Higher-order rogue wave dynamics for a derivative nonlinear Schr\"odinger equation

TL;DR

This paper derives and analyzes higher-order rogue wave solutions for the Chen-Lee-Liu derivative nonlinear Schrödinger equation, revealing how self-steepening effects influence rogue wave localization in nonlinear dispersive media.

Contribution

It provides explicit rogue wave solutions for the CLL-NLS using Darboux transformation, extending the understanding of rogue wave dynamics with self-steepening effects.

Findings

Exact rogue wave solutions derived for CLL-NLS

Localization properties of rogue waves affected by self-steepening

Potential for experimental validation in nonlinear media

Abstract

The the mixed Chen-Lee-Liu derivative nonlinear Schr\"odinger equation (CLL-NLS) can be considered as simplest model to approximate the dynamics of weakly nonlinear and dispersive waves, taking into account the self-steepnening effect (SSE). The latter effect arises as a higher-order correction of the nonlinear Schr\"ordinger equation (NLS), which is known to describe the dynamics of pulses in nonlinear fiber optics, and constiutes a fundamental part of the generalized NLS. Similar effects are decribed within the framework of the modified NLS, also referred to as the Dysthe equation, in hydrodynamics. In this work, we derive fundamental and higher-order solutions of the CLL-NLS by applying the Darboux transformation (DT). Exact expressions of non-vanishing boundary solitons, breathers and a hierarchy of rogue wave solutions are presented. In addition, we discuss the localization characters of such rogue waves, by characterizing their length and width. In particular, we describe how the localization properties of first-order NLS rogue waves can be modified by taking into account the SSE, presented in the CLL-NLS. This is illustrated by use of an analytical and a graphical method. The results may motivate similar analytical studies, extending the family of the reported rogue wave solutions as well as possible experiments in several nonlinear dispersive media, confirming these theoretical results.