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Higher-order modulation effects on solitary wave envelopes in deep water Part 2. Multi-soliton envelopes

Published online by Cambridge University Press:  26 April 2006

T. R. Akylas
T. R. Akylas
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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Abstract

Previous experimental and numerical work indicates that an initially symmetric deep-water wave pulse of uniform frequency and moderately small steepness evolves in an asymmetric manner and eventually separates into distinct wave groups, owing to higher-order modulation effects, not accounted for by the nonlinear Schrödinger equation (NLS). Here perturbation methods are used to provide analytical confirmation of this group splitting on the basis of the more accurate envelope equation of Dysthe (1979). It is demonstrated that an initially symmetric multisoliton wave envelope, consisting of N bound NLS solitons, ultimately breaks up into N separate groups; a procedure is devised for determining the relative speed changes of the individual groups. The case of a bi-soliton (N = 2) is discussed in detail, and the analytical predictions are compared to numerical results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 224 , March 1991 , pp. 417 - 428
Copyright
© 1991 Cambridge University Press

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References

Ablowitz, M. J. & Benney, D. J., 1970 The evolution of multiphase modes for nonlinear dispersive waves. Stud. Appl. Maths 49, 225–238.Google Scholar
Akylas, T. R.: 1989 Higher-order modulation effects on solitary wave envelopes in deep water. J. Fluid Mech. 198, 387–397 (referred to herein as I).Google Scholar
Dysthe, K. B.: 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369, 105–114.Google Scholar
Feir, J. E.: 1967 Discussion: some results from wave pulse experiments. Proc. R. Soc. Lond. A 299, 54–58.Google Scholar
Lo, E. Y.: 1986 Long-time evolution of surface waves in coastal waters. Doctoral dissertation, Department of Civil Engineering, MIT.
Lo, E. & Mei, C. C., 1985 A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation. J. Fluid Mech. 150, 395–416.Google Scholar
Peregrine, D. H.: 1983 Water waves, nonlinear Schrödinger equations and their solutions. J. Austral. Math. Soc. B 25, 16–43.Google Scholar
Satsuma, J. & Yajima, N., 1974 Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media. Suppl. Progr. Theor. Phys. 55, 284–306.Google Scholar
Su, M. Y.: 1982 Evolution of groups of gravity waves with moderate to high steepness. Phys. Fluids 25, 2167–2174.Google Scholar
Zakharov, V. E. & Shabat, A. B., 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69.Google Scholar