Skip to main content
SpringerLink
Log in

Symmetry reduction and exact solutions of the multidimensional monge-ampere equation

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

By using the subgroup structure of the generalized Poincare groupP( 1, 4), we perform a symmetry reduction of the multidimensional Monge-Ampere equation to differential equations with a smaller number of independent variables. On the basis of solutions of the reduced equations, we construct some classes of exact solutions of the equation under consideration.

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. A. V. Pogorelov,Multidimensional Minkowski Problem [in Russian], Moscow, Nauka 1975.

    Google Scholar 

  2. A. V. Pogorelov,Multidimensional Monge-Ampere Equation [in Russian], Moscow, Nauka 1988.

    MATH  Google Scholar 

  3. M. Kozlowski, “The Monge-Ampere equation in affine differential geometry,”Anz. Math. Naturwiss. kl.,126, 21–24 (1990).

    MathSciNet  Google Scholar 

  4. V. V. Kozlov, “Liouville property of invariant measures of fully integrable systems and the Monge-Ampere equation,”Mat. Zametki,53, No. 4, 45–61 (1993).

    MATH  Google Scholar 

  5. Y. Nutku and O. Sarioglu, “An integrable family of Monge-Ampere equations and their multi-Hamiltonian structure,”Phys. Lett.,A173, No. 3, 270–274 (1993).

    MathSciNet  Google Scholar 

  6. S. V. Khabirov, “Nonisoentropic one-dimensional motions of gas constructed by a compact group of the inhomogeneous Monge-Ampere equation,”Mat. Sb.,181, No. 12, 1607–1622 (1990).

    MATH  Google Scholar 

  7. S. V. Khabirov, “Application of contact transformations of the inhomogeneous Monge-Ampere equation in one-dimensional gas dynamics,”Dokl. Akad. Nauk SSSR,310, No. 2, 333–336 (1990).

    MathSciNet  Google Scholar 

  8. W. I. Fushchich and N. I. Serov, “Symmetry and some exact solutions of the multidimensional Monge-Ampere equation,”Dokl. Akad. Nauk SSSR,273, No. 3, 543–546 (1983).

    MathSciNet  Google Scholar 

  9. W. I. Fushchich, W. M. Shtelen, and N. I. Serov,Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics [in Russian], Naukova Dumka, Kiev (1989).

    MATH  Google Scholar 

  10. W. I. Fushchich, W. M. Shtelen, and N. I. Serov,Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics, Kluwer, Dordrecht 1993.

    MATH  Google Scholar 

  11. V. M. Fedorchuk,Continuous Subgroups of the Inhomogeneous De Sitter Group [in Russian], Preprint No. 78.18, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1978).

    Google Scholar 

  12. V. M. Fedorchuk, “Splitting subalgebras of the Lie algebra of generalized Poincare groupP( 1, 4),”Ukr. Mat. Zh.,31, No. 6, 717–722 (1979).

    MATH  MathSciNet  Google Scholar 

  13. V. M. Fedorchuk, “Nonsplitting subalgebras of the Lie algebra of generalized Poincare groupP( 1, 4),”Ukr. Mat. Zh.,33, No. 5, 696–700 (1981).

    MATH  MathSciNet  Google Scholar 

  14. V. M. Fedorchuk and W. I. Fushchich, “On subgroups of generalized Poincare group,” in:Group-Theoretical Methods in Physics, Proceedings of the International Seminar (Zvenigorod, 1979) [in Russian], Vol. 1, Moscow, Nauka (1980), pp. 61–66.

    Google Scholar 

  15. W. I. Fushchich, A. F. Barannik, L. F. Barannik, and V. M. Fedorchuk, “Continuous subgroups of the Poincare groupP( 1, 4),”J. Phys. A: Math. Gen.,18, No. 14, 2893–2899 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  16. W. I. Fushchich, V. M. Fedorchuk, and I. M. Fedorchuk,Subgroup Structure of Generalized Poincare Group and Exact Solutions of Some Nonlinear Wave Equations [in Russian], Preprint No. 86.27, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1986).

    Google Scholar 

  17. V. M. Fedorchuk and I. M. Fedorchuk,Reduction and Exact Solutions of Five-Dimensional Nonlinear Wave Equations [in Russian], Preprint No. 88.21, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1988).

    Google Scholar 

Download references

Authors
  1. V. M. Fedorchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. S. Leibov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fedorchuk, V.M., Leibov, O.S. Symmetry reduction and exact solutions of the multidimensional monge-ampere equation. Ukr Math J 48, 775–783 (1996). https://doi.org/10.1007/BF02384226

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02384226

Keywords

Search

Navigation