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On localized solutions in nonlinear Faraday resonance

Published online by Cambridge University Press:  26 April 2006

E. W. Laedke  and
K. H. Spatschek
E. W. Laedke
Affiliation:
Institut fur Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, D-4000 Düsseldorf, Germany
K. H. Spatschek
Affiliation:
Institut fur Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf, D-4000 Düsseldorf, Germany
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Abstract

The dynamics of a nonlinear modulated cross-wave of resonant frequency ω1 and carrier frequency ω ≈ ω1 is considered. The wave is excited in a long channel of width 6 that contains water of depth d, which is subjected to a vertical oscillation of frequency 2ω. As has been shown by Miles (1984b), the complex amplitude satisfies a cubic Schrödinger equation with weak damping and parametric driving. The stability of its solitary wave solution is considered here in various parameter regions. We find that in a certain regime the solitary wave is stable. Completely new is the result of instability outside this parameter regime. The instability has also been verified numerically. It is shown that the final stage of solitary wave instability is a cnoidal-wave-type solution.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 223 , February 1991 , pp. 589 - 601
Copyright
© 1991 Cambridge University Press

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References

Blaha, R., Laedke, E. W. & Spatschbk, K. H., 1987 Maximum growth rates for packets of waves in water of finite depth. Phys. Fluids 30, 264–266.Google Scholar
Guggenheimer, J. & Holmes, P. H., 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector fields. Springer.
Laedke, E. W. & Spatschek, K. H., 1979 On the applicability of the variation of action method to some one-field solitons. J. Math. Phys. 20, 1838–1841.Google Scholar
Larraza, A. & Puttermann, S., 1984 Theory of non-propagating surface-wave solitons. J. Fluid Mech. 148, 443–449.Google Scholar
Lichtenberg, A. J. & Lieberman, M. A., 1983 Regular and Stochastic Motion, p. 383. Springer.
Makhankov, V. G.: 1978 Dynamics of of classical solitons. Phys. Rep. 35, 1–128.Google Scholar
Miles, J. W.: 1984a Nonlinear Faraday 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
Spatschek, K. H., Pietsch, H., Laedke, E. W. & Eickermann, Th., 1989 On the role of soliton solutions in temporal chaos: examples for plasmas and related systems. In Singular Behavior and Nonlinear Dynamics, pp. 555–564. World Scientific.
Whitham, G. B.: 1974 Linear and Nonlinear Waves, pp. 601–603. Wiley.
Wu, J., Keolian, R. & Rudnick, I., 1984 Observation of a non-propagating hydrodynamic soliton. Phys. Rev. Lett. 52, 1421–1424.Google Scholar
Zakharov, V. E., Kuznetsov, E. A. & Rubenchik, A. M., 1986 Soliton stability. In Solitons, pp. 503–554. Elsevier.