Asymptotic Solutions of the Wave Equation with Degenerate Velocity and with Right-Hand Side Localized in Space and Time

Автор(и)

  • Anatoly Anikin Ishlinsky Institute for Problems in Mechanics RAS, pr. Vernadskogo, 101-1, Moscow, 119526, Russia
    Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Moscow Region, 141701, Russia
  • Sergey Dobrokhotov Ishlinsky Institute for Problems in Mechanics RAS, pr. Vernadskogo, 101-1, Moscow, 119526, Russia
    Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Moscow Region, 141701, Russia
  • Vladimir Nazaikinskii Ishlinsky Institute for Problems in Mechanics RAS, pr. Vernadskogo, 101-1, Moscow, 119526, Russia
    Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Moscow Region, 141701, Russia

DOI:

https://doi.org/10.15407/mag14.04.393

Ключові слова:

хвильове рівняння, асимптотичний розв'язок, канонічний оператор Маслова.

Анотація

Вивчається задача Коші для неоднорідного двовимірного хвильового рівняння зі змінними коефіцієнтами та нульовими початковими даними. Вважається, що права частина локалізована в просторі та часі. Рівняння розглядається в області з межею (берегом). Вважається, що швидкість на березі зникає як квадратний корінь відстані до берега, тобто хвильове рівняння має задану на кривій особливість. Ця крива і визначає межу області, в якій вивчається задача. Основний результат роботи – ефективні асимптотичні формули для розв'язку зазначеної задачі, включаючи окіл берега.

Mathematics Subject Classification: 34E20, 35L05, 35Q35.

Посилання

A.Yu. Anikin, S.Yu. Dobrokhotov, and V.E. Nazaikinskii, Simple asymptotics for a generalized wave equation with degenerating velocity and their applications to the linearized long wave run-up problem, Mat. Zametki 104 (2018), No. 4, 483–504 (Russian); Engl. transl.: Math. Notes 104, No. 3–4 (2018), 471–488.

G.F. Carrier and H.P. Greenspan, Water waves of finite amplitude on a sloping beach, J. Fluid Mech. 4 (1958), No. 1, 97–109.

S.Yu. Dobrokhotov and A.Yu. Anikin, Approximation of solutions of the twodimensional wave equation with variable velocity and localized right-hand side using some “simple” solutions, Mat. Zametki 100 (2016), No. 6, 825–837 (Russian); Engl. transl.: Math. Notes 100 (2016), No. 6, 796–806. https://doi.org/10.1134/S0001434616110195

S.Yu. Dobrokhotov, D.S. Minenkov, V.E. Nazaikinskii, and 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.

S.Yu. Dobrokhotov, D.S. Minenkov, V.E. Nazaikinskii, and 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.

S.Yu. Dobrokhotov and 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, Moscow, 2015, 98–139 (Russian).

S.Yu. Dobrokhotov and V.E. Nazaikinskii, Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation, Mat. Zametki 100 (2016), No. 5, 710–731 (Russian); Engl. transl.: Math. Notes 100 (2016), No. 5–6, 695—713. https://doi.org/10.1134/S0001434616110195

S.Yu. Dobrokhotov, V.E. Nazaikinskii, and B. Tirozzi, Asymptotic solution of the one-dimensional wave equation with localized initial data and with degenerating velocity. I, Russ. J. Math. Phys. 17 (2010), No. 4, 434–447.

S.Yu. Dobrokhotov, V.E. Nazaikinskii, and B. Tirozzi, Asymptotic solutions of the two-dimensional model wave equation with degenerating velocity and localized initial data, Algebra Anal. 22 (2010), No. 6, 67–90 (Russian); Engl. transl.: St. Petersburg Math. J. 22 (2011), No. 6, 895–911. https://doi.org/10.1090/S1061-0022-2011-01175-6

S.Yu. Dobrokhotov, V.E. Nazaikinskii, and 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 (2013), No. 4, 389–401.

S. Dobrokhotov, A. Shafarevich, and B. Tirozzi, Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations, Russ. J. Math. Phys. 15 (2008), No. 2, 192–221.

S.Yu. Dobrokhotov, B. Tirozzi, and A.I. Shafarevich, Representations of rapidly decreasing functions by the Maslov canonical operator, Mat. Zametki 82 (2007), No. 5, 792–796 (Russian); Engl. transl.: Math. Notes 82 (2007), No. 5–6, 713–717. https://doi.org/10.1134/S0001434607110144

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

M.V. Fedoriuk and V.P. Maslov, Semi-Classical Approximation in Quantum Mechanics, Nauka, Moscow, 1976 (Russian); Engl. transl.: Reidel, Dordrecht, 1981.

V. A. Fock, On the canonical transformation in classical and quantum mechanics, Vestn. Leningrad. Univ. 16 (1959), 67-–70 (Russian); Engl. transl.: Acta Phys. Acad. Sci. Hungaricae 27 (1–4) (1969), 219–224.

J.B. Keller and H.B. Keller, Water Wave Run-Up on a Beach, 1964, ONR, Research Report Contract No. NONR-3828(00), Dept. of the Navy, Washington, DC.

A. Kozelkov, V. Efremov, A. Kurkin, E. Pelinovsky, N. Tarasova, and D. Strelets, Three-dimensional numerical simulation of tsunami waves based on Navier-Stokes equation, Science of Tsunami Hazards 36 (2017), No. 4, 45–58.

C.C. Mei, The Applied Dynamics of Ocean Surface Waves, World Sci., Singapore, 1989.

V.E. Nazaikinskii, On the representations of localized functions in R2 by the Maslov canonical operator, Mat. Zametki 96 (2014), No. 1, 88–100 (Russian); Enlg. transl.: Math. Notes 96 (2014), No. 1–2, 99–109. https://doi.org/10.1134/S0001434614070268

V.E. Nazaikinskii, Phase space geometry for a wave equation degenerating on the boundary of the domain, Mat. Zametki 92 (2012), No. 1, 153–156 (Russian); Engl. transl,: Math. Notes 92 (2012), No. 1–2, 144–148. https://doi.org/10.1134/S0001434612070176

V.E. Nazaikinskii, The Maslov canonical operator on Lagrangian manifolds in the phase space corresponding to a wave equation degenerating on the boundary, Mat. Zametki 96 (2014), No. 2, 261–276 (Russian); Engl. transl.: Math. Notes 96 (2014), No. 1–2, 248–260. https://doi.org/10.1134/S0001434614070268

O.A. Oleinik and E.V. Radkevic, Second order equations with nonnegative characteristic form, American mathematical society, Providence, Rhode Iland Plenum Press, New York-London, 1973. https://doi.org/10.1007/978-1-4684-8965-1

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

E.N. Pelinovsky and R.Kh. Mazova, Exact analytical solutions of nonlinear problems of tsunami wave run-up on slopes with different profiles, Natural Hazards 6 (1992), No. 3, 227–249.

S.Ya. Sekerzh-Zen’kovich, Simple asymptotic solution of the Cauchy–Poisson problem for head waves, Russ. J. Math. Phys. 16 (2009) No. 2, 315–322.

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

J.J. Stoker, Water Waves: The Mathematical Theory with Applications, John Wiley and Sons, New York, 1992. https://doi.org/10.1002/9781118033159

C.E. Synolakis, On the roots of f (z) = J0 (z) − iJ1 (z), Quart. Appl. Math. 46, (1988), No. 1, 105–107.

T. Vukašinac and P. Zhevandrov, Geometric asymptotics for a degenerate hyperbolic equation, Russ. J. Math. Phys. 9 (2002), No. 3, 371–381.

S. Wang, B. Le Mehaute, and Chia-Chi Lu, Effect of dispersion on impulsive waves, Marine Geophysical Researchers 9 (1987), 95–111. https://doi.org/10.1007/BF00338252

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Anikin, A.; Dobrokhotov, S.; Nazaikinskii, V. Asymptotic Solutions of the Wave Equation with Degenerate Velocity and with Right-Hand Side Localized in Space and Time. Журн. мат. фіз. анал. геом. 2018, 14, 393-405.

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