Dbar-dressing method for the Gerdjikov–Ivanov equation with nonzero boundary conditions
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, takes the form [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]. Recently, we generalized the Dbar-dressing method to construct Lax pair and -soliton solutions for the coupled GI equation [16].
This method was first proposed by Zakharov and Shabat [17], and further developed by Beals, Coifman, Manakov, Ablowitz and Fokas [18], [19], [20], [21], [22], [23]. At present, there have been a large number of equations that have been successfully studied with the dressing method [24], [25], [26], [27], [28]. Comparatively, there is less work on the integral equation with nonzero boundary conditions via Dbar-dressing method. Especially, to our knowledge, there is still no research work on GI equation (1.2) with nonzero boundary conditions by using Dbar-dressing method.
In this work, we apply Dbar-dressing method to investigate the Lax pair and N-soliton solution for the GI equation (1.2) with the following nonzero boundary where , and are independent of . The paper is organized as follows. In Section 2, starting from a Dbar-problem with non-canonical normalization conditions, we obtain the asymptotic expansion of when and . In Section 3, we construct the Lax pair of the GI equation with nonzero boundary conditions via the Dbar-dressing method and establish the relation between potential and the expansion coefficient of . In Section 4, starting from Dbar-equation, a formula for -soliton solutions of GI equation with nonzero boundary conditions is constructed. As applications of the -soliton formula, we give explicit one- and two-soliton solutions for the GI equation in Section 5.
Section snippets
Dressing for GI equation
Consider a matrix Dbar problem where and are 2 × 2 matrix, the distribution is independent of and . We add non-canonical normalization conditions where
Define a linear operator , we have then satisfies the boundary conditions We consider a new
Lax pair of GI equation with nonzero boundary
In this section, we will construct the Lax pair of GI equation with nonzero boundary conditions. Suppose that [29], then we can get, Now we introduce the following solution space of the Dbar-problem (2.1) as In particular, let and We note that Eq. (3.1) implies that, for , we can come to this conclusion and
-Soliton solutions of GI equation
In this section, we will construct the N-soliton solutions of the GI equation (1.2) still based on the Dbar-equation (2.1).
Proposition 3 Suppose that are discrete spectrum in complex plane . We choose a spectral transform matrix as where is a constant satisfying Then the GI equation (1.2) admits the N-soliton solutions where
Applications of the N-soliton formula
As special applications of the formula (4.2), we present explicit one- and two-soliton solutions for coupled GI equation (1.2).
CRediT authorship contribution statement
Jinghua Luo: Conceptualization, Software, Data creation, Writing - original draft, Visualization, Investigation, Validation, Writing - review and editing. Engui Fan: Methodology, Supervision.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11671095, 51879045).
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