Skip to main content
Log in

Differential invariants of generic hyperbolic Monge-Ampère equations

  • Research Article
  • Published:
Central European Journal of Mathematics

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  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.

    Google Scholar 

  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.

    Google Scholar 

  3. P. Hartman and A. Wintner: “On hyperbolic partial differential equations”, Am. J. Math., Vol. 74, (1952), pp. 834–864.

    Article  MathSciNet  Google Scholar 

  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.

    MATH  Google Scholar 

  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.

    MATH  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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

    MathSciNet  Google Scholar 

  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.

    Google Scholar 

  9. A. Kushner: “Monge-Ampère equations and e-structures”, Dokl. Akad. Nauk, Vol. 361(5), (1998), pp. 595–596.

    MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  11. V.V. Lychagin: “Contact geometry and non-linear second order differential equations”, Russian Math. Surveys, Vol. 34, (1979), pp. 149–180.

    Article  MathSciNet  Google Scholar 

  12. V.V. Lychagin: Lectures on Geometry of Differential Equations, Universita “La Sapienza”, Roma, 1992.

    Google Scholar 

  13. V.V. Lychagin and V.N. Rubtsov: “Local classification of Monge-Ampere equations”, Soviet Math. Doklady, Vol. 272(1), (1983), pp. 34–38.

    MathSciNet  Google Scholar 

  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

    MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  17. M. Matsuda: “Two methods of integrating Monge-Ampère’s equations”, Trans. Amer. Math. Soc., Vol. 150, (1970), pp. 327–343.

    Article  MathSciNet  Google Scholar 

  18. M. Matsuda: “Two methods of integrating Monge-Ampère’s equations. II”, Trans. Amer. Math. Soc., Vol. 166, (1972), pp. 371–386.

    Article  MathSciNet  Google Scholar 

  19. T. Morimoto: “La géométrie des équations de Monge-Ampère”, C. R. Acad. Sci., Paris, Vol. 289, (1979), pp. A-25–A-28.

    MathSciNet  Google Scholar 

  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.

    Google Scholar 

  21. O.P. Tchij: “Contact geometry of hyperbolic Monge-Ampère eqquations”, Lobachevskii Journal of Mathematics, Vol. 4, (1999), pp. 109–162.

    MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  23. D.V. Tunitsky: “Monge-Ampère equations and functors of characteristic connection”, Izv. RAN, Ser. Math., Vol. 65(6), (2001), pp. 173–222.

    Google Scholar 

  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.

    Google Scholar 

  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).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

About this article

Cite this article

Marvan, M., Vinogradov, A.M. & Yumaguzhin, V.A. Differential invariants of generic hyperbolic Monge-Ampère equations. centr.eur.j.math. 5, 105–133 (2007). https://doi.org/10.2478/s11533-006-0043-4

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11533-006-0043-4

Keywords

MSC (2000)

Navigation