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Behavior near the focal points of asymptotic solutions to the Cauchy problem for the linearized shallow water equations with initial localized perturbations. (English) Zbl 1178.35301

Summary: We study the behavior of the wave part of asymptotic solutions to the Cauchy problem for linearized shallow water equations with initial perturbations localized near the origin. The global representation for these solutions based on the generalized Maslov canonical operator was given earlier. The asymptotic solutions are also localized in the neighborhood of certain curves (fronts). The simplification of general formulas and the behavior of asymptotic solutions in a neighborhood of the regular part of fronts was also given earlier.
Here, the behavior of asymptotic solutions in a neighborhood of the focal point of the fronts is discussed in detail and the proof of formulas announced earlier for the wave equation is given. This paper can be regarded as a continuation of the paper of [the first two authors and A. I. Shafarevich, Russ. J. Math. Phys. 15, No. 2, 192–221 (2008; Zbl 1180.35336)].

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
35C15 Integral representations of solutions to PDEs
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
35B40 Asymptotic behavior of solutions to PDEs
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology

Citations:

Zbl 1180.35336
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[1] V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1974) [in Russian]; Graduate Texts in Mathematics 60 (Springer, Berlin, 1978).
[2] V. I. Arnold, Singulariries of Caustics and Wavefronts (Kluwer Academic Publishers, 1990).
[3] V. M. Babich, V. S. Buldyrev, and I.A. Molotkov, Space-Time Ray Method, Linear and Nonlinear Waves (Leningrad University Press, Leningrad, 1985) [in Russian]. · Zbl 0678.35002
[4] V. M. Babich, ”Propagation of Nonstationary Waves and Caustics,” Zap. Nauchn. Sem. POMI 246(32), 228–260 (1958).
[5] V. V. Belov, S.Yu. Dobrokhotov, and T.Ya. Tudorovskiy, J. Engrg. Math. 55(1–4), 183–237 (2006). · Zbl 1110.81080
[6] M. V. Berry, ”Tsunami Asymptotics,” New J. Phys. 7(129), 1–18 (2005).
[7] M. V. Berry, ”Focused Tsunami Waves,” Proc. R. Soc. Ser. A 463, 3055–3071 (2007). · Zbl 1158.86001
[8] V. A. Borovikov and M.Ya. Kelbert, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 173–174 (1984) [Fluid Dynamics 31 (4), (1996)].
[9] V. A. Borovikov, Uniform Stationary Phase Method IEE, Electromagnetic Waves 40 (London, 1994).
[10] S. F. Dotsenko, B.Yu. Sergievskii, and L.V. Cherkasov, ”Space Tsunami Waves Generated by Alternating Displacement of the Ocean Surface,” in Tsunami Research 1 (Moscow, 1986), pp. 7–14.
[11] S.Yu. Dobrokhotov, ”Maslov’s Methods in Linearized Theory of Gravitational Waves on the Liquid Surface,” Dokl. Akad. Nauk SSSR 269, 76–80 (1983) [Sov. Phys. Doklady 28, 229–231 (1983)]. · Zbl 0533.76013
[12] S.Yu. Dobrokhotov, V.M. Kuzmina, and P. N. Zhevandrov, ”Asymptotic of the Solution of the Cauchy-Poisson Problem in a Layer of Nonconstant Thickness,” Mat. Zametki 53(6), 141–145 (1993) [Math. Notes 53 (6), 657–660 (1993)].
[13] S.Yu. Dobrokhotov, V. P. Maslov, P. N. Zhevandrov, and A. I. Shafarevich, ”Asymptotic Fast-Decreasing Solution of Linear, Strictly Hyperbolic Systems with Variable Coefficients,” Math. Notes 49(4), 355–365 (1991). · Zbl 0735.35089
[14] S. Dobrokhotov, S. Sekerzh-Zenkovich, B. Tirozzi, and T. Tudorovski, ”Asymptotic Theory of Tsunami Waves: Geometrical Aspects and the Generalized Maslov Representation,” Publications of the Kyoto Research Mathematical Institute, 118–152 (2006).
[15] S. Dobrokhotov, S. Sekerzh-Zenkovich, B. Tirozzi, and T. Tudorovski, ”Description of Tsunami Propagation Based on the Maslov Canonical Operator,” Doklady Mathematics 74(1), 592–596 (2006). · Zbl 1152.35089
[16] S. Dobrokhotov, S. Sekerzh-Zenkovich, B. Tirozzi, and B. Volkov, ”Explicit Asymptotics for Tsunami Waves in Framework of the Piston Model,” Russ. J. Earth Sciences 8(ES403), 1–12 (2006).
[17] S.Yu. Dobrokhotov, A. I. Shafarevich, and B. Tirozzi, ”Localized Wave and Vortical Solutions of Linear Hyperbolic Systems and Their Application to the Linear Shallow Water Equations,” Russ. J. Math. Phys. 15(2), 192–221 (2008). · Zbl 1180.35336
[18] S.Yu. Dobrokhotov, A. I. Shafarevich, and B. Tirozzi, ”Representations of Rapidly Decreasing Functions by the Maslov Canonical Operator,” Mat. Zametki 82(5), 713–717 (2007) [Math. Notes 82 (5), 713–717 (2007)]. · Zbl 1344.58014
[19] S.Yu. Dobrokhotov, S. Sinitsyn, and B. Tirozzi, ”Asymptotics of Localized Solutions of the One-Dimensional Wave Equation with Variable Velocity. I. The Cauchy Problem,” Russ. J. Math. Phys. 14(1), 28–56 (2007). · Zbl 1146.35055
[20] S.Yu. Dobrokhotov and P.N. Zhevandrov, ”Asymptotic Expansions and the Maslov Canonical Operator in the Linear Theory of Water Waves. I. Main Constructions and Equations for Surface Gravity Waves,” Russ. J. Math. Phys. 10, 1–31 (2003). · Zbl 1065.76026
[21] F. V. Dolzhanskii, V. A. Krymov, and D.Yu. Manin, ”Stability and Vortex Structures of Quasi-Two-Dimensional Shear Flows,” Soviet Physics Uspekhi 33, 495–520 (1990).
[22] A. S. Krukovskii and D. S. Lukin, ”Construction of Uniform Diffraction Theory Based on Methods of Boundary and Angle Catastrophes,” Radiotechnics and Electronics 43(9), 1044–1060 (1998).
[23] V. P. Maslov, Perturbation Theory and Asymptotic Methods (Moscow Univ., Moscow, 1965); Théorie des perturbations et méthodes asymptotiques (Dunod, Gauthier-Villars, Paris, 1972).
[24] V. P. Maslov, Operational Methods (Nauka, Moscow, 1973; Mir, Moscow, 1973).
[25] V. P. Maslov, The Complex WKB Method for Nonlinear Equations (Nauka, Moscow: 1977) [in Russian]; English transl.: The Complex WKB Method for Nonlinear Equations. I. Linear Theory (Birkhäuser, Basel-Boston-Berlin, 1994).
[26] V. P. Maslov and M.V. Fedoryuk, Quasiclassical Approximation for the Equations of Quantum Mechanics (Nauka, Moscow, 1976) [in Russian]; Semi-Classical Approximation in Quantum Mechanics, Mathematical Physics and Applied Mathematics 7, Contemp. Math. 5 (D. Reidel, Dordrecht, 1981).
[27] V. P. Maslov and M.V. Fedoryuk, ”Logarithmic Asymptotics of Fast Decaying Solutions to Petrovskii Type Hyperbolic Systems,” Math. Notes 45(5), 50–62 (1989). · Zbl 0698.35014
[28] V. P. Maslov and G. A. Ome’janov, Geometric Asymptotics for Nonlinear PDE. I, Transl. Math. Monogr. 202 (Amer. Math. Society, Providence, 2001).
[29] C. C. Mei, The Applied Dynamics of Ocean Surface Waves (World Scientific, Singapore, 1989). · Zbl 0991.76003
[30] A. N. Nayfeh, Perturbation Methods (Wiley, New York, 1973). · Zbl 0265.35002
[31] E. N. Pelinovski, Hydrodynamics of Tsunami Waves (Nizhnii Novgorod, 1996) [in Russian].
[32] S.Ya. Sekerzh-Zenkovich, ”Simple Asymptotic Solution of the Cauchy-Poisson Problem for Head Waves,” Russ. J. Math. Phys. 16(2), 315–322 (2009). · Zbl 1179.35248
[33] J. J. Stoker, Water Waves (Interscience, New York, 1957).
[34] M. I. Vishik and L.A. Lusternik, ”Regular Degeneration and Boundary Layer for Linear Differential Equations with Small Parameter,” Uspekhi Mat. Nauk 12(5), 3–122 (1957) [Amer. Math. Soc. Transl. (2) 20, 239–364 (1962)]. · Zbl 0087.29602
[35] S. Wang, ”The Propagation of the Leading Wave,” in ASCE Specialty Conference on Coastal Hydrodynamics (University of Delaware, 1987), pp. 1657–1670.
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