Extensions Of Super KdV Equation And AKNS Equation And The Exp Method For MNW Equation

As the main component of nonlinear dynamics,soliton plays an important role in solving non-linear evolution equations.Exp-function method and inverse scattering transformation method are two effective methods to solve non-linear partial differential equations.One of the simplest forms of the exp-function method(the simplest exp-function method)can solve the middle expression expansion problem to some extent.The inverse scattering transformation is divided into two parts: the direct scattering and the inverse scattering.The direct scattering part uses the similarity transformation of linear spectral problem,the translation matrix and so on to obtain the exact solution of the equation;the inverse scattering part determines the scattering data by solving Riemann problem or integral equation,and the soliton solution can be obtained in the case of no reflection potentials.On the one hand,the super-KdV equation and AKNS equation are integrably extension from the perspective of embedding coefficient function and connecting mixed spectrum,and the inverse scattering method is extended to the super-KdV equation with variable coefficient and AKNS equation with mixed spectrum.On the other hand,the simplest exp-function method is applied to the high-order MNW equation to solve its exact solution in two cases.The main work of this paper is as follows:Firstly,the super-KdV equation with arbitrary variable coefficient is obtained by embedding the coefficient function.By using Kulish and Zeitlin's method,the inverse scattering transform is extended to the super-KdV equation with variable coefficient.Based on the one dimensional Glassmann algebra,the exact solution of super-KdV equation is obtained through scattering analysis,and the exact solution is reduced to soliton solution in the case of no reflection potentials.Inspired by the inverse scattering transform of super-KdV equation,a method of solving for the inverse scattering transform of the supersymmetric KdV equation is presented.Secondly,by extending the spectral parameter of AKNS equation corresponding to linear spectrum problem,AKNS equation is integrably extended and a new mixed spectrum AKNS equation is obtained.In the case of time-varying spectral parameters,the inverse scattering transform is extended to the mixed spectrum AKNS equation.In the case of no reflection potential,the exact solution obtained by inverse scattering transformation is reduced,from which the N-soliton solution of mixed spectrum AKNS equation is obtained,and the dynamic evolution of the single soliton solution is simulated.Lastly,the MNW equation of high-order is solved by the simplest exp-function method,and two exact solutions are obtained.The process shows that the simplest exp-function method solves the so-called middle expression expansion problem to a certain extent and provides a simpler but effective mathematical tool for constructing exact solutions of nonlinear evolution equations in some research fields including the fluid field.