Inverse scattering and soliton solutions of nonlocal complex reverse-spacetime mKdV equations

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Abstract

The paper deals with the inverse scattering transforms for nonlocal complex reverse-spacetime multicomponent integrable modified Korteweg–de Vries (mKdV) equations. We establish associated Riemann–Hilbert problems and determine their solutions by the Sokhotski–Plemelj formula. The inverse scattering problems consist of Gelfand–Levitan–Marchenko type equations for the generalized matrix Jost solutions and the recovery formula for the potential. When reflection coefficients are zero, the corresponding Riemann–Hilbert problems yield soliton solutions to the nonlocal complex reverse-spacetime mKdV equations.

Introduction

Nonlocal integrable equations have been studied very recently, including nonlocal scalar nonlinear Schrödinger (NLS) equations [3], [4] and nonlocal scalar modified Korteweg–de Vries (mKdV) equations [5], [17]. Their inverse scattering transforms were established under zero or nonzero boundary conditions [2], [4], [17]. The N-soliton solutions were generated from the Riemann–Hilbert problems [22], [34], via Darboux transformations [16], [23], and through the Hirota bilinear method [14]. A few multicomponent generalizations [5], [7], [13], [22], [31] were also presented and analyzed.

We will apply the Riemann–Hilbert technique [30] to study the inverse scattering transforms and particularly generate soliton solutions. Various integrable equations, such as the multiple wave interaction equations [30], the general coupled nonlinear Schrödinger equations [32], the Harry Dym equation [33], and the generalized Sasa–Satsuma equation [10], have been studied by analyzing associated Riemann–Hilbert problems. In this paper, we would like to propose a kind of multicomponent nonlocal complex reverse-spacetime mKdV equations, and construct their inverse scattering transforms and soliton solutions through establishing associated Riemann–Hilbert problems.

The rest of the letter is structured as follows. In Section 2, we make a kind of nonlocal group reductions to generate nonlocal complex reverse-spacetime mKdV equations. In Section 3, we establish associated Riemann–Hilbert problems and determine their solutions by the Sokhotski–Plemelj formula to present the inverse scattering transforms. In Section 4, we construct soliton solutions from the reflectionless transforms, whose Riemann–Hilbert problems have the identity jump matrix. In the last section, we give a conclusion and some concluding remarks.

Section snippets

Nonlocal complex reverse-spacetime mKdV equations

Let n be an arbitrary natural number. Assume that λ stands for a spectral parameter, and u, a 2n-dimensional potential u=u(x,t)=(p,qT)T,p=(p1,p2,,pn),q=(q1,q2,,qn)T.Let us consider the multicomponent AKNS matrix spectral problems (see, e.g., [26]): iϕx=Uϕ=U(u,λ)ϕ,iϕt=Vϕ=V(u,λ)ϕ,with the Lax pair U=λΛ+P,V=λ3Ω+Q.The involved four matrices are defined by Λ=diag(α1,α2In), Ω=diag(β1,β2In), P=0pq0,and Q=βαλ20pq0βα2λpqipxiqxqpβα3i(pqxpxq)pxx+2pqpqxx+2qpqi(qpxqxp),where α1,α2,β1,β2 are

Inverse scattering transforms

Let q be determined by (2.10). In what follows, we discuss the scattering and inverse scattering for the nonlocal complex reverse-spacetime mKdV equations (2.11) by the Riemann–Hilbert approach [30] (see also [6], [11] for the local case). The results will be the basis for generating soliton solutions later.

Soliton solutions

Let NN be arbitrary. Assume that s11 has N zeros {λk,1kN}, and sˆ11 has other N zeros {λˆk,1kN}. We also assume that all these zeros, λk and λˆk,1kN, are geometrically simple. Then, each of kerT+(λk), 1kN, contains only a single basis column vector, denoted by vk, 1kN; and each of kerT(λˆk), 1kN, a single basis row vector, denoted by vˆk, 1kN: T+(λk)vk=0,vˆkT(λˆk)=0,1kN.

The Riemann–Hilbert problems with the identity jump matrix, the canonical normalization conditions in

Concluding remarks

This paper proposed a class of nonlocal complex reverse-spacetime multicomponent modified Korteweg–de Vries (mKdV) equations from a kind of nonlocal group reductions, and constructed their inverse scattering transforms. The basic tool is the Riemann–Hilbert approach to matrix spectral problems. We determined solutions to the Riemann–Hilbert problems by applying the Sokhotski–Plemelj formula, and systematically presented soliton solutions to the nonlocal complex reverse-spacetime mKdV equations,

Acknowledgments

The work was supported in part by NSFC under the grants 11975145 and 11972291, NSF under the grant DMS-1664561, and the Natural Science Foundation for Colleges and Universities in Jiangsu Province (17 KJB 110020).

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