$\bar{\partial }$-dressing method for the coupled Gerdjikov–Ivanov equation

Abstract

We apply the $\bar{\partial }$-dressing method to study a coupled Gerdjikov–Ivanov (GI) equation. Compared with results on the classical coupled GI equation, more general spatial and a time singular spectral problems associated with GI equation are derived from a local 2 × 2 matrix $\bar{\partial }$-equation via two linear constraint equations. A more general GI hierarchy with source is proposed by using recursive operator. The $N$-solitons of the coupled GI equation are constructed still based the $\bar{\partial }$-equation by choosing a special spectral transformation matrix. Further the explicit one- and two-soliton solutions are obtained. The results on the GI equation can be recovered from the conclusion given above as special reductions.

Introduction

The nonlinear Schrödinger equation is one of the most important soliton equations, which comes from a wide range of practical backgrounds, such as water wave theory, plasma physics, quantum field theory, and other fields [1], [2], [3]. In recent years, several versions of derivative nonlinear Schrodinger equations were also introduced to investigate the effects of high-order perturbations [4], [5]. Among them, there are three celebrated derivative Schrödinger equations, including Kaup–Newell equation [6], Chen–Lee–Liu equation [7] and Gerdjikov–Ivanov (GI) equation [8].

It is well known that the GI equation is an important integrable model in mathematics and physics, it takes the form

$$i{q}_t+q_{xx}-i{q}^2{\bar{q}}_x+\frac{1}{2}{\left|q\right|}^{4}q=0,$$

which is a special reduction case when $r=-q^{\ast }$ for the following coupled GI equation

$$i{q}_t+q_{xx}+i{q}^2{r}_x+\frac{1}{2}{q}^3{r}^2=0,$$$$i{r}_t-r_{xx}+i{r}^2{q}_x-\frac{1}{2}{q}^2{r}^3=0.$$

It has been shown that both GI equation (1.1) and coupled GI equation (1.2) are integrable in the Liouville sense by means of the trace identity [9]. In recent years, the GI equation has been discussed extensively. For example, the Darboux transformation [10], the nonlinearization [11], [12], the similarity reduction, the bifurcation theory [13], [14], the Riemann–Hilbert method [15]. Comparatively, there is less work on coupled GI equation (1.2) including the Darboux transformation [9], Rogue wave solutions [16], multi-soliton solutions by Hirota method [17].

However, to our knowledge, there is still no research work on both GI equation (1.1) and coupled GI equation (1.2) by using $\bar{\partial }$-dressing method. This method was first proposed by Zakharov and Shabat [18], and further developed by Beals, Coifman, Manakov, Ablowitz and Fokas [19], [20], [21], [22], [23], [24]. At present, there have been a large number of equations that have been successfully studied with the dressing method [25], [26], [27], [28], [29], [30].

In this work, we apply $\bar{\partial }$-dressing method to investigate the Lax pair, GI hierarchy with source and N-soliton solution for the coupled GI equation (1.2). The paper is organized as follows. In Section 2, starting from a $\bar{\partial }$-equation, we present a new Lax pair with singular dispersion relation for the coupled GI equation (1.2) using the $\bar{\partial }$-dressing method. In Section 3, based the relation between $\bar{\partial }$-dressing transformation matrix and potential matrix, we derive a coupled GI hierarchy with source, which contains the classical coupled GI hierarchy as a special case. In Section 4, starting from $\bar{\partial }$-equation, a formula for $N$-soliton solutions of coupled GI equation (1.2) is constructed. As applications of the $N$-soliton formula, we give explicit one- and two-soliton solutions for the coupled GI equation in Section 5.