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 Mat. Zametki, 2017, Volume 102, Issue 6, Pages 828–835 (Mi mz11716)

On the Asymptotics of a Bessel-Type Integral Having Applications in Wave Run-Up Theory

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: Rapidly oscillating integrals of the form
\begin{equation*} I(r,h)=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{\tfrac ih F(r\cos\phi)} G(r\cos\phi)  d\phi, \end{equation*}
where $F(r)$ is a real-valued function with nonvanishing derivative, arise when constructing asymptotic solutions of problems with nonstandard characteristics such as the Cauchy problem with spatially localized initial data for the wave equation with velocity degenerating on the boundary of the domain; this problem describes the run-up of tsunami waves on a shallow beach in the linear approximation. The computation of the asymptotics of this integral as $h\to0$ encounters difficulties owing to the fact that the stationary points of the phase function $F(r\cos\phi)$ become degenerate for $r=0$. For this integral, we construct an asymptotics uniform with respect to $r$ in terms of the Bessel functions $\mathbf{J}_0(z)$ and $\mathbf{J}_1(z)$ of the first kind.

Keywords: rapidly oscillating integral, degeneration of stationary points, uniform asymptotics, Bessel function, wave equation.

 Funding Agency Grant Number Russian Science Foundation 16-11-10282 This work was supported by the Russian Science Foundation under grant 16-11-10282.

DOI: https://doi.org/10.4213/mzm11716

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English version:
Mathematical Notes, 2017, 102:6, 756–762

Bibliographic databases:

Document Type: Article
UDC: 517.9

Citation: S. Yu. Dobrokhotov, V. E. Nazaikinskii, “On the Asymptotics of a Bessel-Type Integral Having Applications in Wave Run-Up Theory”, Mat. Zametki, 102:6 (2017), 828–835; Math. Notes, 102:6 (2017), 756–762

Citation in format AMSBIB
\Bibitem{DobNaz17} \by S.~Yu.~Dobrokhotov, V.~E.~Nazaikinskii \paper On the Asymptotics of a Bessel-Type Integral Having Applications in Wave Run-Up Theory \jour Mat. Zametki \yr 2017 \vol 102 \issue 6 \pages 828--835 \mathnet{http://mi.mathnet.ru/mz11716} \crossref{https://doi.org/10.4213/mzm11716} \elib{http://elibrary.ru/item.asp?id=30737867} \transl \jour Math. Notes \yr 2017 \vol 102 \issue 6 \pages 756--762 \crossref{https://doi.org/10.1134/S0001434617110141} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000418838500014} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85039455205} 

• http://mi.mathnet.ru/eng/mz11716
• https://doi.org/10.4213/mzm11716
• http://mi.mathnet.ru/eng/mz/v102/i6/p828

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This publication is cited in the following articles:
1. 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”, Math. Notes, 104:4 (2018), 471–488
2. Dobrokhotov S.Yu. Tolstova O.L. Sekerzh-Zenkovich S.Ya. Vargas C.A., “Influence of the Elastic Base of a Basin on the Propagation of Waves on the Water Surface”, Russ. J. Math. Phys., 25:4 (2018), 459–469
3. S. I. Kabanikhin, O. I. Krivorotko, “An algorithm for source reconstruction in nonlinear shallow-water equations”, Comput. Math. Math. Phys., 58:8 (2018), 1334–1343
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