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Super compact equation for water waves

Published online by Cambridge University Press:  12 September 2017

A. I. Dyachenko Open the ORCID record for A. I. Dyachenko [Opens in a new window] ,
D. I. Kachulin  and
V. E. Zakharov
A. I. Dyachenko*
Affiliation:
Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia Novosibirsk State University, 630090, Novosibirsk-90, Russia
D. I. Kachulin
Affiliation:
Novosibirsk State University, 630090, Novosibirsk-90, Russia
V. E. Zakharov
Affiliation:
Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia Novosibirsk State University, 630090, Novosibirsk-90, Russia Department of Mathematics, University of Arizona, Tucson, AZ 857201, USA Physical Institute of RAS, Leninskiy prospekt, 53, Moscow, 119991, Russia Space Research Institute of RAS, 84/32 Profsoyuznaya Str, Moscow, 117997, Russia
*
†Email address for correspondence: alexd@itp.ac.ru
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Abstract

Mathematicians and physicists have long been interested in the subject of water waves. The problems formulated in this subject can be considered fundamental, but many questions remain unanswered. For instance, a satisfactory analytic theory of such a common and important phenomenon as wave breaking has yet to be developed. Our knowledge of the formation of rogue waves is also fairly poor despite the many efforts devoted to this subject. One of the most important tasks of the theory of water waves is the construction of simplified mathematical models that are applicable to the description of these complex events under the assumption of weak nonlinearity. The Zakharov equation, as well as the nonlinear Schrödinger equation (NLSE) and the Dysthe equation (which are actually its simplifications), are among them. In this article, we derive a new modification of the Zakharov equation based on the assumption of unidirectionality (the assumption that all waves propagate in the same direction). To derive the new equation, we use the Hamiltonian form of the Euler equation for an ideal fluid and perform a very specific canonical transformation. This transformation is possible due to the ‘miraculous’ cancellation of the non-trivial four-wave resonant interaction in the one-dimensional wave field. The obtained equation is remarkably simple. We call the equation the ‘super compact water wave equation’. This equation includes a nonlinear wave term (à la NLSE) together with an advection term that can describe the initial stage of wave breaking. The NLSE and the Dysthe equations (Dysthe Proc. R. Soc. Lond. A, vol. 369, 1979, pp. 105–114) can be easily derived from the super compact equation. This equation is also suitable for analytical studies as well as for numerical simulation. Moreover, this equation also allows one to derive a spatial version of the water wave equation that describes experiments in flumes and canals.

JFM classification

Mathematical Foundations: Hamiltonian theory Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave breaking
Type
Papers
Information
Journal of Fluid Mechanics , Volume 828 , 10 October 2017 , pp. 661 - 679
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Copyright
© 2017 Cambridge University Press 

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Dyachenko et al. supplementary movie 1

Collision of two breathers

Download Dyachenko et al. supplementary movie 1(Video)
Video 21.1 MB

Dyachenko et al. supplementary movie 2

Freak wave pre-breaking

Download Dyachenko et al. supplementary movie 2(Video)
Video 6.9 MB

Dyachenko et al. supplementary movie 3

Freak wave pre-breaking zoomed

Download Dyachenko et al. supplementary movie 3(Video)
Video 5.3 MB

Dyachenko et al. supplementary movie 4

Breathers in a flume

Download Dyachenko et al. supplementary movie 4(Video)
Video 37.6 MB