On the so-called rogue waves in the nonlinear Schr\"odinger equation

TL;DR

This paper investigates the connection between rogue water waves and homoclinic solutions of the nonlinear Schrödinger equation, concluding that these solutions are more observable and common than the rare rogue waves, and explores alternative mechanisms.

Contribution

The study analyzes the phase space structure of homoclinic orbits in NLS and argues that approximate homoclinic solutions are more observable and prevalent than rogue waves.

Findings

Approximate homoclinic orbits are the most observable solutions.

These solutions likely correspond to common deep ocean waves.

Rogue waves may result from other mechanisms like initial data dependence or finite time blow-up.

Abstract

The mechanism of a rogue water wave is still unknown. One popular conjecture is that the Peregrine wave solution of the nonlinear Schr\"odinger equation (NLS) provides a mechanism. A Peregrine wave solution can be obtained by taking the infinite spatial period limit to the homoclinic solutions. In this article, from the perspective of the phase space structure of these homoclinic orbits in the infinite dimensional phase space where the NLS defines a dynamical system, we exam the observability of these homoclinic orbits (and their approximations). Our conclusion is that these approximate homoclinic orbits are the most observable solutions,and they should correspond to the most common deep ocean waves rather than the rare rogue waves. We also discuss other possibilities for the mechanism of a rogue wave: rough dependence on initial data or finite time blow up.