Geodesic
Integrability of truncated Hugoniot–Maslov chains for trajectories of mesoscale vortices on shallow water
Teoretičeskaâ i matematičeskaâ fizika, Tome 125 (2000) no. 3, pp. 491-518 Cet article a éte moissonné depuis la source Math-Net.Ru

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The problem of trajectories of “large” (mesoscale) shallow-water vortices manifests integrability properties. The Maslov hypothesis states that such vortices can be generated using solutions with weak pointlike singularities of the type of the square root of a quadratic form; such square-root singular solutions may describe the propagation of mesoscale vortices in the atmosphere (typhoons and cyclones). Such solutions are necessarily described by infinite systems of ordinary differential equations (chains) in the Taylor coefficients of solutions in the vicinities of singularities. A proper truncation of the “vortex chain” for a shallow-water system is a system of 17 nonlinear equations. This system becomes the Hill equation when the Coriolis force is constant and almost becomes the physical pendulum equations when the Coriolis force depends on the latitude. In a rough approximation, we can then explicitly describe possible trajectories of mesoscale vortices, which are analogous to oscillations of a rotating solid body swinging on an elastic thread. 
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S. Yu. Dobrokhotov. Integrability of truncated Hugoniot–Maslov chains for trajectories of mesoscale vortices on shallow water. Teoretičeskaâ i matematičeskaâ fizika, Tome 125 (2000) no. 3, pp. 491-518. http://geodesic.mathdoc.fr/item/TMF_2000_125_3_a6/

[1] V. P. Maslov, UMN, 35:2 (1980), 252–253

[2] G. G. Chernyi, Gazovaya dinamika, Nauka, M., 1986

[3] A. G. Kulikovskii, E. I. Sveshnikova, Nelineinye volny v uprugikh sredakh, Moskovskii litsei, M., 1998

[4] Dzh. B. Uizem, Lineinye i nelineinye volny, Mir, M., 1977 | MR

[5] F. V. Shugaev, L. S. Shtemenko, Propagation and Reflection of Shock Waves, World Sci., Singapore, 1998 | MR | Zbl

[6] V. P. Maslov, “O rasprostranenii udarnoi volny v izoentropicheskom nevyazkom gaze”, Itogi nauki i tekhniki. Sovrem. problemy matematiki, 8, ed. R. V. Gamkrelidze, VINITI, M., 1977, 199–271 | MR

[7] V. P. Maslov, G. A. Omelyanov, Sib. matem. zhurnal, 24:5 (1983), 172–182 | MR | Zbl

[8] M. A. Grinfeld, Prikladnaya matematika i mekhanika, 42:5 (1978), 883–898 | MR | Zbl

[9] S. Yu. Dobrokhotov, Russ. J. Math. Phys., 6:2 (1999), 137–173 ; 3, 282–313 | MR | Zbl | MR | Zbl

[10] V. G. Danilov, V. P. Maslov, V. M. Shelkovich, TMF, 114:1 (1998), 3–55 | DOI | MR | Zbl

[11] A. M. Obukhov, Izv. AN SSSR. Ser. geogr., 13:4 (1949), 281–306 | MR

[12] F. V. Dolzhanskii, V. A. Krymov, D. Yu. Manin, UFN, 160:7 (1990), 1–47 | DOI

[13] G. S. Golitsyn, Vvedenie v fiziku planetarnykh atmosfer, Gidrometeoizdat, L., 1973

[14] V. A. Gordin, Matematicheskie zadachi gidrodinamicheskogo prognoza pogody: analiticheskie aspekty, Gidrometeoizdat, L., 1987 | MR

[15] J. Pedlosky, Geophysical Fluid Dynamics, Springer, Berlin, 1982

[16] A. P. Khain, G. G. Sutyrin, Tropicheskie tsiklony i ikh vzaimodeistvie s okeanom, Gidrometeoizdat, L., 1983

[17] L. D. Landau, E. M. Lifshits, Gidrodinamika, Nauka, M., 1986 | MR

[18] G. M. Reznik, J. Fluid Mech., 240 (1992), 405–432 | DOI | MR | Zbl

[19] V. M. Gryanik, Okeanologiya, 26 (1988), 174

[20] V. V. Bulatov, Yu. V. Vladimirov, V. G. Danilov, S. Yu. Dobrokhotov, DAN, 338:1 (1994), 102–105 | MR | Zbl

[21] S. Yu. Dobrokhotov, TMF, 112:1 (1997), 47–66 | DOI | MR | Zbl

[22] S. Yu. Dobrokhotov, Tsepochki Gyugonio–Maslova uravneniya Khilla i fizicheskogo mayatnika dlya traektorii mezomasshtabnykh vikhrei na “melkoi vode”, Preprint No 663, Inst. probl. mekhaniki RAN, M., 2000

[23] V. P. Maslov, Teoriya vozmuschenii i asimptoticheskie metody, MGU, M., 1965 | MR

[24] V. M. Babich, Algebra i analiz, 3:5 (1991), 1–37 | MR

[25] N. N. Bogolyubov, Yu. A. Mitropolskii, Matematicheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974 | MR

[26] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, “Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki”, Itogi nauki i tekhniki. Sovrem. problemy matematiki. Fundamentalnye napravleniya, 3, VINITI, M., 1985 | MR

[27] V. F. Butuzov, A. B. Vasileva, Asimptoticheskie razlozheniya dlya singulyarno vozmuschennykh uravnenii, Nauka, M., 1973 | MR | Zbl

[28] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[29] V. N. Zhikharev, O neobkhodimykh usloviyakh suschestvovaniya i edinstvennosti tipa resheniya so slaboi rasprostranyayuscheisya osobennostyu, sosredotochennoi v tochke, dlya uravnenii gidrodinamiki v sluchae dvukh prostranstvennykh peremennykh, Dep. v VINITI No B86/8148, VINITI, M., 1986

[30] J. F. Colombeau, Elementary Introduction to New Generalized Functions, North Holland, 1985 | MR | Zbl

[31] Yu. V. Egorov, UMN, 45:5 (1990), 3–40 | MR | Zbl

[32] V. A. Arkadev, A. K. Pogrebkov, M. K. Polivanov, TMF, 53:2 (1982), 163–180 ; 54:1 (1983), 23–37 | MR | MR | Zbl

[33] V. G. Danilov, G. A. Omel'yanov, Truncation of a Chain Hugoniot-type Conditions for Shock Waves and its Justification for the Hopf Equation, Preprint ESI 502, The Ervin Schrödinger Int. Inst. for Mat. Phys., Vienna, 1997 | MR | Zbl

[34] R. Ravindran, P. Prasad, Appl. Math. Lett., 3:2(3) (1990), 107–109 | DOI | MR

[35] M. I. Vishik, A. V. Fursikov, Matematicheskie problemy statisticheskoi mekhaniki, Nauka, M., 1980 | MR

[36] I. A. Kibel, N. E. Kochin, N. V. Roze, Teoreticheskaya gidromekhanika, Fizmatgiz, M., 1963

[37] V. E. Zakharov, E. A. Kuznetsov, UFN, 167:11 (1997), 1137–1167 | DOI

[38] N. N. Moiseev, V. V. Rumyantsev, Dinamika tela s polostyami, soderzhaschimi zhidkost, Nauka, M., 1965 | Zbl

[39] V. F. Zaitsev, A. D. Polyanin, Spravochnik po obyknovennym differentsialnym uravneniyam. Tochnye resheniya, Fizmatlit, M., 1995 | MR

[40] L. V. Ovsyannikov, Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 | MR

[41] V. Buslaev, A. Grigis, Turning Points for Adiabatically Perturbed Periodic Equations, Preprint Univ. Paris-13, 1998 | MR

[42] A. D. Bryuno, Stepennaya geometriya v algebraicheskikh i differentsialnykh uravneniyakh, Nauka, M., 1998 | MR | Zbl

[43] Y. Wang, G. J. Holland, J. Atmos. Sci., 53 (1996), 411–427 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[44] L. N. Gutman, Vvedenie v nelineinuyu teoriyu mezomaterialnykh protsessov, Gidrometeoizdat, L., 1969

[45] V. N. Maslennikova, M. E. Bogovskii, Matematicheskoe modelirovanie, 6:9 (1994), 26–32 | MR | Zbl

[46] P. G. Saffman, Vortex dynamics, Cambrige Univ. Press, Cambrige, 1992 | MR | Zbl

[47] J.-L. Lions, R. Temam, S. Wang, Commun. Pure Appl. Math., 1:8 (1997), 707–752 | 3.0.CO;2-A class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[48] V. A. Bogomolov, Izv. AN SSSR. Fizika atmosfery i okeana, 15:1 (1979), 29–36 | MR