Skip to main content
SpringerLink
Log in

On reduction and exact solutions of one class of nonlinear elliptic systems

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We study the system of two equations of elliptic type with two nonlinearities depending on the sum of squares of sought-for functions. We obtain conditions on nonlinearities which allow us to reduce the system to one equation. We also find parametric families of exact solutions, both radially symmetric and anisotropic with respect to spatial variables, described by elementary or harmonic functions. In case of controlled nonlinearity we specify wide class of realizable exact solutions expressed via harmonic functions.

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. Drivotin, O.I., Ovsyannikov, D.A. Modeling of Intensive Beams of Charged Particles (St.-Petersburg State University, St.-Petersburg, 2003) [in Russian].

    Google Scholar 

  2. Vedenyapin, V., Sinitsyn, A., Dulov, E. Kinetic Boltzmann–Vlasov and Related Equations (Elsevier, Amsterdam, 2011).

    MATH  Google Scholar 

  3. Drivotin, O. I., Ovsyannikov, D.A. “Solutions to the Vlasov Equation for Charged Particle Beam in Magnetic Field”, Izv. IGU. Ser.Matem. 6, No. 4, 2–22 (2013) [in Russian].

    MATH  Google Scholar 

  4. Polyanin, A. D., Zaitsev, V. F. Handbook of NonlinearMathematical Physics Equations: Exact Solutions (Fizmatlit, Moscow, 2002) [in Russian].

    Book  Google Scholar 

  5. Ben Abdallah, N., Degond, P., Mehats, F. “Mathematical Model ofMagnetic Insulation”, Physics of Plasmas 5, 1522–1534 (1998).

    Article  MathSciNet  Google Scholar 

  6. http://eqworldipmnetru/ru/solutions/syspde/spde3107pdf

  7. Pukhnachev, V. V. “Exact Solutions of the Equations of Motion for an Incompressible Viscoelastic Maxwell Medium”, Journal of AppliedMechanics and Technical Physics 50, No. 2, 181–187 (2009).

    Article  MathSciNet  Google Scholar 

  8. Ibragimov, N. Kh., Rudenko, O. V. “Principle of an A priori Use of Symmetries in the Theory of Nonlinear Waves”, Acoustical Physics 50, No. 4, 406–419 (2004).

    Article  Google Scholar 

  9. Vyazmina, E. A., Polyanin, A. D. “New Classes of Exact Solutions of Nonlinear Diffusion-Kinetic Equations and Systems of General Form”, Theor. Foundations of Chemical Technology 40, No. 6, 1–10 (2006) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of System Dynamics and Control Theory, ul. Lermontova 134, Irkutsk, 664033, Russia

    A. A. Kosov & E. I. Semyonov

Authors
  1. A. A. Kosov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. I. Semyonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Kosov.

Additional information

Original Russian Text © A.A. Kosov, E.I. Semyonov, 2015, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2015, No. 12, pp. 43–54.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kosov, A.A., Semyonov, E.I. On reduction and exact solutions of one class of nonlinear elliptic systems. Russ Math. 59, 36–45 (2015). https://doi.org/10.3103/S1066369X1512004X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X1512004X

Keywords

Search

Navigation