Eur. Phys. J. Plus (2020) 135: 717
https://doi.org/10.1140/epjp/s13360-020-00714-z

Regular Article

Inverse scattering transform for the complex short-pulse equation by a Riemann–Hilbert approach

¹ Department of Mathematics, University at Buffalo, 14260, Buffalo, NY, USA
² Department of Applied Mathematics and Statistics, Montclair State University, 07043, Montclair, NJ, USA
³ School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, 78539, Edinburg, TX, USA

^a bprinari@buffalo.edu

Received: 31 May 2020
Accepted: 26 August 2020
Published online: 12 September 2020

Abstract

In this paper, we develop the inverse scattering transform (IST) for the complex short-pulse equation (CSP) on the line with zero boundary conditions at space infinity. The work extends to the complex case the Riemann–Hilbert approach to the IST for the real short-pulse equation proposed by A. Boutet de Monvel, D. Shepelsky and L. Zielinski in 2017. As a byproduct of the IST, soliton solutions are also obtained. Unlike the real SPE, in the complex case discrete eigenvalues are not necessarily restricted to the imaginary axis, and, as consequence, smooth 1-soliton solutions exist for any choice of discrete eigenvalue $k_1\in \mathbb{C}$ with $\mathrm{Im}{k}_1<|\mathrm{Re}{k}_1|$. The 2-soliton solution is obtained for arbitrary eigenvalues $k_1,k_2$, providing also the breather solution of the real SPE in the special case $k_2=-k_1^{\ast }$.