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

This article is cited in 17 scientific papers (total in 17 papers)

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

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Received: 13.09.2010

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
\by S.~Yu.~Dobrokhotov, V.~E.~Naza{\v\i}kinski{\v\i}, B.~Tirozzi
\paper Asymptotic solutions of the two-dimensional model wave equation with degenerating velocity and localized initial data
\jour Algebra i Analiz
\yr 2010
\vol 22
\issue 6
\pages 67--90
\jour St. Petersburg Math. J.
\yr 2011
\vol 22
\issue 6
\pages 895--911

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    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. V. P. Maslov, “Two-fluid picture of supercritical phenomena”, Theoret. and Math. Phys., 180:3 (2014), 1096–1129  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    9. Maslov V.P., “New Construction of Classical Thermodynamics and Ud-Statistics”, Russ. J. Math. Phys., 21:2 (2014), 256–284  crossref  mathscinet  zmath  isi  elib  scopus
    10. Nazaikinskii V.E., “Maslov's Canonical Operator For Degenerate Hyperbolic Equations”, Russ. J. Math. Phys., 21:2 (2014), 289–290  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  crossref  mathscinet  isi  elib
    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  mathnet  crossref  crossref  mathscinet  isi  elib
    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  mathnet  crossref  crossref  mathscinet  isi  elib
    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  crossref  mathscinet  isi  scopus
    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  crossref  isi
    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  mathnet  crossref  crossref  isi  elib
    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  mathnet  crossref
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