High-order soliton matrix for an extended nonlinear Schr\"{o}dinger equation

TL;DR

This paper develops a mathematical framework using Riemann-Hilbert problems and Darboux transformations to construct and analyze high-order soliton solutions of the extended nonlinear Schrödinger equation, relevant for optical fiber wave propagation.

Contribution

It introduces a novel method for constructing high-order soliton matrices for the ENLS equation using RHP, dressing, and Darboux transformations, enabling detailed analysis of soliton dynamics.

Findings

Explicit N-soliton solutions derived

Collision dynamics and asymptotic behaviors analyzed

Control of soliton parameters via spectral parameters

Abstract

The extended nonlinear Schr\"{o}dinger (ENLS) equation with third-order term and fourth-order term which describes the wave propagation in the optical fibers is more accurate than the NLS equation. A study of high-order soliton matrix is presented for an ENLS equation in the framework of the Riemann-Hilbert problem (RHP). Through a standard dressing procedure and the generalized Darboux transformation (gDT), soliton matrix for simple zeros and elementary high-order zeros in the RHP for the ENLS equation are constructed. Then the N-soliton solutions and high-order soliton solutions for the ENLS equation can be determined. Moreover, collision dynamics along with the asymptotic behavior for the two-solitons and long-time asymptotic estimations for the high-order one-soliton are concretely analyzed. For the given spectral parameters, we can control the propagation direction, velocity, width and other physical quantities of solitons by adjusting the free parameters of ENLS equation.