Bright and Dark Solitons on the Surface of Finite-Depth Fluid Below the Modulation Instability Threshold

TL;DR

This paper derives a high-order nonlinear Schrödinger equation for finite-depth fluids and demonstrates the existence of bright solitons below the modulation instability threshold, expanding understanding of wave behaviors in such conditions.

Contribution

It introduces a finite-depth counterpart to Dysthe's equation and shows it admits bright soliton solutions below the instability threshold, which was not possible with standard NLSE.

Findings

Bright solitons exist below the modulation instability threshold in finite-depth fluids.

The generalized equation admits both bright and dark soliton solutions.

Bright solitons coexist with dark solitons observed experimentally.

Abstract

We use the high-order nonlinear Schr\"{o}dinger equation (NLSE) derived to model the evolution of slowly modulated wave trains with narrow spectrum on the surface of ideal finite-depth fluid. This equation is the finite-depth counterpart of celebrated Dysthe's equation, which is usually used for the same purpose in the case of infinite depth. We demonstrate that this generalized equation admits bright soliton solutions for depths below the modulation instability threshold $kh\approx 1.363$ ($k$ being the carrier wave number and $h$ the undisturbed fluid depth), which is not possible in the case of standard NLSE. These bright solitons can exist along with the dark solitons that have recently been observed in a water wave tank [Phys. Rev. Lett. 110, 124101 (2013)].