The Matrix Exponential Solutions To The Isospectral AKNS Hierarchy Via The Triplet Matrix Inverse Scattering Transformation

The systematic method of solving soliton equations is inverse scattering transform.Up to now,all the classic methods such as Hirota method,Wronski technique,Darboux transform,Backlund transform and nonlinearization of the Lax pairs are related to it.The primary advantage of this approach is that it is not only an effective way of differential equations,differential-difference equations and difference equations,but also the whole soliton equation hierarchies.In this paper,the matrix exponential solutions to the isospectral AKNS hierarchy are obtained via the triplet matrix inverse scattering transformation.Concretely,the classic Gel'fand-Levitan-Marchenko(GLM)equation of AKNS hierarchy is revised and triplet matrix(A,B,C)is introduced first.Then the potentials of the AKNS equation are recovered as the matrix exponential form.They are proved to be the solutions of the second order AKNS equation.As the application of these solutions,the one-soliton,two-soliton,and three-soliton solutions are derived by choosing different kind of matric in them.Finally,this method are promoted to the AKNS hierarchy by mathematical induction.The classic inverse scattering transformation are simplied and more kinds of solution are gained.