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2007 | 5 | 1 | 105-133
Tytuł artykułu

Differential invariants of generic hyperbolic Monge-Ampère equations

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper basic differential invariants of generic hyperbolic Monge-Ampère equations with respect to contact transformations are constructed and the equivalence problem for these equations is solved.
Wydawca
Czasopismo
Rocznik
Tom
5
Numer
1
Strony
105-133
Opis fizyczny
Daty
wydano
2007-03-01
online
2007-03-01
Twórcy
Bibliografia
  • [1] D.V. Alekseevsky, A.M. Vinogradov and V.V. Lychagin: “Basic ideas and concepts of differential geometry”, In: Geometry, I, Encyclopaedia Math. Sci., Vol. 28, Springer, Berlin, 1991, pp. 1–264.
  • [2] A. Frölicher and A. Nijenhuis: “Theory of vector valued differential forms. Part I: Derivations in the graded ring of differential forms,” Indag. Math., Vol. 18, (1956), pp. 338–359.
  • [3] P. Hartman and A. Wintner: “On hyperbolic partial differential equations”, Am. J. Math., Vol. 74, (1952), pp. 834–864. http://dx.doi.org/10.2307/2372229
  • [4] I.S. Krasil’shchik, V.V. Lychagin and A.M. Vinogradov: Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986.
  • [5] I.S. Krasil’shchik and A.M. Vinogradov (Ed.): Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Translations of Mathematical Monographs, Vol. 182, American Mathematical Society, Providence RI, 1999.
  • [6] B.S. Kruglikov: “Some classificational problems in four-dimensional geometry: distributions, almost complex structures and the generalized Monge-Ampère equations”, Math. Sbornik, Vol. 189(11), (1998), pp. 61–74 (in Russian); English translation in Sb. Math., Vol. 186(11–12), (1998), pp. 1643–1656; e-print: http://xxx.lanl.gov/abs/dg-ga/9611005.
  • [7] B.S. Kruglikov: “Symplectic and contact Lie algebras with application to the Monge-Ampère equations”, Trudy Mat. Inst. Steklova, Vol. 221, (1998), pp. 232–246 (in Russian); English translation in Proc. Steklov Math. Inst., Vol. 221(2), (1998), pp. 221–235; e-print: http://xxx.lanl.gov/abs/dg-ga/9709004
  • [8] B.S. Kruglikov: “Classification of Monge-Ampère equations with two variables”, In: Geometry and Topology of Caustics - CAUSTICS’ 98 (Warsaw), Banach Center Publications, Vol. 50, Polish Acad. Sci., Warsaw, 1999, pp. 179–194.
  • [9] A. Kushner: “Monge-Ampère equations and e-structures”, Dokl. Akad. Nauk, Vol. 361(5), (1998), pp. 595–596.
  • [10] H. Lewy: “Über das Anfangswertproblem bei einer hyperbolischen nichtlinearen partiellen Differentialgleichung zweiter Ordnung mit zwei unabhängigen Verànderlichen”, Math. Annalen, Vol. 98, (1928), pp. 179–191. http://dx.doi.org/10.1007/BF01451588
  • [11] V.V. Lychagin: “Contact geometry and non-linear second order differential equations”, Russian Math. Surveys, Vol. 34, (1979), pp. 149–180. http://dx.doi.org/10.1070/RM1979v034n01ABEH002873
  • [12] V.V. Lychagin: Lectures on Geometry of Differential Equations, Universita “La Sapienza”, Roma, 1992.
  • [13] V.V. Lychagin and V.N. Rubtsov: “Local classification of Monge-Ampere equations”, Soviet Math. Doklady, Vol. 272(1), (1983), pp. 34–38.
  • [14] V.V. Lychagin and V.N. Rubtsov: “On the Sophus Lie theorems for Monge-Ampere equations”, Belorussian Acad. Sci. Doklady, Vol. 27(5, (1983), pp. 396–398
  • [15] V.V. Lychagin, V.N. Rubtsov and I.V. Chekalov: “A classification of Monge-Ampere equations”, Ann. Sc. Ecole Norm. Sup., Vol. 4(26), (1993), pp. 281–308.
  • [16] M. Marvan, A.M. Vinogradov and V.A. Yumaguzhin: “Differential invariants of generic hyperbolic Monge-Ampère equations”, Russian Acad. Sci. Dokl. Math., Vol. 405, (2005), pp. 299–301 (in Russian); English translation in: Doklady Mathematics, Vol. 72, (2005), pp. 883–885.
  • [17] M. Matsuda: “Two methods of integrating Monge-Ampère’s equations”, Trans. Amer. Math. Soc., Vol. 150, (1970), pp. 327–343. http://dx.doi.org/10.2307/1995496
  • [18] M. Matsuda: “Two methods of integrating Monge-Ampère’s equations. II”, Trans. Amer. Math. Soc., Vol. 166, (1972), pp. 371–386. http://dx.doi.org/10.2307/1996056
  • [19] T. Morimoto: “La géométrie des équations de Monge-Ampère”, C. R. Acad. Sci., Paris, Vol. 289, (1979), pp. A-25–A-28.
  • [20] T. Morimoto: “Monge-Ampère equations viewed from contact geometry”, In: Symplectic Singularities and Geometry of Gauge Fields (Warsaw, 1995), Banach Center Publ., Vol. 39, Polish Acad. Sci., Warsaw, 1997, pp. 105–121.
  • [21] O.P. Tchij: “Contact geometry of hyperbolic Monge-Ampère eqquations”, Lobachevskii Journal of Mathematics, Vol. 4, (1999), pp. 109–162.
  • [22] D.V. Tunitsky: “On the global solvability of hyperbolic Monge-Ampère equations”, Izv. Ross. Akad. Nauk Ser. Mat., Vol. 61(5), (1997), pp. 177–224 (in Russian); English translation in: Izv. Math, Vol. 61(5), (1997), pp. 1069–1111.
  • [23] D.V. Tunitsky: “Monge-Ampère equations and functors of characteristic connection”, Izv. RAN, Ser. Math., Vol. 65(6), (2001), pp. 173–222.
  • [24] A.M. Vinogradov: “Scalar differential invariants, diffieties and characteristic classes”, In: Mechanics, Analysis and Geometry: 200 Years after Lagrange, M. Francaviglia, North-Holland, 1991, pp. 379–414.
  • [25] A.M. Vinogradov and V.A. Yumaguzhin: “Differential invariants of webs on 2-dimensional manifolds”, Mat. Zametki, Vol. 48(1), (1990), pp. 46–68 (in Russian).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-006-0043-4
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