Inverse scattering transform for the complex coupled short‐pulse equation

Abstract

In this paper, we develop the Riemann-Hilbert approach to the inverse scattering transform (IST) for the complex coupled short-pulse equation on the line with zero boundary conditions at space infinity, which is a generalization of recent work on the scalar real shortpulse equation (SPE) and complex short-pulse equation (cSPE). As a byproduct of the IST, soliton solutions are also obtained. As is often the case, the zoology of soliton solutions for the coupled system is richer than in the scalar case, and it includes both fundamental solitons (the natural, vector generalization of the scalar case), and fundamental breathers (a superposition of orthogonally polarized fundamental solitons, with the same amplitude and velocity but having different carrier frequencies), as well as composite breathers, which still correspond to a minimal set of discrete eigenvalues but cannot be reduced to a simple superposition of fundamental solitons. Moreover, it is found that the same constraint on the discrete eigenvalues which leads to regular, smooth one-soliton solutions in the complex SPE, also holds in the coupled case, for both a single fundamental soliton and a single fundamental breather, but not, in general, in the case of a composite breather.

Direct and inverse scattering problems of the modified Sawada–Kotera equation: Riemann–Hilbert approach

“…More than 50 years ago, the inverse scattering transform (IST) [1], also called the nonlinear Fourier transform method, was introduced to solve the initial problems of the integrable evolution equations on the line, which is one of the most remarkable developments in integrable systems. Many integrable evolution equations with the second-order Lax pairs have been solved by IST, such as the nonlinear Schrödinger equation, the KdV equation [1], the sine-Gordon equation [2], the modified KdV equation [3], the non-local mKdV equation [4], the Camassa-Holm equation [5][6][7][8], the short-pulse equation [9], etc. Integrable equations with the third-order Lax pairs can also be studied by IST, for example, Zakharov [10], Kaup [11], McKean [12] and Deift et al [13] .…”

Soliton interactions and Yang–Baxter maps for the complex coupled short‐pulse equation

“…Below, we give a succinct overview of the IST for the ccSPE as developed in Ref. [46]. The ccSPE (2) with 𝜎 = 1 possess the following Lax pair:…”

“…[42][43][44][45], and the IST was developed in Ref. [46]. The main goal of this work is to study interactions of vector solitons of the focusing ccSPE.…”

“…In Section 2, we give a brief overview of the IST for the ccSPE as developed in Ref. [46], and of its one-soliton solutions, which include fundamental solitons, fundamental breathers, and composite breathers, depending on the rank and structure of the norming constant associated with the soliton. We also discuss in detail the case of self-symmetric discrete eigenvalues, and derive the explicit expression of a self-symmetric soliton.…”

Darboux transformation and solitonic solution to the coupled complex short pulse equation

“…As far as we know, the study of the CCSP equation has only a few. Most recently, an inverse scattering analysis is done for the CCSP equation with only the vanishing boundary condition [19]. Therefore, there is no systematic analysis for various soliton solutions for the CCSP equation, which motivates the present study.…”