Dark solitons for an extended quintic nonlinear Schr\"odinger equation: Application to water waves at $kh = 1.363$

TL;DR

This paper investigates dark solitons in an extended quintic nonlinear Schrödinger equation relevant to water waves at a critical parameter value, revealing two types of gray solitons supported by analytical and numerical methods.

Contribution

It introduces a new model for water waves at a specific parameter and characterizes two types of gray solitons, supported by analytical derivations and numerical validation.

Findings

Existence of approximate dark soliton solutions from an effective KdV equation

Identification of two types of gray solitons: fast and slow

Numerical simulations confirm analytical predictions

Abstract

We study the existence, formation and dynamics of gray solitons for an extended quintic nonlinear Schr\"odinger (NLS) equation. The considered model finds applications to water waves, when the characteristic parameter $kh$ - where $k$ is the wavenumber and $h$ is the undistorted water's depth - takes the critical value $kh=1.363$. It is shown that this model admits approximate dark soliton solutions emerging from an effective Korteweg-de Vries equation and that two types of gray solitons exist: fast and slow, with the latter being almost stationary objects. Analytical results are corroborated by direct numerical simulations.