An initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with a $4\times4$ Lax pair

TL;DR

This paper formulates and analyzes an initial-boundary value problem for a coupled focusing-defocusing complex short pulse equation using a $4\times 4$ Lax pair and the unified transform method, providing a way to solve it via a Riemann-Hilbert problem.

Contribution

It introduces a novel approach to solve the coupled focusing-defocusing complex short pulse equation on the half-line using a $4\times 4$ Lax pair and Riemann-Hilbert problem formulation.

Findings

Solution expressed via a $4\times 4$ matrix Riemann-Hilbert problem.

Spectral functions satisfy a global relation.

Method applicable to ultra-short optical pulse propagation models.

Abstract

In this paper we investigate the coupled focusing-defocusing complex short pulse equation, which describe the propagation of ultra-short optical pulses in cubic nonlinear media. Through the unified transform method, the initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with $4\times 4$ Lax pair on the half-line are to be analyzed. Assuming that the solution $\{q_1(x,t),q_2(x,t)\}$ of the coupled focusing-defocusing complex short pulse equation exists, we show that $\{q_{1,x}(x,t),q_{2,x}(x,t)\}$ can be expressed in terms of the unique solution of a $4\times 4$ matrix Riemann-Hilbert problem formulated in the complex $\lambda$-plane. Thus, the solution $\{q_1(x,t),q_2(x,t)\}$ can be obtained by integration with respect to $x$. Moreover, we also get that some spectral functions are not independent and satisfy the so-called global relation.