Some special solutions to the Hyperbolic NLS equation

TL;DR

This paper numerically investigates special bi-periodic solutions of the Hyperbolic NLS equation, revealing complex patterns and analyzing their stability, contributing new computational methods and insights into wave dynamics in deep water models.

Contribution

It introduces a numerical approach using the Petviashvili method to compute non-localized standing wave solutions and explores their complex spatial patterns and stability.

Findings

Revealed non-trivial spatial patterns of standing waves

Demonstrated stability properties of perturbed solutions

Developed symbolic dynamics description for solutions

Abstract

The Hyperbolic Nonlinear Schrodinger equation (HypNLS) arises as a model for the dynamics of three-dimensional narrowband deep water gravity waves. In this study, the Petviashvili method is exploited to numerically compute bi-periodic time-harmonic solutions of the HypNLS equation. In physical space they represent non-localized standing waves. Non-trivial spatial patterns are revealed and an attempt is made to describe them using symbolic dynamics and the language of substitutions. Finally, the dynamics of a slightly perturbed standing wave is numerically investigated by means a highly acccurate Fourier solver.