Some special solutions to the Hyperbolic NLS equation

Laurent Vuillon; Denys Dutykh; Francesco Fedele

doi:10.1016/j.cnsns.2017.09.018

Some special solutions to the Hyperbolic NLS equation

Laurent Vuillon, Denys Dutykh, Francesco Fedele

Mathematics

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6 Scopus citations

Abstract

The Hyperbolic Nonlinear SCHRÖDINGER equation (HypNLS) arises as a model for the dynamics of three–dimensional narrow-band deep water gravity waves. In this study, the symmetries and conservation laws of this equation are computed. The PETVIASHVILI method is then 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 accurate FOURIER solver.

Original language	British English
Pages (from-to)	202-220
Number of pages	19
Journal	Communications in Nonlinear Science and Numerical Simulation
Volume	57
DOIs	https://doi.org/10.1016/j.cnsns.2017.09.018
State	Published - Apr 2018

Keywords

Deep water
Gravity waves
Ground states
Hyperbolic equations
NLS equation
Wave patterns

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10.1016/j.cnsns.2017.09.018

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title = "Some special solutions to the Hyperbolic NLS equation",

abstract = "The Hyperbolic Nonlinear SCHR{\"O}DINGER equation (HypNLS) arises as a model for the dynamics of three–dimensional narrow-band deep water gravity waves. In this study, the symmetries and conservation laws of this equation are computed. The PETVIASHVILI method is then 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 accurate FOURIER solver.",

keywords = "Deep water, Gravity waves, Ground states, Hyperbolic equations, NLS equation, Wave patterns",

author = "Laurent Vuillon and Denys Dutykh and Francesco Fedele",

note = "Publisher Copyright: {\textcopyright} 2017 Elsevier B.V.",

year = "2018",

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T1 - Some special solutions to the Hyperbolic NLS equation

AU - Vuillon, Laurent

AU - Dutykh, Denys

AU - Fedele, Francesco

PY - 2018/4

Y1 - 2018/4

N2 - The Hyperbolic Nonlinear SCHRÖDINGER equation (HypNLS) arises as a model for the dynamics of three–dimensional narrow-band deep water gravity waves. In this study, the symmetries and conservation laws of this equation are computed. The PETVIASHVILI method is then 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 accurate FOURIER solver.

AB - The Hyperbolic Nonlinear SCHRÖDINGER equation (HypNLS) arises as a model for the dynamics of three–dimensional narrow-band deep water gravity waves. In this study, the symmetries and conservation laws of this equation are computed. The PETVIASHVILI method is then 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 accurate FOURIER solver.

KW - Deep water

KW - Gravity waves

KW - Ground states

KW - Hyperbolic equations

KW - NLS equation

KW - Wave patterns

UR - http://www.scopus.com/inward/record.url?scp=85030173473&partnerID=8YFLogxK

U2 - 10.1016/j.cnsns.2017.09.018

DO - 10.1016/j.cnsns.2017.09.018

M3 - Article

AN - SCOPUS:85030173473

SN - 1007-5704

VL - 57

SP - 202

EP - 220

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

ER -

Some special solutions to the Hyperbolic NLS equation

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