Geodesic
Asymptotic solutions of the wave equation with degenerate velocity and with right-hand side localized in space and time
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018) no. 4, pp. 393-405 
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. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018) no. 4, pp. 393-405. http://geodesic.mathdoc.fr/item/JMAG_2018_14_4_a0/
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     title = {Asymptotic solutions of the wave equation with degenerate velocity and with right-hand side localized in space and time},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
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     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2018_14_4_a0/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We study the Cauchy problem for the inhomogeneous two-dimensional wave equation with variable coefficients and zero initial data. The right-hand side is assumed to be localized in space and time. The equation is considered in a domain with a boundary (shore). The velocity is assumed to vanish on the shore as a square root of the distance to the shore, that is, the wave equation has a singularity on the curve. This curve determines the boundary of the domain where the problem is studied. The main result of the paper is efficient asymptotic formulas for the solution of this problem, including the neighborhood of the shore.

[1] Math. Notes, 104:3–4 (2018), 471–488 | DOI | DOI | MR | MR

[2] G. F. Carrier, H. P. Greenspan, “Water waves of finite amplitude on a sloping beach”, J. Fluid Mech., 4:1 (1958), 97–109 | DOI | MR | Zbl

[3] Math. Notes, 100:6 (2016), 796–806 | DOI | DOI | MR | Zbl

[4] S. Yu. Dobrokhotov, D. S. Minenkov, V. E. Nazaikinskii, B. Tirozzi, “Functions of noncommuting operators in an asymptotic problem for a 2D wave equation with variable velocity and localized right-hand side”, Oper. Theory Adv. Appl., 228, Birkhauser, Basel, 2013, 95–126 | MR

[5] S. Yu. Dobrokhotov, D.S. Minenkov, V.E. Nazaikinskii, B. Tirozzi, “Simple exact and asymptotic solutions of the 1D run-up problem over a slowly varying (quasiplanar) bottom”, Theory and Applications in Mathematical Physics, World Sci., Singapore, 2015, 29–47 | DOI | MR

[6] S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Asymptotics of localized wave and vortex solutions of linearized shallow water equations”, Actual Problems of Mechanics, the book of papers dedicated to the 50th anniversary of the Ishlinsky Institute for Problems in Mechanics RAS, Nauka, M., 2015, 98–139 (Russian)

[7] Math. Notes, 100:5–6 (2016), 695–713 | DOI | DOI | MR | Zbl

[8] S. Yu. Dobrokhotov, V. E. Nazaikinskii, B. Tirozzi, “Asymptotic solution of the one-dimensional wave equation with localized initial data and with degenerating velocity. I”, Russ. J. Math. Phys., 17:4 (2010), 434–447 | DOI | MR | Zbl

[9] St. Petersburg Math. J., 22:6 (2011), 895–911 | DOI | MR | Zbl

[10] S. Yu. Dobrokhotov, V. E. Nazaikinskii, B. Tirozzi, “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 | DOI | MR | Zbl

[11] S. Dobrokhotov, A. Shafarevich, B. Tirozzi, “Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations”, Russ. J. Math. Phys., 15:2 (2008), 192–221 | DOI | MR | Zbl

[12] Math. Notes, 82:5–6 (2007), 713–717 | DOI | DOI | MR | Zbl

[13] S. F. Dotsenko, B. Yu. Sergievskii, L. V. Cherkasov, “Space tsunami waves generated by alternating displacement of the ocean surface”, Tsunami Research, 1 (1986), 7–14

[14] Reidel, Dordrecht, 1981 | MR | Zbl

[15] Acta Phys. Acad. Sci. Hungaricae, 27:1–4 (1969), 219–224 | DOI | MR

[16] J. B. Keller, H. B. Keller, Water Wave Run-Up on a Beach, ONR, Research Report Contract No. NONR-3828-00, Washington, DC, 1964

[17] A. Kozelkov, V. Efremov, A. Kurkin, E. Pelinovsky, N. Tarasova, D. Strelets, “Three-dimensional numerical simulation of tsunami waves based on Navier-Stokes equation”, Science of Tsunami Hazards, 36:4 (2017), 45–58

[18] C. C. Mei, The Applied Dynamics of Ocean Surface Waves, World Sci., Singapore, 1989 | Zbl

[19] Math. Notes, 96:1–2 (2014), 99–109 | DOI | DOI | MR | Zbl

[20] Math. Notes, 92:1–2 (2012), 144–148 | DOI | DOI | MR | Zbl

[21] Math. Notes, 96:1–2 (2014), 248–260 | DOI | DOI | MR | Zbl

[22] O. A. Oleinik, E. V. Radkevic, Second order equations with nonnegative characteristic form, American mathematical society, Providence, Rhode Iland; Plenum Press, 1973 | MR

[23] E. N. Pelinovskii, Hydrodynamics of Tsunami Waves, Inst. Prikl. Fiz., Nizhnii Novgorod, 1996 (Russian)

[24] E. N. Pelinovsky, R. Kh. Mazova, “Exact analytical solutions of nonlinear problems of tsunami wave run-up on slopes with different profiles”, Natural Hazards, 6:3 (1992), 227–249 | DOI

[25] S. Ya. Sekerzh-Zen'kovich, “Simple asymptotic solution of the Cauchy–Poisson problem for head waves”, Russ. J. Math. Phys., 16:2 (2009), 315–322 | DOI | MR | Zbl

[26] Y. I. Shokin, L. B. Chubarov, An. G. Marchuk, K. V. Simonov, Computational Experiment in the Tsunami Problem, Nauka, Novosibirsk, 1989 (Russian)

[27] J. J. Stoker, Water Waves: The Mathematical Theory with Applications, John Wiley and Sons, New York, 1992 | MR

[28] C. E. Synolakis, “On the roots of $f(z)=J_0(z)-iJ_1(z)$”, Quart. Appl. Math., 46:1 (1988), 105–107 | DOI | MR | Zbl

[29] T. Vukašinac, P. Zhevandrov, “Geometric asymptotics for a degenerate hyperbolic equation”, Russ. J. Math. Phys., 9:3 (2002), 371–381 | MR | Zbl

[30] S. Wang, B. Le Mehaute, Chia-Chi Lu, “Effect of dispersion on impulsive waves”, Marine Geophysical Researchers, 9 (1987), 95–111 | DOI