Dbar-dressing method for the Gerdjikov–Ivanov equation with nonzero boundary conditions

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Highlights

  • The ̄-dressing method to study the GI equation with nonzero boundary.

  • A spatial and a time spectral problem associated with GI equation are derived.

  • N-solitons of the GI equation with nonzero boundary are constructed.

Abstract

We apply the Dbar-dressing method to study a Gerdjikov–Ivanov (GI) equation with nonzero boundary at infinity. A spatial and a time spectral problem associated with GI equation are derived with a asymptotic expansion method. The N-soliton solutions of the GI equation are constructed based the Dbar-equation by choosing a special spectral transformation matrix. Further the explicit one- and two-soliton solutions are obtained.

Introduction

The nonlinear Schrödinger equation iqt+qxx+2|q|2q=0,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] iqt+qxxiq2q̄x+12|q|4q=0,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 N-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 q(x,t)ϱe32iq04t+iq02x,x±,where |ϱ|=q0>0, and ϱ are independent of x,t. 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 ψ when z and z0. 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 ψ. In Section 4, starting from Dbar-equation, a formula for N-soliton solutions of GI equation with nonzero boundary conditions is constructed. As applications of the N-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 ̄ψ(x,t,z)=ψ(x,t,z)r(z),z{0},where ψ(x,t,z) and r(z) are 2 × 2 matrix, the distribution r(z) is independent of x and t. We add non-canonical normalization conditions ψ(x,t,z)eiθ(x,t,z)σ3,z;ψ(x,t,z)izσ3Q0eiθ(x,t,z)σ3,z0where Q0=0ϱϱ̄0,θ(x,t,z)=λ(z)k(z)[x+(2k(z)2q02)t], λ(z)=12(z+q02z),k(z)=12(zq02z).

Define a linear operator Λ, we have Λψψˆ=ψeiθ(x,t,z)σ3,then ψˆ satisfies the boundary conditions ψˆ(x,t,z)I,z,ψˆ(x,t,z)izσ3Q0,z0.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 ψRL1({0})L({0}) [29], then we can get, ψˆ(IRCz)=0ψˆ=0.Now we introduce the following solution space of the Dbar-problem (2.1) as F={ψ|ψ(x,t,z)=ψ(x,t,z)r,z0}.In particular, let ψ(x,t,z)F and N(ψˆ)=Iizσ3Q0 We note that Eq. (3.1) implies that, for ψ1(x,t,z),ψ2(x,t,z) F, we can come to this conclusion N(ψˆ1(x,t,z))=N(ψˆ2(x,t,z))ψˆ1(x,t,z)=ψˆ2(x,t,z)ψ1(x,t,z)=ψ2(x,t,z)and

N-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 ζj are 2N1+N2 discrete spectrum in complex plane . We choose a spectral transform matrix R as r(z)=j=12N1+N2π0cj[δ(zζj)+δ(z+ζj)]c̄j[δ(zζ̄j)+δ(z+ζ̄j)]0,where cj is a constant satisfying cN1+j=ϱ̄ϱ(q02ζ̄j2)c̄j,j=1,,N1,ζj=zj,ζN1+j=q02z̄j,j=1,,N1;ζ2N1+n=ωn,n=1,,N2. Then the GI equation (1.2) admits the N-soliton solutions qN(x,t)=ϱ+2idetMaugdetM,where Maug

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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