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Super compact equation for water waves. / Dyachenko, A. I.; Kachulin, D. I.; Zakharov, V. E.
In: Journal of Fluid Mechanics, Vol. 828, 10.10.2017, p. 661-679.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Super compact equation for water waves
AU - Dyachenko, A. I.
AU - Kachulin, D. I.
AU - Zakharov, V. E.
N1 - Publisher Copyright: © 2017 Cambridge University Press.
PY - 2017/10/10
Y1 - 2017/10/10
N2 - 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.
AB - 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.
KW - Hamiltonian theory
KW - surface gravity waves
KW - wave breaking
KW - EVOLUTION
KW - FREE-SURFACE HYDRODYNAMICS
KW - GRAVITY-WAVES
KW - FLUID
KW - BREATHERS
KW - MODEL
KW - DEEP-WATER
UR - http://www.scopus.com/inward/record.url?scp=85034797961&partnerID=8YFLogxK
U2 - 10.1017/jfm.2017.529
DO - 10.1017/jfm.2017.529
M3 - Article
AN - SCOPUS:85034797961
VL - 828
SP - 661
EP - 679
JO - Journal of Fluid Mechanics
JF - Journal of Fluid Mechanics
SN - 0022-1120
ER -
ID: 14726139