Skip to main content
SpringerLink
Log in

Nonautonomous Evolution Equation of Monge–Ampere Type with Two Space Variables

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

There are the exact solutions to nonautonomous evolution equation with two space variables the right side of which contains the Monge–Ampere operator. The solutions with additive and multiplicative separation of variables are found. The reductions of the given equation to ordinary differential equations (ODEs) are considered. The classic and generalized self-similar solutions and the solutions with functional separation of variables are obtained. In particular, it is shown that the given equation can be reduced to ODE if the coefficient at the time derivative can be represented in the form of product of the functions depending on time, space variables, and sought function. The reductions of the given equation to two-dimensional partial differential equations are also determined.

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

REFERENCES

  1. A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, 2nd ed. (Chapman and Hall–CRC Press, Boca Raton, Fla., 2012).

    MATH  Google Scholar 

  2. B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Euqations and Their Applications to Gas Dynamics (Nauka, Moscow, 1978).

    Google Scholar 

  3. S. V. Khabirov, “Nonisentropic one-dimensional gas motions constructed by means of the contact group of the nonhomogeneous Monge–Ampére equation,” Math. USSR Sb. 71, 447–462 (1992). https://doi.org/10.1070/SM1992v071n02ABEH001405

    Article  MathSciNet  MATH  Google Scholar 

  4. O. N. Shablovskii, “Parametric solutions for the Monge–Ampére equation and gas flow with variable entropy,” Vestn. Tomsk. Gos. Univ. Mat. Mekh., No. 1, 105–118 (2015). https://doi.org/10.17223/19988621/33/11

  5. A. V. Pogorelov, Multidimensional Monge–Ampére Equation (Nauka, Moscow, 1988).

    MATH  Google Scholar 

  6. CRC Handbook of Lie Groups to Differential Equations, Vol. 1: Symmetries, Exact Solutions, and Conservation Laws, Ed. by N. H. Ibragimov (CRC Press, Boca Raton, Fla., 1994).

    MATH  Google Scholar 

  7. I. V. Rakhmelevich, “On the solutions of the Monge–Ampére equation with power-law non-linearity with respect to first derivatives,” Vestn. Tomsk. Gos. Univ. Mat. Mekh., No. 4, 33–43 (2016). https://doi.org/10.17223/19988621/42/4

  8. I. V. Rakhmelevich, “Modified two-dimensional Monge–Ampére equation,” Vestn. Voronezhsk. Gos. Univ. Ser. Fiz. Mat., No. 3, 159–168 (2017).

  9. I. V. Rakhmelevich, “Multi-dimensional Monge–Ampére equation with power nonlinearities on the first derivatives,” Vestn. Voronezhsk. Gos. Univ. Ser. Fiz. Mat., No. 2, 86–98 (2020).

  10. V. A. Galaktionov and S. R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics (Chapman and Hall/CRC Press, Boca Raton, Fla., 2006).

    Book  MATH  Google Scholar 

  11. C. E. Gutierres, The Monge–Ampére Equation (Birkhäuser, Boston, 2001).

    Book  Google Scholar 

  12. M. E. Taylor, Partial Differential Equations III: Nonlinear Equations, Applied Mathematical Sciences, Vol. 117 (Springer, New York, 1996). https://doi.org/10.1007/978-1-4757-4190-2

  13. O. Leibov, “Reduction and exact solutions of the Monge–Ampere equation,” J. Nonlinear Math. Phys. 4, 146–148 (1997). https://doi.org/10.2991/jnmp.1997.4.1-2.21

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. W. I. Fushchych, “The symmetry and exact solutions of some multidimensional nonlinear equations of mathematical physics,” Sci. Works 3, 175–179 (2001).

    Google Scholar 

  16. K.-S. Chou and X.-J. Wang, “A logarithmic Gauss curvature flow and the Minkowski problem,” Ann. Inst. Henri Poincare C 17, 733–751 (2000). https://doi.org/10.1016/S0294-1449(00)00053-6

    Article  MathSciNet  MATH  Google Scholar 

  17. A. D. Polyanin and A. I. Zhurov, Methods of Variable Separation and Exact Solutions to Nonlinear Equations of Mathematical Physics (Izd-vo Inst. Problem Mekh. Ross. Akad. Nauk, Moscow, 2020).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Research Lobachevsky State University of Nizhny Novgorod, 603950, Nizhny Novgorod, Russia

    I. V. Rakhmelevich

Authors
  1. I. V. Rakhmelevich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. V. Rakhmelevich.

Ethics declarations

The author declares that he has no conflicts of interest.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rakhmelevich, I.V. Nonautonomous Evolution Equation of Monge–Ampere Type with Two Space Variables. Russ Math. 67, 52–64 (2023). https://doi.org/10.3103/S1066369X23020044

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X23020044

Keywords:

Search

Navigation