Approximate Peregrine Solitons in Dispersive Nonlinear Wave Equations

TL;DR

This paper extends the validity of the nonlinear Schrödinger (NLS) approximation from Sobolev spaces to more complex function spaces, enabling the use of Peregrine solutions to model rogue waves in diverse dispersive nonlinear systems.

Contribution

It generalizes the NLS approximation validity to functions combining periodic and non-periodic components, facilitating the application of Peregrine solutions to complex wave systems.

Findings

Extended NLS approximation validity to mixed Sobolev spaces.

Enabled modeling of rogue waves in complex dispersive systems.

Provided a framework for applying Peregrine solutions beyond standard settings.

Abstract

The purpose of this short note is to explain how the existing results on the validity of the NLS approximation can be extended from Sobolev spaces $H^s(\mathbb{R})$ to the spaces of functions $u = v + w$ where $v \in H_{{per}}^s$ and $w \in H^s(\mathbb{R})$. This allows us to use the Peregrine solution of the NLS equation to find freak or rogue wave dynamics in more complicated systems.