On a deformation of the nonlinear Schr\"odinger equation

TL;DR

This paper investigates a deformation of the nonlinear Schrödinger equation, exploring its solitary wave solutions, interactions, and rogue waves, revealing sharper and larger amplitude phenomena compared to the classical NLS.

Contribution

It introduces a new deformation of the NLS equation, analyzes its solitary wave solutions, and studies their interactions and rogue wave behavior through numerical simulations.

Findings

Solitary waves exhibit soliton-like behavior with near elastic collisions.

Deformed NLS solutions during interactions are sharper and have larger amplitudes.

Rogue waves in the deformed NLS show increased amplitude and sharper structure.

Abstract

We study a deformation of the nonlinear Schr\"odinger equation recently derived in the context of deformation of hierarchies of integrable systems. This systematic method also led to known integrable equations such as the Camassa-Holm equation. Although this new equation has not been shown to be completely integrable, its solitary wave solutions exhibit typical soliton behaviour, including near elastic collisions. We will first focus on standing wave solutions, which can be smooth or peaked, then, with the help of numerical simulations, we will study solitary waves, their interactions and finally rogue waves in the modulational instability regime. Interestingly the structure of the solution during the collision of solitary waves or during the rogue wave events are sharper and have larger amplitudes than in the classical NLS equation.