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Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations. (English) Zbl 1180.35336

Linear hyperbolic systems of differential equations for an m-D vector function are considered. There is a large discussion on the asymptotics of solutions to the associated Cauchy problem with fast-decaying or localized initial data. The aim is to present explicit formulas for the asymptotic solutions of the Cauchy problem. Authors concentrate and prove results for the Cauchy problem with localized initial data for the linearized shallow water equations. A main point of proofs is the possibility of representation of fast decaying functions by means of Maslov canonical operator over a specific Lagrange manifold.

MSC:

35L45 Initial value problems for first-order hyperbolic systems
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
47G30 Pseudodifferential operators
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B47 Vortex flows for incompressible inviscid fluids
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[1] V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1974) [in Russian]; Graduate Texts in Mathematics 60, Ed. by S. Axler, F. W. Gehring, and K. A. Ribet (Springer, Berlin, 1978).
[2] V. I. Arnol’d, Singularities of Caustics and Wave Fronts (Kluwer, Dordrecht, 1990).
[3] V. M. Babich, V. S. Buldyrev, and I. A. Molotkov, The Space-Time Ray Method, Linear and Nonlinear Waves (Leningrad. Univ., Leningrad, 1985) [in Russian]. · Zbl 0678.35002
[4] V. V. Belov, S. Yu. Dobrokhotov, and T. Ya. Tudorovskiy, ”Operator Separation of Variables for Adiabatic Problems in Quantum and Wave Mechanics,” J. Engrg. Math. 55(1–4), 183–237 (2006). · Zbl 1110.81080
[5] M. V. Berry, ”Tsunami Asymptotics,” New J. of Phys. 7(129), 1–18 (2005).
[6] V. A. Borovikov and M. Ya. Kelbert, ”The Field near the Wave Front in a Cauchy-Poisson Problem,” Izv. AN SSSR Ser. Mekh. Zhidk. Gaza (2), 173–174 (1984) [Fluid Dynamics 19 (2), 321–323 (1984)].
[7] L. M. Brekhovskikh and O. A. Godin, Acoustics of Layered Media II, Point Source and Bounded Beam (Springer, 1992). · Zbl 0753.76003
[8] S. Yu. Dobrokhotov, ”Maslov’s Methods in the Linearized Theory of Gravitational Waves on a Liquid Surface,” Dokl. Akad. Nauk SSSR 269(1), 76–80 (1983) [Sov. Phys.-Doklady 28, 229–231 (1983)]. · Zbl 0533.76013
[9] S. Yu. Dobrokhotov, P. N. Zhevandrov, and V. M. Kuzmina, ”Asymptotic Behavior of the Solution of the Cauchy-Poisson Problem in a Layer of Nonconstant Thickness,” Mat. Zametki 53(6), 141–145 (1993) [Math. Notes 53 (5–6), 657–660 (1993)].
[10] S. Yu. Dobrokhotov, P. N. Zhevandrov, V. P. Maslov, and A. I. Shafarevich, ”Asymptotic Rapidly Decreasing Solutions of Linear Strictly Hyperbolic Systems with Variable Coefficients,” Mat. Zametki 49(4), 31–46 (1991) [Math. Notes 49 (3–4), 355–365 (1991)]. · Zbl 0735.35089
[11] S. Dobrokhotov, S. Sekerzh-Zenkovich, B. Tirozzi, and T. Tudorovski, ”Asymptotic Theory of Tsunami Waves: Geometrical Aspects and the Generalized Maslov Representation,” in Publications of Kyoto Institute (RIMS), Kokyuroku 1457, Vol. 4, 118–153 (2006).
[12] S. Dobrokhotov, S. Sekerzh-Zen’kovich, B. Tirozzi, and T. Tudorovski, ”Description of Tsunami Propagation Based on the Maslov Canonical Operator,” Dokl. Akad. Nauk 409(2) 1–5 (2006) [Dokl. Math. 74 (1), 592–596 (2006)]. · Zbl 1152.35089
[13] 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(4), ES4003, 1–12 (2006).
[14] S. Yu. Dobrokhotov, A. I. Shafarevich, and B. Tirozzi, ”The Cauchy-Riemann Conditions and Localized Asymptotic Solutions of Linearized Equations of Shallow Water Theory,” Prikl. Mat. Mekh. 69(5), 804–809 (2005) [J. Appl. Math. Mech. 69 (5), 720–725 (2005)]. · Zbl 1100.76506
[15] S. Yu. Dobrokhotov, A. I. Shafarevich, and B. Tirozzi, ”Representations of Rapidly Decaying Functions by the Maslov Canonical Operator,” Mat. Zametki 82(5), 792–796 (2007) [Math. Notes 82 (5–6), 713–717 (2007)]. · Zbl 1344.58014
[16] 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), 1–31 (2003). · Zbl 1065.76026
[17] F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin, ”Stability and Vortex Structures of Quasi-Two-Dimensional Shear Flows,” Uspekhi Fiz. Nauk 160(7), 1–47 (1990) [Sov. Phys. Usp. 33, 495–520 (1990)].
[18] V. Guillemin and S. Sternberg, Geometric Asymptotics (Amer. Math. Soc., Providence, 1977). · Zbl 0364.53011
[19] V. Ivrii, ”Linear Hyperbolic Equations,” in Modern Problems of Mathematics, Fundamental Research 33 (VINITI, Moscow), pp. 157–250 [Encyclopaedia of Mathematical Sciences, 33, Springer-Verlag, Berlin, 1993, pp. 149–235].
[20] V. V. Kozlov, The General Theory of Vortices (Udm. Gos. Univ., Izhevsk, 1998) [in Russian].
[21] Yu. A. Kravtsov and Yu. I. Orlov, Geometrical Optics of Inhomogeneous Media (Nauka, Moscow, 1980; Springer, Berlin, 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, (Moskov. Univ., Moscow, 1965) [in Russian]; Théorie des perturbations et méthodes asymptotiques (Dunod, Gauthier-Villars, Paris, 1972). · Zbl 0653.35002
[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; Birkhäuser, Basel-Boston-Berlin, 1994). · Zbl 0449.58001
[26] V. P. Maslov and M. V. Fedoryuk, Semiclassical Approximation in Quantum Mechanics (Nauka, Moscow: 1976; D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981). · Zbl 0449.58002
[27] V. P. Maslov and M. V. Fedoryuk, ”Logarithmic Asymptotics of Rapidly Decreasing Solutions of Equations That Are Hyperbolic in the Sense of Petrovskii,” Mat. Zametki 45(5), 50–62 (1989) [Math. Notes 45 (5–6), 382–391 (1989)].
[28] V. P. Maslov and G. A. Omel’yanov, Geometric Asymptotics for Nonlinear PDE. I (Amer. Math. Society, Providence, RI, 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] G. Panatti, H. Spohn, and S. Teufel, ”Effective Dynamics for Bloch Electrons: Peierls Substitution and Beyond,” Comm. Math. Phys. 242, 547–578 (2003). · Zbl 1058.81020
[32] J. Pedlosky, Geophysical Fluid Dynamics (Berlin-Heidelberg-New York, Springer, 1982). · Zbl 0429.76001
[33] E. N. Pelinovskii, Hydrodynamics of Tsunami Waves (IPF RAN, Nizhnii Novgorod, 1996) [in Russian].
[34] G. M. Reznik and V. Zeitlin, ”Interaction of Free Rossby Waves with Semi-Transparent Equatorial Waveguide. Part 1. Wave Triads,” Phys. D 226(1), 55–79 (2007). · Zbl 1112.76014
[35] F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Vols. 1–2 (Plenum, New York-London, 1980).
[36] B. R. Vainberg [Vaĭnberg], Asymptotic Methods in Equations of Mathematical Physics (Moscow University, Moscow, 1982; Gordon and Breach, New York, 1989).
[37] M. I. Vishik and L. A. Lyusternik, ”Regular Degeneration and Boundary Layer for Linear Differential Equations with Small Parameter,” Uspekhi Mat. Nauk 12(5), 3–122 (1957) [American Math. Society Transl. (2) 20, 239–364 (1962)]. · Zbl 0087.29602
[38] V. I. Yudovich, The Linearization Method in Hydrodynamical Stability Theory (Rostov Univ., Rostov, 1984); English transl. in Transl. Math. Monogr. 74 (American Mathematical Society, Providence).
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