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Three-wave resonance and averaged equations of the interaction of two waves in media described by a cubic Schrödinger equation

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Abstract

A numerical solution of a cubic Schrödinger equation established the phenomenon of Mach reflection of Stokes waves from a vertical wall [1]. In [2], this phenomenon was interpreted as the resonance interaction of three waves. This was based on the derivation of averaged equations of the interaction of two waves, the region of interaction of the incident and reflected waves being treated as a self-similar solution of these equations. The present paper establishes the possibility of describing these solutions by the relations of three-wave resonance; the mathematical significance of the resonance as splitting into two waves is revealed; and the properties of the averaged system are investigated.

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References

  1. D. K. P. Yue and C. C. Mei, “Forward diffraction of Stokes waves by a thin wedge,”J. Fluid Mech.,99, 33 (1980).

    Google Scholar 

  2. I. B. Bakholdin, “Averaged equations and discontinuities describing the propagation of Stokes waves with slowly varying parameters,”Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 5, 113 (1989).

    Google Scholar 

  3. A. V. Gurevich and L. P. Pitaevskii, “Nonstationary structure of a collisionless shock wave,”Zh. Eksp. Teor. Fiz.,65, 590 (1973).

    Google Scholar 

  4. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii.The Theory of Solitons: The Inverse Scattering Method [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  5. M. V. Pavlov, “Nonlinear Schrödinger equation and the Bogolyubov-Whitham method of averaging,”Teor. Mat. Fiz.,71, 351 (1987).

    Google Scholar 

  6. B. A. Dubrovin and S. P. Novikov, “Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory,”Usp. Mat. Nauk,44, 29 (1989).

    Google Scholar 

  7. I. B. Bakholdin, “Discontinuities of variables characterizing the propagation of solitary waves in a fluid layer,”Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 3, 87 (1984).

    Google Scholar 

  8. G. B. Whitham,Linear and Nonlinear Waves, Wiley-Interscience, New York (1974).

    Google Scholar 

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Authors and Affiliations

  1. Moscow

    I. B. Bakholdin

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  1. I. B. Bakholdin
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Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 107–116, January–February, 1992.

I thank A. G. Kulikovskii and A. A. Barmin for helpful discussions and also colleagues at the Mechanics Section of the V. A. Steklov Mathematics Institute of the USSR Akademy of Sciences for critical comments that stimulated interest in the paper.

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Bakholdin, I.B. Three-wave resonance and averaged equations of the interaction of two waves in media described by a cubic Schrödinger equation. Fluid Dyn 27, 81–87 (1992). https://doi.org/10.1007/BF01054177

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  • DOI: https://doi.org/10.1007/BF01054177

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