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 Tr. Mat. Inst. Steklova, 2015, Volume 289, Pages 163–177 (Mi tm3614)

Envelope solitary waves and dark solitons at a water–ice interface

A. T. Il'ichev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: The article is devoted to the study of some self-focusing and defocusing features of monochromatic waves in basins with horizontal bottom under an ice cover. The form and propagation of waves in such basins are described by the full 2D Euler equations. The ice cover is modeled by an elastic Kirchhoff–Love plate and is assumed to be of considerable thickness so that the inertia of the plate is taken into account in the formulation of the model. The Euler equations involve the additional pressure from the plate that is freely floating at the surface of the fluid. Obviously, the self-focusing is closely connected with the existence of so-called envelope solitary waves, for which the envelope speed (group speed) is equal to the speed of filling (phase speed). In the case of defocusing, solitary envelope waves are replaced by so-called dark solitons. The indicated families of solitary waves are parametrized by the wave propagation speed and bifurcate from the quiescent state. The dependence of the existence of envelope solitary waves and dark solitons on the basin's depth is investigated.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.1134/S0371968515020090

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English version:
Proceedings of the Steklov Institute of Mathematics, 2015, 289, 152–166

Bibliographic databases:

UDC: 532.59

Citation: A. T. Il'ichev, “Envelope solitary waves and dark solitons at a water–ice interface”, Selected issues of mathematics and mechanics, Collected papers. In commemoration of the 150th anniversary of Academician Vladimir Andreevich Steklov, Tr. Mat. Inst. Steklova, 289, MAIK Nauka/Interperiodica, Moscow, 2015, 163–177; Proc. Steklov Inst. Math., 289 (2015), 152–166

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3614
• https://doi.org/10.1134/S0371968515020090
• http://mi.mathnet.ru/eng/tm/v289/p163

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. T. Il'ichev, Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza, 2016, no. 3, 37–47
2. V. V. Markov, G. B. Sizykh, “Exact solutions of the Euler equations for some two-dimensional incompressible flows”, Proc. Steklov Inst. Math., 294 (2016), 283–290
3. A. T. Il'ichev, A. P. Chugainova, “Spectral stability theory of heteroclinic solutions to the Korteweg–de Vries–Burgers equation with an arbitrary potential”, Proc. Steklov Inst. Math., 295 (2016), 148–157
4. Il'ichev A.T., “Solitary wave packets beneath a compressed ice cover”, Fluid Dyn., 51:3 (2016), 327–337
5. A. T. Il'ichev, “Stability of solitary waves in membrane tubes: A weakly nonlinear analysis”, Theoret. and Math. Phys., 193:2 (2017), 1593–1601
6. A. Il'ichev, “Physical parameters of envelope solitary waves at a water-ice interface”, Mathematical Methods and Computational Techniques in Science and Engineering II, AIP Conf. Proc., 1982, ed. N. Bardis, Amer. Inst. Phys., 2018, 020036-1
7. A. T. Il'ichev, “Envelope solitary waves at a water-ice interface: the case of positive initial tension”, Math. Montisnigri, 43 (2018), 49–57
8. Il'ichev A.T., Tomashpolskii V.J., “Characteristic Parameters of Nonlinear Surface Envelope Waves Beneath An Ice Cover Under Pre-Stress”, Wave Motion, 86 (2019), 11–20
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