# Justification of Peregrine soliton from full water waves

## TL;DR

This paper rigorously justifies the Peregrine soliton as an approximate solution of the full water wave equations in a specific regime, connecting it to rogue wave phenomena and proving long-term existence of small initial data solutions.

## Contribution

It provides a rigorous derivation of the NLS and Peregrine soliton from full water wave equations in a non-tangential decay regime, with long-time existence results.

## Key findings

- Justification of NLS as an envelope equation for water waves
- Rigorous derivation of Peregrine soliton from water wave system
- Long-time existence of solutions with small initial data

## Abstract

The Peregrine soliton $Q(x,t)=e^{it}(1-\frac{4(1+2it)}{1+4x^2+4t^2})$ is an exact solution of the 1d focusing nonlinear schr\"{o}dinger equation (NLS) $iB_t+B_{xx}=-2|B|^2B$, having the feature that it decays to $e^{it}$ at the spatial and time infinities, and with a peak and troughs in a local region. It is considered as a prototype of the rogue waves by the ocean waves community. The 1D NLS is related to the full water wave system in the sense that asymptotically it is the envelope equation for the full water waves. In this paper, working in the framework of water waves which decay non-tangentially, we give a rigorous justification of the NLS from the full water waves equation in a regime that allows for the Peregrine soliton. As a byproduct, we prove long time existence of solutions for the full water waves equation with small initial data in space of the form $H^s(\mathbb{R})+H^{s'}(\mathbb{T})$, where $s\geq 4, s'>s+\frac{3}{2}$.