# Recurrence in the high-order nonlinear Schr\"odinger equation: a low dimensional analysis

## TL;DR

This paper investigates the dynamics of deepwater waves modeled by the high-order nonlinear Schrödinger equation, using a three-wave truncation to analyze recurrence phenomena and classify solution topologies.

## Contribution

It introduces a low-dimensional model for the HONLS, validates it against simulations, and clarifies the role of fourth-order terms in wave recurrence and spectral shifts.

## Key findings

- Validated three-wave truncation against numerical simulations.
- Identified the impact of fourth-order terms on wave dynamics.
- Classified solution topologies related to recurrence phenomena.

## Abstract

We study a three-wave truncation of the high-order nonlinear Schr\"odinger equation for deepwater waves (HONLS, also named Dysthe equation). We validate our approach by comparing it to numerical simulation, distinguish the impact of the different fourth-order terms and classify the solutions according to their topology. This allows us to properly define the temporary spectral upshift occurring in the nonlinear stage of Benjamin-Feir instability and provides a tool for studying further generalizations of this model.