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On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves

Published online by Cambridge University Press:  26 April 2006

Vladimir P. Krasitskii
Vladimir P. Krasitskii*
Affiliation:
P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, 23 Krasikova, 117218 Moscow, Russia
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Abstract

Many studies of weakly nonlinear surface waves are based on so-called reduced integrodifferential equations. One of these is the widely used Zakharov four-wave equation for purely gravity waves. But the reduced equations now in use are not Hamiltonian despite the Hamiltonian structure of exact water wave equations. This is entirely due to shortcomings of their derivation. The classical method of canonical transformations, generalized to the continuous case, leads automatically to reduced equations with Hamiltonian structure. In this paper, attention is primarily paid to the Hamiltonian reduced equation describing the combined effects of four- and five-wave weakly nonlinear interactions of purely gravity waves. In this equation, for brevity called five-wave, the non-resonant quadratic, cubic and fourth-order nonlinear terms are eliminated by suitable canonical transformation. The kernels of this equation and the coefficients of the transformation are expressed in explicit form in terms of expansion coefficients of the gravity-wave Hamiltonian in integral-power series in normal variables. For capillary–gravity waves on a fluid of finite depth, expansion of the Hamiltonian in integral-power series in a normal variable with accuracy up to the fifth-order terms is also given.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 272 , 10 August 1994 , pp. 1 - 20
Copyright
Copyright © Cambridge University Press 1994

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References

Broer, L. J. F. 1974 On the Hamiltonian theory of surface waves. Appl. Sci. Res. 29, 430–446.Google Scholar
Caponi, E. A., Saffman, P. G. & Yuen, H. C. 1982 Instability and confined chaos in a nonlinear dispersive wave system. Phys. Fluids 25, 2159–2166.Google Scholar
Crawford, D. R., Saffman, P. G. & Yuen, H. C. 1980 Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves. Wave Motion 2, 1–16.Google Scholar
Hasselmann, K. 1962 On the nonlinear energy transfer in a gravity wave spectrum. Part 1. J. Fluid Mech. 12, 481–500.Google Scholar
Krasitskii, V. P. 1990 Canonical transformations in a theory of weakly nonlinear waves with a nondecay dispersion law. Sov. Phys. JETP (English transl.) 71 (5), 921–927.Google Scholar
Krasitskii, V. P. 1991 Canonical transformations and reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. In Proc. Nonlinear Water Waves Workshop (ed. Peregrine, D. H.), pp 66–74. University of Bristol.Google Scholar
Krasitskii, V. P. & Kalmykov, V. A. 1993 On four-wave reduced equation for surface gravity waves. Izv. Russian Acad. Sci. Atmos. Ocean Phys. 29, 237–243 (in Russian).Google Scholar
Milder, D. M. 1990 The effects of truncation on surface-wave Hamiltonians. J. Fluid Mech. 216, 249–262.Google Scholar
Miles, J. W. 1977 On Hamilton's principle for surface waves. J. Fluid Mech. 83, 153–158.Google Scholar
Mmiles, J. & Salmon, R. 1985 Weakly dispersive nonlinear gravity waves. J. Fluid Mech. 157, 519–531.Google Scholar
Stiassnie, M. & Shemer, L. 1984 On modification of the Zakharov equation for surface gravity waves. J. Fluid Mech. 143, 47–67.Google Scholar
Stiassnie, M. & Shemer, L. 1987 Energy computations for evolution of class I and II instabilities of Stokes waves. J. Fluid Mech. 174, 229–312.Google Scholar
Tick, L. J. 1959 A non-linear random model of gravity waves. Part 1. J. Math. Mech. 8, 643–652.Google Scholar
West, B. J. 1981 Deep Water Gravity Waves. Lecture Notes in Physics, vol. 146. Springer.Google Scholar
Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67–229.Google Scholar
Zakharov, V. E. 1966 On instability of waves in nonlinear dispersive media. Sov. Phys. JETP (in Russian) 51, 1107–1114.Google Scholar
Zzakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of deep fluid. J. Appl. Mech. Tech. Phys. (English transl.) 2, 190–194.Google Scholar
Zakharov, V. E. 1974 Hamiltonian formalism for waves in nonlinear dispersive media. Izv. Vysshykh Uchebnykh Zavedenii Radiofiz. (in Russian) 17, 431–453.Google Scholar
Zakharov, V. E. 1984 Kolmogorov spectra in weak turbulence problems. In Handbook of Plasma Physics, vol. 2, pp. 3–36. Elsevier.Google Scholar
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