Skip to main content
SpringerLink
Log in

A Contact Linearization Problem for Monge-Ampère Equations and Laplace Invariants

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

We solve a problem of contact linearization for non-degenerate regular Monge-Ampère equations. In order to solve the problem we construct tensor invariants of equations with respect to contact transformations and generalize the classical Laplace invariants.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, I.M., Juras, M.: Generalized Laplace invariants and the method of Darboux. Duke Math. J. 89, 351–375 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Forsyth, A.R.: Theory of Differential Equations. Part 4. Partial Differential Equations, vol. 6. Cambridge University Press, Cambridge (1906). 596 pp.

    Google Scholar 

  3. Hunter, J.K., Saxton, R.: Dynamics of director fields. SIAM J. Appl. Math. 51(6), 1498–1521 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kruglikov, B.S.: On some classification problems in four-dimensional geometry: distributions, almost complex structures, and the generalized Monge-Ampère equations. Mat. Sb. 189(11), 61–74 (1998)

    MathSciNet  Google Scholar 

  5. Kruglikov, B.S.: Symplectic and contact Lie algebras with application to the Monge-Ampère equations. Tr. Mat. Inst. Steklova 221, 232–246 (1998)

    MathSciNet  Google Scholar 

  6. Kruglikov, B.S.: Classification of Monge-Ampère equations with two variables. In: Geometry and Topology of Caustics, CAUSTICS’98, Warsaw, pp. 179–194. Polish Acad. Sci., Warsaw (1999)

    Google Scholar 

  7. Kushner, A.G.: Chaplygin and Keldysh normal forms of Monge-Ampère equations of variable type. Math. Zametki 52(5), 63–67 (1992)

    MathSciNet  Google Scholar 

  8. Kushner, A.G.: Classification of mixed type Monge-Ampère equations. In: Geometry in Partial Differential Equations, pp. 173–188. World Scientific, Singapore (1993)

    Google Scholar 

  9. Kushner, A.G.: Symplectic geometry of mixed type equations. Am. Math. Soc. Transl. 2, 131–142 (1995)

    MathSciNet  Google Scholar 

  10. Kushner, A.G.: Monge–Ampère equations and e-structures. Dokl. Math. 58(1), 103–104 (1998)

    MathSciNet  Google Scholar 

  11. Kushner, A.G.: Almost product structures and Monge-Ampère equations. Lobachevskii J. Math. 23, 151–181 (2006). http://ljm.ksu.ru

    MATH  MathSciNet  Google Scholar 

  12. Kushner, A.G., Lychagin, V.V., Rubtsov, V.N.: Contact Geometry and Nonlinear Differential Equations. Cambridge University Press, Cambridge (2007). 496 pp.

    MATH  Google Scholar 

  13. Lie, S.: Ueber einige partielle Differential-Gleichungen zweiter Orduung. Math. Ann. 5, 209–256 (1872)

    Article  MathSciNet  Google Scholar 

  14. Lie, S.: Begrundung einer Invarianten-Theorie der Beruhrungs-Transformationen. Math. Ann. 8, 215–303 (1874)

    Article  MathSciNet  Google Scholar 

  15. Lychagin, V.V.: Contact geometry and second-order nonlinear differential equations. Russ. Math. 34, 137–165 (1979)

    MATH  MathSciNet  Google Scholar 

  16. Lychagin, V.V.: Lectures on Geometry of Differential Equations. 1, 2. La Sapienza, Rome (1993)

  17. Lychagin, V.V., Rubtsov, V.N.: On Sophus Lie theorems for Monge-Ampère equations. Dokl. Akad. Nauk BSSR 27(5), 396–398 (1983)

    MATH  MathSciNet  Google Scholar 

  18. Morimoto, T.: La géométrie des équations de Monge-Ampére. C.R. Acad. Sci. Paris A-B 289(1), A25–A28 (1979)

    MathSciNet  Google Scholar 

  19. Morozov, O.I.: Contact equivalence of the generalized Hunter-Saxton equation and the Euler-Poisson equation, pp. 1–3. Preprint (2004). arXiv: math-ph/0406016

  20. Tunitskii, D.V.: On the contact linearization of Monge-Ampère equations. Izv. Ross. Akad. Nauk Ser. Mat. 60(2), 195–220 (1996)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Russian Academy of Sciences, Institute of Control Sciences, Profsoyuznaya, 65, lab. 6, 117997, Moscow, Russia

    Alexei G. Kushner

Authors
  1. Alexei G. Kushner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexei G. Kushner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kushner, A.G. A Contact Linearization Problem for Monge-Ampère Equations and Laplace Invariants. Acta Appl Math 101, 177–189 (2008). https://doi.org/10.1007/s10440-008-9195-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-008-9195-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation