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.
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, Trudy Mat. Inst. Steklova, 289, MAIK Nauka/Interperiodica, Moscow, 2015, 163–177; Proc. Steklov Inst. Math., 289 (2015), 152–166
\Bibitem{Ili15}
\by A.~T.~Il'ichev
\paper Envelope solitary waves and dark solitons at a water--ice interface
\inbook Selected issues of mathematics and mechanics
\bookinfo Collected papers. In commemoration of the 150th anniversary of Academician Vladimir Andreevich Steklov
\serial Trudy Mat. Inst. Steklova
\yr 2015
\vol 289
\pages 163--177
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm3614}
\crossref{https://doi.org/10.1134/S0371968515020090}
\elib{https://elibrary.ru/item.asp?id=23738467}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2015
\vol 289
\pages 152--166
\crossref{https://doi.org/10.1134/S0081543815040094}
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https://www.mathnet.ru/eng/tm3614
https://doi.org/10.1134/S0371968515020090
https://www.mathnet.ru/eng/tm/v289/p163
This publication is cited in the following 11 articles:
Hess M., “Validity of the Nonlinear Schrodinger Approximation For Quasilinear Dispersive Systems With More Than One Derivative”, Math. Meth. Appl. Sci., 45:3 (2022), 1725–1751
A. T. Il'ichev, “Effective wavelength of envelope waves on the water surface beneath an ice sheet: small amplitudes and moderate depths”, Theoret. and Math. Phys., 208:3 (2021), 1182–1200
A. T. Il'ichev, V. J. Tomashpolskii, “Characteristic parameters of nonlinear surface envelope waves beneath an ice cover under pre-stress”, Wave Motion, 86 (2019), 11–20
T.-T. Jia, Y.-T. Gao, G.-F. Deng, L. Hu, “Quintic time-dependent-coefficient derivative nonlinear Schrodinger equation in hydrodynamics or fiber optics: bilinear forms and dark/anti-dark/gray solitons”, Nonlinear Dyn., 98:1 (2019), 269–282
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
A. T. Il'ichev, “Envelope solitary waves at a water-ice interface: the case of positive initial tension”, Math. Montisnigri, 43 (2018), 49–57
A. T. Il'ichev, “Stability of solitary waves in membrane tubes: A weakly nonlinear
analysis”, Theoret. and Math. Phys., 193:2 (2017), 1593–1601
A. T. Il'ichev, Izv. Ross. Akad. Nauk Mekh. Zhidk. Gaza, 2016, no. 3, 37–47
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
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
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