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Asymptotic solutions of the linear shallow-water equations with localized initial data. (English) Zbl 1325.76036

Summary: A new asymptotic method for solving Cauchy problems with localized initial data (perturbation) for the linearized shallow-water equation is suggested. The solution is decomposed into two parts: waves and vortices. Metamorphosis of the profile takes place for the wave part: it is localized in the neighborhood of the initial point and later on it is localized in the neighborhood of the front (1-D curve on the plane). Initially, the front is a smooth curve, but as the time increases turning (focal) and self-intersection points might appear. The vortical part is localized in the neighborhood of a point moving along the trajectory of the basic velocity vector field. Both parts are described by very simple formulae taking into account the form of the initial perturbation. These formulae are expressed by means of elementary functions for some special choice of the initial data. Due to the profile-metamorphosis phenomenon the method leading to the final formulae is not elementary. It makes use of semiclassical approximations and ray expansions. It is based on the generalization of the construction known as the Maslov canonical operator and boundary-layer expansions.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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[1] Dolzhanskii FV, Krymov VA, Manin DYu (1990) Stability and vortex structures of quasi-two-dimensional shear flows. Sov Phys Uspekhi 33: 495–520
[2] Mei CC (1989) The applied dynamics of ocean surface waves. World Scientific, Singapore · Zbl 0991.76003
[3] Pedlosky J (1982) Geophysical fluid dynamics. Springer, New York · Zbl 0429.76001
[4] Shokin YuI, Chubarov LB, Marchuk AG, Simonov KV (1989) Numerical experiment in tsunami problem. ”Nauka”, Siberian Division, Novosibirsk
[5] Dobrokhotov S, Sekerzh-Zenkovich S, Tirozzi B, Tudorovski T (2006) Description of tsunami propagation based on the Maslov canonical operator. Dokl Math 74(1): 592–596 · Zbl 1152.35089
[6] Dobrokhotov SYu, Shafarevich AI, Tirozzi B (2005) The Cauchy–Riemann conditions and localized asymptotic solutions of linearized equations of shallow water theory. J Appl Math Mech 69(5): 720–725 · Zbl 1100.76506
[7] Dobrokhotov SYu, Sekerzh-Zenkovich SYa, Tirozzi B, Volkov B (2006) Explicit asymptotics for tsunami waves in framework of the piston model. Russ J Earth Sci 8(ES403): 1–12
[8] Dobrokhotov SYu, Shafarevich AI, Tirozzi B (2008) Localized wave and vortical solutions to linear hyperbolic systems and their application to the linear shallow water equations. Russ J Math Phys 15(2): 192–221 · Zbl 1180.35336
[9] Dobrokhotov SYu, Tirozzi B, Vargas CA (2009) Behavior near the focal points of asymptotic solutions to the Cauchy problem for the linearized shallow water equations with initial localized perturbations. Russ J Math Phys 16(2): 228–245 · Zbl 1178.35301
[10] Babich VM, Buldyrev VS, Molotkov IA (1985) Space-time ray method, linear and nonlinear waves. Leningrad University Press, Leningrad. Cambridge University Press (English translation, 1999)
[11] Kravtsov YuA, Orlov YuI (1999) Caustics, catastrophes and wave fields. Springer, Berlin
[12] Maslov VP, Fedoriuk MV (1981) Semi-classical approximation in quantum mechanics. In: Mathematical physics and applied mathematics 7, contemporary mathematics 5. D. Reidel Publishing Co. ix, Dordrecht, 301 pp
[13] Vainberg BR (1982) Asymptotic methods in equations of mathematical physics. Moskow University Publications, Moscow, 293 pp (English translation: Vaĭnberg BR (1989) Asymptotic methods in equations of mathematical physics. Gordon and Breach Science Publishers, New York, 498 pp)
[14] Maslov VP (1965) Perturbation theory and asymptotic methods. Moscow University Publications, Moscow, 549 pp. // Maslov VP (1972) Théorie des perturbations et méthodes asymptotiques. Dunod, Gauthier-Villars xii, Paris, 384 pp
[15] Dobrokhotov SYu, Zhevandrov PN (2003) 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 · Zbl 1065.76026
[16] Arnold VI (1990) Singularities of caustics and wavefronts. Kluwer, Dordrecht
[17] Stoker JJ (1957) Water waves. Interscience, New York
[18] Berry MV (2007) Focused tsunami waves. Proc R Soc A 463: 3055–3071 · Zbl 1158.86001
[19] Nayfeh AN (1973) Perturbation methods. Wiley, New York · Zbl 0265.35002
[20] Vishik MI, Lusternik LA (1957) Regular degeneration and boundary layer for linear differential equations with small parameter. Uspekhi Mat Nauk 12(5):3–122 (English translation: (1962) Am Math Soc Transl 20(2):239–364
[21] Maslov VP (1977) The complex WKB method for nonlinear equations. Nauka, Moscow (English translation: Maslov VP (1994) The complex WKB method for nonlinear equations. I. Linear theory. Birkhäuser, Basel)
[22] Babich VM (1958) Propagation of nonstationary waves and caustics. Zapiski Nauchnyh Seminarov LOMI 246(32): 228–260
[23] Berry MV (2005) Tsunami asymptotics. New J Phys 7(129): 1–18
[24] Brekhovskikh LM, Godin OA (1992) Acoustics of layered media II, point source and bounded beams. Springer, New York · Zbl 0753.76003
[25] Kiselev AP (1980) Generation of modulated vibrations, in generation of modulated vibrations in nonhomogeneous media. Mathematical questions of wave propagation theory, 11 . Zapiski Nauchnyh Seminarov LOMI 104: 111–122
[26] Maslov VP, Fedorjuk MV (1989) Logarithmic asymptotics of fast decaying solutions to Petrovskii type hyperbolic systems. Math Notes 45(5): 50–62 · Zbl 0698.35014
[27] Dobrokhotov SYu, Maslov VP, Zhevandrov PN, Shafarevich AI (1991) Asymptotic fast-decreasing solution of linear, strictly hyperbolic systems with variable coefficients. Math Notes 49(4): 355–365 · Zbl 0735.35089
[28] Pelinovski EN (1996) Hydrodynamics of tsunami waves. Applied Physics Institute Press, Nizhnii Novgorod (in Russian)
[29] Borovikov VA, Kelbert MA (1984) The field near the wave front in a Cauchy–Poisson problem. Fluid Dyn 31(4): 321–323 · Zbl 0551.76015
[30] Dobrokhotov SYu, Kuzmina VM, Zhevandrov PN (1993) Asymptotic of the solution of the Cauchy–Poisson problem in a layer of non constant thickness. Math Notes 53(6): 657–660 · Zbl 0810.35084
[31] Dotsenko SF, Sergievskii BYu, Cherkasov LV (1986) Space tsunami waves generated by alternating displacement of the ocean surface. Tsunami Res 1: 7–14
[32] Wang S (1987) The propagation of the leading wave. In: ASCE specialty conference on coastal hydrodynamics, University of Delaware, June 29–July 1, pp 657–670
[33] Sekerzh-Zenkovich SYa (2009) Simple asymptotic solution to the Cauchy–Poisson problem for leading waves. Russ J Math Phys 16(2): 215–222
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