Riemann-Hilbert approach for a mixed coupled nonlinear Schr\"odinger system and its soliton solutions

TL;DR

This paper applies the Riemann-Hilbert method to derive explicit soliton solutions for an integrable mixed coupled nonlinear Schrödinger system, relevant for optical fiber pulse propagation, and discusses its generalizations.

Contribution

It introduces a Riemann-Hilbert approach to explicitly solve the mCNLS system and extends the analysis to a generalized multi-component NLS system.

Findings

Explicit N-soliton solutions for the mCNLS system derived

One- and two-soliton solutions expressed explicitly

Discussion of a generalized multi-component NLS system

Abstract

In this work, we examine the integrable mixed coupled nonlinear Schr\"odinger (mCNLS) system, which describe the propagation of an optical pulse in a birefringent optical fiber. By the Riemann-Hilbert(RH) approach, the N-soliton solutions of the mCNLS system can be expressed explicitly when the jump matrix of a specific RH problem is a $3\times3$ unit matrix. As a special example, the expression of one- and two-soliton are displayed explicitly. More generally, as a promotion, an integrable generalized multi-component NLS system with its linear spectral problem be discussed. It is hoped that our results can help enrich the nonlinear dynamical behaviors of the mCNLS.