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 Mat. Zametki, 2018, Volume 104, Issue 4, Pages 483–504 (Mi mz11884)

Simple Asymptotics for a Generalized Wave Equation with Degenerating Velocity and Their Applications in the Linear Long Wave Run-Up Problem

A. Yu. Anikinab, S. Yu. Dobrokhotovab, V. E. Nazaikinskiiab

a Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow
b Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region

Abstract: Asymptotic solutions of the wave equation degenerating on the boundary of the domain (where the wave propagation velocity vanishes as the square root of the distance from the boundary) can be represented with the use of a modified canonical operator on a Lagrangian submanifold, invariant with respect to the Hamiltonian vector field, of the nonstandard phase space constructed by the authors in earlier papers. The present paper provides simple expressions in a neighborhood of the boundary for functions represented by such a canonical operator and, in particular, for the solution of the Cauchy problem for the degenerate wave equation with initial data localized in a neighborhood of an interior point of the domain.

Keywords: wave equation, nonstandard characteristics, run-up on a sloping beach, localized source, near-boundary asymptotics.

 Funding Agency Grant Number Russian Science Foundation 16-11-10282 This work was supported by the Russian Science Foundation under grant 16-11-10282 in accordance with paragraph 17.7 of Sec. 3.1. and paragraph (7) of Sec. 3.2 of the work schedule for this grant for the year 2017.

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DOI: https://doi.org/10.4213/mzm11884

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English version:
Mathematical Notes, 2018, 104:4, 471–488

Bibliographic databases:

UDC: 517.9

Citation: A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Simple Asymptotics for a Generalized Wave Equation with Degenerating Velocity and Their Applications in the Linear Long Wave Run-Up Problem”, Mat. Zametki, 104:4 (2018), 483–504; Math. Notes, 104:4 (2018), 471–488

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz11884
• https://doi.org/10.4213/mzm11884
• http://mi.mathnet.ru/eng/mz/v104/i4/p483

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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. Anatoly Anikin, Sergey Dobrokhotov, Vladimir Nazaikinskii, “Asymptotic solutions of the wave equation with degenerate velocity and with right-hand side localized in space and time”, Zhurn. matem. fiz., anal., geom., 14:4 (2018), 393–405
2. Dobrokhotov S.Yu., Nazaikinskii V.E., “Asymptotic Localized Solutions of the Shallow Water Equations Over a Nonuniform Bottom”, AIP Conference Proceedings, 2048, eds. Pasheva V., Popivanov N., Venkov G., Amer Inst Physics, 2018, 040026
3. A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “Asymptotics, Related to Billiards with Semi-Rigid Walls, of Eigenfunctions of the $\nabla D(x)\nabla$ Operator in Dimension 2 and Trapped Coastal Waves”, Math. Notes, 105:5 (2019), 789–794
4. Anikin A.Yu., Dobrokhotov S.Yu., Nazaikinskii V.E., Tsvetkova A.V., “Asymptotic Eigenfunctions of the Operator Delta D(X)Delta Defined in a Two-Dimensional Domain and Degenerating on Its Boundary and Billiards With Semi-Rigid Walls”, Differ. Equ., 55:5 (2019), 644–657
5. A. Yu. Anikin, D. S. Minenkov, “On the Run-Up for Two-Dimensional Shallow Water in the Linear Approximation”, Math. Notes, 106:2 (2019), 163–171
6. S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Nonstandard Lagrangian Singularities and Asymptotic Eigenfunctions of the Degenerating Operator $-\frac{d}{dx}D(x)\frac{d}{dx}$”, Proc. Steklov Inst. Math., 306 (2019), 74–89
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