Approximate rogue wave solutions of the forced and damped Nonlinear Schr\"odinger equation for water waves

TL;DR

This paper investigates how wind and dissipation influence rogue wave formation in water waves by transforming the forced and damped NLS equation into a standard form and deriving approximate solutions.

Contribution

It introduces a transformation method to approximate rogue wave solutions in the forced/damped NLS equation considering wind and dissipation effects.

Findings

Wind and dissipation significantly affect rogue wave formation.

Approximate solutions reveal the influence of external forces on wave dynamics.

Transformation approach simplifies analysis of complex water wave models.

Abstract

We consider the effect of the wind and the dissipation on the nonlinear stages of the modulational instability. By applying a suitable transformation, we map the forced/damped Nonlinear Schr\"odinger (NLS) equation into the standard NLS with constant coefficients. The transformation is valid as long as |{\Gamma}t| \ll 1, with {\Gamma} the growth/damping rate of the waves due to the wind/dissipation. Approximate rogue wave solutions of the equation are presented and discussed. The results shed some lights on the effects of wind and dissipation on the formation of rogue waves.