Inverse Scattering Transform And Bilinear Method For Two Generalized AKNS Equation Hierarchies

It is a worthy researching subject that constructing nonlinear differential equations and constructing multi-soliton solutions in soliton theory which all have important significance no matter in both theory and application. Inverse scattering method and bilinear method are two main methods. Between them, inverse scattering transform has a strong theory but can solve the whole equation hierarchy. Though the bilinear method is a direct algebraic method, it can not be easily uesed to bilinearize the considered equation. It is based on the existing AKNS equation hierarchies, this paper will construct two new generalized AKNS equation hierarchies and obtain multi-soliton solutions of the equation hierarchies by using the inverse scattering transform method and bilinear method. This process of solving the equation hierarchies hints the relationship between the inverse scattering transform method and the bilinear method.The main results of this paper can be summed up as: Firstly, two new and general AKNS equation hierarchies are derived in Chapter two. On one hand, the spectral parameter is genearized by introduing coefficient functions so that the corresponding linear spectral problem has more generalization. Then the first kind of new generalized AKNS equation hierarchy is constructed from this. On the other hand, the nonlinear term in the time evolution expression of the spectral is generalized by introducting coefficient functions, so as to construct the second kind of new and generalized AKNS equation hierarchy.Secondly, in Chapter three, the inverse scattering transform method is generalized and applied to solve these two new generalized AKNS equation hierarchies derived in Chapter two. Specifically speaking, the direct scattering analysis is firstly given to these two generalized AKNS equation hierarchies. Then the scattering data are obtined via the translation transformation. Finally, new multi-soliton solutions are obtained by utilizing inverse scattering transform method and the GLM equation to solve these two AKNS equation hierarchies.Finally, in Chapter four, the appropriate transformations are taken to derive the blinear forms of the two new and generalized AKNS equation hierarchies in Chapter two. An then the two generalized AKNS equation hierarhies can be solved. As a result, the same single-soliton solutions, double-soliton solutions are obtained. Also, the general experssion of the multi-soliton solutions is deduced. At the same time, it can be conclude from the solving process that there has certain relation between the inverse scattering transform method and bilinear method. It is difficult to use bilinear method to solve equation hierarchies, but the solution obtained by the inverse scattering transform method can provide hint for bilinear method.