Generalized Matrix Exponential Solutions to the AKNS Hierarchy

Received Dec 4, 2017; Revised Jan 8, 2018; Accepted Jan 14, 2018

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Many nonlinear models are studied and shown to possess hierarchies, and recursion operators play a crucial role in constructing hierarchies of soliton equations [1]. Associated with the variational derivative, recursion operators have been developed to formulate the Hamiltonian structures proving the integrability of soliton hierarchies [2, 3]. Recursion operators also have a tight correlation with one-soliton solutions. For example, the Korteweg-de Vries (KdV) hierarchy $$\begin{array}{}\text{(1)}& u_t=T^n{u}_x,\hspace{1em}n=\mathrm{0,1},\mathrm{2},\dots ,\end{array}$$possesses the following one-soliton solution: $$\begin{array}{}\text{(2)}& u=\frac{k^{\mathrm{2}}}{\mathrm{2}}{\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}}^{\mathrm{2}}\frac{k\left(x+k^{\mathrm{2}n}t+\delta \right)}{\mathrm{2}},\hspace{1em}n=\mathrm{0,1},\mathrm{2},\dots ,\end{array}$$where $T={\partial }^{\mathrm{2}}+\mathrm{4}u+\mathrm{2}{u}_x{\partial }^{-\mathrm{1}}$ is the recursion operator of the KdV hierarchy, $\delta$ is a constant, and $\partial =\partial /\partial x$, $\partial {\partial }^{-\mathrm{1}}={\partial }^{-\mathrm{1}}\partial =\mathrm{1}$. Notice that the dispersion relation of (2) $$\begin{array}{}\text{(3)}& kx+k^{\mathrm{2}n+\mathrm{1}}t\end{array}$$is linked closely with the order of the recursion operator $T$.

Since the discovery of scattering behavior of solitons [4] and the IST [5], solitons have received much attention. The IST has been also well developed and widely used to solve nonlinear equations [1, 6]. It can be used to solve not only normal soliton equations but also unusual soliton equations such as equations with self-consistent sources [7], nonisospectral equations [8, 9], and equations with steplike finite-gap backgrounds [10] and on quasi-periodic backgrounds [11]. Recently, Ablowitz and Musslimani developed the IST for the integrable nonlocal nonlinear Schrödinger (NLS) equation [12].

The Sylvester equation $$\begin{array}{}\text{(4)}& AM-MB=C\end{array}$$is one of the most well-known matrix equations. It appears frequently in many areas of applied mathematics and plays a central role, in particular, in systems and control theory, signal processing, filtering, model reduction, image restoration, and so on....

$$\begin{array}{}\hspace{1em}\end{array}$$$$\begin{array}{}\hspace{1em}\end{array}$$

$$\begin{array}{}\hspace{1em}\hspace{1em}\end{array}$$

$$\begin{array}{c}\hspace{1em}\end{array}$$$$\begin{array}{c}\hspace{1em}\\ \hspace{1em}\end{array}$$

$$\begin{array}{}\left\{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\right\}\end{array}$$

$$\begin{array}{c}\hspace{0.17em}\hspace{0.17em}\end{array}$$

$$\begin{array}{c}\hspace{0.17em}\hspace{0.17em}\end{array}$$

$$\begin{array}{c}\hspace{0.17em}\end{array}$$

$\hspace{0.17em}\hspace{0.17em}$

$$\begin{array}{}\hspace{0.17em}\end{array}$$

$\hspace{0.17em}\hspace{0.17em}$

$\hspace{0.17em}\hspace{0.17em}$

$$\begin{array}{}\left[\left(\hspace{0.17em}\right)\right]\left[\hspace{0.17em}\right]\end{array}$$

$\hspace{0.17em}\hspace{0.17em}$