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 Tr. Mat. Inst. Steklova, 2013, Volume 281, Pages 170–187 (Mi tm3469)

Homogenization and dispersion effects in the problem of propagation of waves generated by a localized source

V. V. Grushinab, S. Yu. Dobrokhotovac, S. A. Sergeevac

a Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Russia
b Moscow State Institute of Electronics and Mathematics — Higher School of Economics, Moscow, Russia
c Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia

Abstract: We construct asymptotic solutions to the wave equation with velocity rapidly oscillating against a smoothly varying background and with localized initial perturbations. First, using adiabatic approximation in the operator form, we perform homogenization that leads to a linearized Boussinesq-type equation with smooth coefficients and weak “anomalous” dispersion. Then, asymptotic solutions to this and, as a consequence, to the original equations are constructed by means of a modified Maslov canonical operator; for initial perturbations of special form, these solutions are expressed in terms of combinations of products of the Airy functions of a complex argument. On the basis of explicit formulas obtained, we analyze the effect of fast oscillations of the velocity on the solution fronts and solution profiles near the front.

DOI: https://doi.org/10.1134/S0371968513020143

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English version:
Proceedings of the Steklov Institute of Mathematics, 2013, 281, 161–178

Bibliographic databases:

UDC: 517.9

Citation: V. V. Grushin, S. Yu. Dobrokhotov, S. A. Sergeev, “Homogenization and dispersion effects in the problem of propagation of waves generated by a localized source”, Modern problems of mechanics, Collected papers. Dedicated to Academician Andrei Gennad'evich Kulikovskii on the occasion of his 80th birthday, Tr. Mat. Inst. Steklova, 281, MAIK Nauka/Interperiodica, Moscow, 2013, 170–187; Proc. Steklov Inst. Math., 281 (2013), 161–178

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3469
• https://doi.org/10.1134/S0371968513020143
• http://mi.mathnet.ru/eng/tm/v281/p170

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Erratum

This publication is cited in the following articles:
1. V. V. Grushin, S. Yu. Dobrokhotov, “Homogenization in the Problem of Long Water Waves over a Bottom Site with Fast Oscillations”, Math. Notes, 95:3 (2014), 324–337
2. V. V. Grushin, S. Yu. Dobrokhotov, S. A. Sergeev, “Correction to the paper “Homogenization and dispersion effects in the problem of propagation of waves generated by a localized source” (Proc. Steklov Inst. Math. 281, 161–178 (2013))”, Proc. Steklov Inst. Math., 288 (2015), 265–265
3. Dobrokhotov S.Yu., Nazaikinskii V.E., Tirozzi B., “on a Homogenization Method For Differential Operators With Oscillating Coefficients”, Dokl. Math., 91:2 (2015), 227–231
4. Dobrokhotov S.Yu., Grushin V.V., Sergeev S.A., Tirozzi B., “Asymptotic theory of linear water waves in a domain with nonuniform bottom with rapidly oscillating sections”, Russ. J. Math. Phys., 23:4 (2016), 455–474
5. D. A. Karaeva, A. D. Karaev, V. E. Nazaikinskii, “Homogenization method in the problem of long wave propagation from a localized source in a basin over an uneven bottom”, Differ. Equ., 54:8 (2018), 1057–1072
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