Researches On Three Problems Of Inverse Scattering Transform And Exponential Funtion Method

In soliton theory,inverse scattering transform method and exponential function method are important methods developed in recent years for solving nonlinear partial differential equations.The inverse scattering transform method is firstly to obtain scattering data at the time t=0 through direct scattering,then to determine the time dependence of the scattering data varying with time t,finally to obtain solutions of the given nonlinear partial differential equations by the reconstruction of potentials.The exponential function method is firstly to suppose the rational ansatz of exponential functions of a given nonlinear partial differential equation,then to determine the values of undetermined parameters in the ansatz through balancing the highest order derivative term with the highest nonlinear term and collecting the coefficients of exponential function with same power.On the one hand,this dissertation studies the issue of how to generalize the inverse scattering transform method to two new nonisospectral AKNS equations,the spectral parameters of which depending on sine function and rational expression,respectively.On the other hand,this dissertation studies the issues of how to solve the "middle expression expansion" problem appeared in the process of using the exponential function method and how to determine the simplest ansatz of nonlinear lattice equation when using exponential function method.The main work of this dissertation includes:Firstly,two new nonisospectral AKNS equations,the spectral parameters of which respectively depending on sine function and rational expression,are derived.Then,the inverse scattering transform method is generalized to solve the two AKNS equations,respectively.As a result,new exact solutions and new N-soliton solutions of two nonisospectral equations are obtained.Furthermore,local spatial structures and dynamic evolution behavior of some obtained solutions are simulated.Secondly,a direct algorithm of exponential function method is presented by giving new form of ansatz of the exponential function method.As two examples of the algorithm,we apply it to the KdV equation and the Jimbo-Miwa equation.The examples show that our algorithm can solve,to a greater extent,the problem of "middle expression expansion" appeared in the process of using the exponential function method.Finally,a theorem and its proof of the simplest ansatz of the exponential function method for solving a class of nonlinear lattice equations with variable coefficients by defining the positive power and the negative power of rational exponential function.Applying the theorem,we can omit the process of determining the ansatz by banlancing the highest derivative term and the highest nonlinear term of the given equation.Thus,the exponential function method for solving the nonlinear lattice equation is improved.As an example,we use the simplest ansatz to solve the variable-coefficient mKdV lattice equation.The effectiveness of the simplest ansatz is shown in the example.