S. Yu. Dobrokhotov, V. E. Nazaǐkinskiǐ, B. Tirozzi, “Asymptotic solutions of the two-dimensional model wave equation with degenerating velocity and localized initial data”, Algebra i Analiz, 22:6 (2010), 67–90; St. Petersburg Math. J., 22:6 (2011), 895

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Algebra i Analiz, 2010, Volume 22, Issue 6, Pages 67–90 (Mi aa1214)

This article is cited in 23 scientific papers (total in 23 papers)

Research Papers

Asymptotic solutions of the two-dimensional model wave equation with degenerating velocity and localized initial data

S. Yu. Dobrokhotov^ab, V. E. Nazaĭkinskiĭ^ab, B. Tirozzi^c

^a Moscow Institute of Physics and Technology, Moscow, Russia
^b A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia
^c University of Rome "La Sapienza", Rim, Italy

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Abstract: The Cauchy problem is considered for the two-dimensional wave equation with velocity $c=\sqrt x_1$ on the half-plane $\{x_1\geq0,\ x_2\}$, with initial data localized in a neighborhood of the point $(1,0)$. This problem serves as a model problem in the theory of beach run-up of long small-amplitude surface waves excited by a spatially localized instantaneous source. The asymptotic expansion of the solution is constructed with respect to a small parameter equal to the ratio of the source linear size to the distance from the $x_2$-axis (the shoreline). The construction involves Maslov's canonical operator modified to cover the case of localized initial conditions. The relationship of the solution with the geometrical optics ray diagram corresponding to the problem is analyzed. The behavior of the solution near the $x_2$-axis is studied. Simple solution formulas are written out for special initial data.

Keywords: wave equation with degenerating velocity, asymptotic expansion, wave front, singular Lagrangian manifold, run-up.

Received: 13.09.2010

English version:
St. Petersburg Mathematical Journal, 2011, Volume 22, Issue 6, Pages 895–911
DOI: https://doi.org/10.1090/S1061-0022-2011-01175-6

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Document Type: Article

Language: Russian

Citation: S. Yu. Dobrokhotov, V. E. Nazaǐkinskiǐ, B. Tirozzi, “Asymptotic solutions of the two-dimensional model wave equation with degenerating velocity and localized initial data”, Algebra i Analiz, 22:6 (2010), 67–90; St. Petersburg Math. J., 22:6 (2011), 895–911

Citation in format AMSBIB

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https://www.mathnet.ru/eng/aa/v22/i6/p67

This publication is cited in the following 23 articles:

V.E. Nazaikinskii, “On the Phase Spaces for a Class of Boundary-Degenerate Equations”, Russ. J. Math. Phys., 31:4 (2024), 713
V. E. Nazaikinskii, “On an elliptic operator degenerating on the boundary”, Funct. Anal. Appl., 56:4 (2022), 324–326
S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. I. Shafarevich, “Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations”, Russian Math. Surveys, 76:5 (2021), 745–819
Kuznetsov S.V., “Abnormal Dispersion of Fundamental Lamb Modes in Fg Plates: II-Symmetric Versus Asymmetric Variation”, Z. Angew. Math. Phys., 72:2 (2021), 73
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
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
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
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
S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. A. Tolchennikov, “Uniform Asymptotics of the Boundary Values of the Solution in a Linear Problem on the Run-Up of Waves on a Shallow Beach”, Math. Notes, 101:5 (2017), 802–814
An. G. Marchuk, “The assessment of tsunami heights above the parabolic bottom relief within the wave-ray approach”, Num. Anal. Appl., 10:1 (2017), 17–27
Lozhnikov D.A., Nazaikinskii V.E., “Method For the Analysis of Long Water Waves Taking Into Account Reflection From a Gently Sloping Beach”, Pmm-J. Appl. Math. Mech., 81:1 (2017), 21–28
Minenkov D.S., “Asymptotics Near the Shore For 2D Shallow Water Over Sloping Planar Bottom”, Proceedings of the International Conference Days on Diffraction (Dd) 2017, eds. Motygin O., Kiselev A., Goray L., Suslina T., Kazakov A., Kirpichnikova A., IEEE, 2017, 240–243
S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Characteristics with Singularities and the Boundary Values of the Asymptotic Solution of the Cauchy Problem for a Degenerate Wave Equation”, Math. Notes, 100:5 (2016), 695–713
V. E. Nazaikinskii, “On the Representation of Localized Functions in $\mathbb R^2$ by Maslov's Canonical Operator”, Math. Notes, 96:1 (2014), 99–109
V. E. Nazaikinskii, “The Maslov Canonical Operator on Lagrangian Manifolds in the Phase Space Corresponding to a Wave Equation Degenerating on the Boundary”, Math. Notes, 96:2 (2014), 248–260
V. P. Maslov, “Two-fluid picture of supercritical phenomena”, Theoret. and Math. Phys., 180:3 (2014), 1096–1129
Maslov V.P., “New Construction of Classical Thermodynamics and Ud-Statistics”, Russ. J. Math. Phys., 21:2 (2014), 256–284
Nazaikinskii V.E., “Maslov's Canonical Operator For Degenerate Hyperbolic Equations”, Russ. J. Math. Phys., 21:2 (2014), 289–290
Sipailo P.A., “On the Numerical Simulation of the Propagation of the Wave Front of a Tsunami Wave in a Pool of Variable Depth with Run-Up on the Beach”, Russ. J. Math. Phys., 20:3 (2013), 383–386
Dobrokhotov S.Yu., Nazaikinskii V.E., Tirozzi B., “Two-Dimensional Wave Equation with Degeneration on the Curvilinear Boundary of the Domain and Asymptotic Solutions with Localized Initial Data”, Russ. J. Math. Phys., 20:4 (2013), 389–401

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