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Asymptotic Solution of the Cauchy Problem with Localized Initial Data for a Wave Equation with Small Dispersion Effects

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Abstract

An asymptotic solution of the Cauchy problem for a multidimensional wave equation with a localized initial function is constructed taking into account small dispersion effects. The three-dimensional case and the construction of the asymptotics for it are considered separately. With a special choice of the initial function in the three-dimensional case, the asymptotics is presented explicitly using the Airy and Scorer functions. As an example where dispersion effects arise, we consider the homogenization of a wave equation with a rapidly oscillating coefficient and construct an asymptotic solution of such an equation.

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ACKNOWLEDGMENTS

The author is grateful to S.Yu. Dobrokhotov, A.I. Shafarevich, A.A.Tolchennikov, and A.Yu. Anikin for valuable advice and guidance.

Funding

The work was supported by the Ministry of Science and Higher Education of the Russian Federation, state registration no. AAAA-A20-120011690131-7.

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526, Russia

    S. A. Sergeev

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  1. S. A. Sergeev
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Correspondence to S. A. Sergeev.

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Translated by V. Potapchouck

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Sergeev, S.A. Asymptotic Solution of the Cauchy Problem with Localized Initial Data for a Wave Equation with Small Dispersion Effects. Diff Equat 58, 1376–1395 (2022). https://doi.org/10.1134/S00122661220100081

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  • DOI: https://doi.org/10.1134/S00122661220100081

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