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

Research Papers

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

S. Yu. Dobrokhotovab, V. E. Nazaĭkinskiĭab, B. Tirozzic

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

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.

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English version:
St. Petersburg Mathematical Journal, 2011, 22:6, 895–911

Bibliographic databases:

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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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. Dobrokhotov S.Yu., Nazaikinskii V.E., Tirozzi B., “Asymptotic solution of the one-dimensional wave equation with localized initial data and with degenerating velocity. I”, Russian J. Math. Phys., 17:4 (2010), 434–447
2. V. E. Nazaikinskii, “Degenerate Wave Equation with Localized Initial Data: Asymptotic Solutions Corresponding to Various Self-Adjoint Extensions”, Math. Notes, 89:5 (2011), 749–753
3. V. E. Nazaikinskii, “Phase Space Geometry for a Wave Equation Degenerating on the Boundary of the Domain”, Math. Notes, 92:1 (2012), 144–148
4. 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
5. 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
6. 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
7. 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
8. V. P. Maslov, “Two-fluid picture of supercritical phenomena”, Theoret. and Math. Phys., 180:3 (2014), 1096–1129
9. Maslov V.P., “New Construction of Classical Thermodynamics and Ud-Statistics”, Russ. J. Math. Phys., 21:2 (2014), 256–284
10. Nazaikinskii V.E., “Maslov's Canonical Operator For Degenerate Hyperbolic Equations”, Russ. J. Math. Phys., 21:2 (2014), 289–290
11. 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
12. 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
13. 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
14. 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
15. 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
16. 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
17. 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
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