A dynamic equation for water waves in one horizontal dimension

A. I. Dyachenko; V. E. Zakharov

doi:10.1016/j.euromechflu.2011.08.001

A dynamic equation for water waves in one horizontal dimension

A. I. Dyachenko
, V. E. Zakharov

Research output: Contribution to journal › Article › peer-review

28 Scopus citations

Abstract

We apply a canonical transformation to a water wave equation to remove cubic nonlinear terms and to drastically simplify fourth-order terms in the Hamiltonian. This transformation explicitly uses the vanishing exact four-wave interaction for water gravity waves for a 2D potential fluid. After transformation, the well-known but cumbersome Zakharov equation is drastically simplified and can be written in X-space in a compact form. This new equation is very suitable for analytical studies and numerical simulations.

Original language	English (US)
Pages (from-to)	17-21
Number of pages	5
Journal	European Journal of Mechanics, B/Fluids
Volume	32
Issue number	1
DOIs	https://doi.org/10.1016/j.euromechflu.2011.08.001
State	Published - Mar 2012
Externally published	Yes

Keywords

Free surface
Gravity waves
Zakharov equation

ASJC Scopus subject areas

Mathematical Physics
General Physics and Astronomy

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10.1016/j.euromechflu.2011.08.001

Cite this

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title = "A dynamic equation for water waves in one horizontal dimension",

abstract = "We apply a canonical transformation to a water wave equation to remove cubic nonlinear terms and to drastically simplify fourth-order terms in the Hamiltonian. This transformation explicitly uses the vanishing exact four-wave interaction for water gravity waves for a 2D potential fluid. After transformation, the well-known but cumbersome Zakharov equation is drastically simplified and can be written in X-space in a compact form. This new equation is very suitable for analytical studies and numerical simulations.",

keywords = "Free surface, Gravity waves, Zakharov equation",

author = "Dyachenko, \{A. I.\} and Zakharov, \{V. E.\}",

note = "Funding Information: This work was supported by a grant from the Government of the Russian Federation for Support of Scientific Research under the Direction of Leading Scientists in Russian Universities ( N11.G34.31.0035 , leading scientist V.E. Zakharov, GOU VPO Novosibirsk State University). We also received support from the US Army Corps of Engineers Grant W912-BU-08-P-0143 , ONR Grant N00014-10-1-0991 , NSF Grant DMS 0404577 , Grant NOPP “TSA-a two scale approximation for wind-generated ocean surface waves”, RFBR Grants 09-01-00631 and 09-05-13605 , the Fundamental Problems in Nonlinear Dynamics Program (RAS Presidium), and a Leading Scientific Schools of Russia grant.",

year = "2012",

month = mar,

doi = "10.1016/j.euromechflu.2011.08.001",

language = "English (US)",

volume = "32",

pages = "17--21",

journal = "European Journal of Mechanics, B/Fluids",

issn = "0997-7546",

publisher = "Elsevier B.V.",

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T1 - A dynamic equation for water waves in one horizontal dimension

AU - Dyachenko, A. I.

AU - Zakharov, V. E.

N1 - Funding Information: This work was supported by a grant from the Government of the Russian Federation for Support of Scientific Research under the Direction of Leading Scientists in Russian Universities ( N11.G34.31.0035 , leading scientist V.E. Zakharov, GOU VPO Novosibirsk State University). We also received support from the US Army Corps of Engineers Grant W912-BU-08-P-0143 , ONR Grant N00014-10-1-0991 , NSF Grant DMS 0404577 , Grant NOPP “TSA-a two scale approximation for wind-generated ocean surface waves”, RFBR Grants 09-01-00631 and 09-05-13605 , the Fundamental Problems in Nonlinear Dynamics Program (RAS Presidium), and a Leading Scientific Schools of Russia grant.

PY - 2012/3

Y1 - 2012/3

N2 - We apply a canonical transformation to a water wave equation to remove cubic nonlinear terms and to drastically simplify fourth-order terms in the Hamiltonian. This transformation explicitly uses the vanishing exact four-wave interaction for water gravity waves for a 2D potential fluid. After transformation, the well-known but cumbersome Zakharov equation is drastically simplified and can be written in X-space in a compact form. This new equation is very suitable for analytical studies and numerical simulations.

AB - We apply a canonical transformation to a water wave equation to remove cubic nonlinear terms and to drastically simplify fourth-order terms in the Hamiltonian. This transformation explicitly uses the vanishing exact four-wave interaction for water gravity waves for a 2D potential fluid. After transformation, the well-known but cumbersome Zakharov equation is drastically simplified and can be written in X-space in a compact form. This new equation is very suitable for analytical studies and numerical simulations.

KW - Free surface

KW - Gravity waves

KW - Zakharov equation

UR - https://www.scopus.com/pages/publications/84856286112

UR - https://www.scopus.com/pages/publications/84856286112#tab=citedBy

U2 - 10.1016/j.euromechflu.2011.08.001

DO - 10.1016/j.euromechflu.2011.08.001

M3 - Article

SN - 0997-7546

VL - 32

SP - 17

EP - 21

JO - European Journal of Mechanics, B/Fluids

JF - European Journal of Mechanics, B/Fluids

IS - 1

ER -

A dynamic equation for water waves in one horizontal dimension

Abstract

Keywords

ASJC Scopus subject areas

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