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Bifurcations and nonlinear dynamics of surface waves in Faraday resonance

Published online by Cambridge University Press:  26 April 2006

H. Friedel ,
E. W. Laedke  and
K. H. Spatschek
H. Friedel
Affiliation:
Institut für Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
E. W. Laedke
Affiliation:
Institut für Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
K. H. Spatschek
Affiliation:
Institut für Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
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Abstract

The nonlinear dynamics of nonlinear modulated cross-waves of resonant frequency ω1 and carrier frequency ω ≈ ω1 is investigated. In a long channel of width b, that contains fluid of depth d and which is subjected to a vertical oscillation of frequency 2ω, the wave can appear in solitary form. As has been shown previously, the solitary wave is only stable in a certain parameter regime; depending on damping and driving amplitudes the wave becomes unstable. The nonlinear development of the instabilities of solitary waves is the central problem of this paper. It is shown how instabilities are saturated following generic routes to chaos in time with spatially coherent structures. Finally, the case of time-modulated driving amplitudes is also considered. In most cases it appears that nonlinear waves of simple spatial structures take part in the nonlinear dynamics, but a few cases of spatial chaos are also reported.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 284 , 10 February 1995 , pp. 341 - 358
Copyright
© 1995 Cambridge University Press

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References

Barashenkov, I. V., Bogdan, M. M. & Korobov, V. I. 1991 Stability diagram of the phase-locked solitons in the parametrically driven, damped nonlinear Schrödinger equation. Europhys. Lett. 15, 113–118.Google Scholar
Chen, X.-N. & Wei, R.-J. 1994 Dynamic behaviour of a non-propagating soliton under a periodically modulated oscillation. J. Fluid Mech. 259, 291–303.Google Scholar
Ciliberto, S. & Gollub, J. P. 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 391–398.Google Scholar
Denardo, B., Wright, W., Putterman, S. & Larraza, A. 1990 Observation of a kink soliton on the surface of a liquid. Phys. Rev. Lett. 64, 1518–1521.Google Scholar
Ezerskii, A. B., Rabinovich, M. I., Rentov, V. P. & Starobinets, M. 1986 Spatiotemporal chaos in the parametric excitation of a capillary ripple. Sov. Phys. JETP 64, 1228–1236.Google Scholar
Funakoshi, M. & Inoue, S. 1987 Surface waves due to resonant horizontal oscillation. J. Fluid Mech. 192, 219–247.Google Scholar
Funakoshi, M. & Inoue, S. 1990 Bifurcations of limit cycles in surface waves due to resonant forcing. Fluid Dyn. Res. 5, 225–271.Google Scholar
Funakoshi, M. & Inoue, S. 1992 Stable periodic orbits of equations for resonantly forced water waves. J. Phys. Soc. Japan 61, 3411–3412.Google Scholar
Grauer, R. & Birnir, B. 1992 The center manifold and bifurcations of damped and driven sine-Gordon breathers. Physica D 56, 165–184.Google Scholar
Guthart, G. S. & Wu, T. Y.-T. 1991 Observation of a standing kink cross wave parametrically excited. Proc. R. Soc. Lond. A 434, 435–440.Google Scholar
Kambe, J. & Umeki, M. 1990 Nonlinear dynamics of two-mode interactions in parametric excitation of surface waves. J. Fluid Mech. 212, 373–393.Google Scholar
Laedke, E. W. & Spatschek, K. H. 1991 On localized solutions in nonlinear Faraday resonance. J. Fluid Mech. 223, 589–601.Google Scholar
Larraza, A. & Putterman, S. 1984 Theory of non-propagating surface-wave solitons. J. Fluid Mech. 148, 443–449.Google Scholar
Miles, J. W. 1984a Nonlinear Farady resonance. J. Fluid Mech. 146, 285–302.Google Scholar
Miles, J. W. 1984b Parametrically excited solitary waves. J. Fluid Mech. 148, 451–460.Google Scholar
Miles, J. W. 1994 A note on slowly modulated Faraday waves (preprint).
Miles, J. W. & Henderson, D. 1990 Parametrically forced surface waves. Ann. Rev. Fluid Mech. 22, 143–165.Google Scholar
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279–303.Google Scholar
Nozaki, K. & Bekki, N. 1983 Chaos in a perturbed nonlinear Schrödinger equation. Phys. Rev. Lett. 50, 1226–1229.Google Scholar
Sasaki, K. 1993 Standing-wave solitons on an interface between layered fluids in a channel. J. Phys. Soc. Japan 62, 2675–2684.Google Scholar
Simonelli, F. & Gollub, J. P. 1989 Surface mode interactions: effects of symmetry and degeneracy. J. Fluid Mech. 199, 471–494.Google Scholar
Spatschek, K. H. 1994 Soliton systems in the presence of amplification, irregularities, and damping. In Nonlinear coherent structures in physics and biology (ed. K. H. Spatschek & F. G. Mertens). Plenum, in press.
Spatschek, K.H., Pietsch, H., Laedke, E.W. & Eickermann, Th. 1990 Chaotic behaviour in time in nonlinear Schrödinger systems. In Nonlinear World, Vol. 2 (ed. V. G. Baryakhtar et al.), pp. 978–1001. World Scientific.
Umeki, M. 1991a Parametric dissipative nonlinear Schrödinger equation. J. Phys. Soc. Japan 60, 146–167.Google Scholar
Umeki, M. 1991b Faraday resonance in rectangular geometry. J. Fluid Mech. 227, 161–192.Google Scholar
Wei, R., Wang, B., Mao, Y., Miao, G. & Zheng, X. 1990 Further investigation of nonpropagating solitons and their transition to chaos. J. Acoust. Soc. Am. 88, 469–472.Google Scholar
Wu, J., Keolian, R. & Rudnick, I. 1984 Observation of non-propagating hydrodynamic soliton. Phys. Rev. Lett. 52, 1421–1424.Google Scholar
Yamada, T & Nozaki, K. 1989 Effects of dissipative perturbation on bound-state solitons of nonlinear Schrödinger equation. J. Phys. Soc. Japan 58, 1944–1947.Google Scholar