firstly,a nonlinear time-fractional partial differential equation with initial boundary conditions is solved by the variable separation method.As a result,some new exact solutions are obtained.Then two examples are used to show that the variable separation method can provide an effective algorithm to solve some other nonlinear time-fractional partial differential equations.Secondly,in the framework of the inverse scattering transform theory,we construct several new generalized AKNS systems with Lax integrability starting from the generalized AKNS spectral problems and their generalized evolution equations,then based on the time dependence of scattering data of generalized AKNS spectral problems,the systematic analysis of the time dependence is carried out and the exact solutions of these generalized AKNS systems are constructed.In the case of reflectionless potentrals,the obtained exact solutions are reduced to soliton solutions.Besides,the dynamic properties,spatial structures and the characteristics of singularity in the propagations of soliton solutions are studied through figures.