---
abstract: |
    The Hyperbolic Nonlinear Schrödinger equation () 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 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.

    **:** Hyperbolic equations; NLS equation; wave patterns; deep water waves
address:
- 'LAMA, UMR 5127 CNRS, Université de Savoie, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France'
- 'University College Dublin, School of Mathematical Sciences, Belfield, Dublin 4, Ireland LAMA, UMR 5127 CNRS, Université de Savoie, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex, France'
- 'School of Civil and Environmental Engineering & School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, USA'
author:
- Laurent Vuillon
- 'Denys Dutykh$^*$'
- Francesco Fedele
bibliography:
- 'biblio.bib'
title: Some special solutions to the Hyperbolic NLS equation
---

[^1]

Introduction
============

The celebrated cubic Nonlinear Schrödinger (NLS) equation is one of the most important equations 