Skip to main content
SpringerLink
Log in

The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface

  • Theoretical and Mathematical Physics
  • Published:
Technical Physics Aims and scope Submit manuscript

Abstract

We continue the study of the second Painlevé equation within the framework of the electrostatic probe theory. The integrability conditions for the equation are found for the partial absorption of charged particles by the probe surface. A sets of solutions with the asymptotics y ∼ ν/x for x → +∞ is constructed numerically in a wide range of the free parameter ν. Also, solutions (related to those mentioned above) for half-integer and integer ν, including solutions representable in asymptotic form at x → +∞ through the Airy function y ∼ cAi(x) in the limit ν → 0, are found. The results are discussed from the standpoint of the isomonodromic deformation method.

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. V. I. Gromak, Differentsial’nye Uravneniya 18, 753 (1982).

    MATH  MathSciNet  Google Scholar 

  2. H. Flaschka and A. C. Newell, Commun. Math. Phys. 76, 65 (1980).

    Article  MathSciNet  Google Scholar 

  3. A. A. Kapaev and V. Yu. Novokshenov, Dokl. Akad. Nauk SSSR 290, 590 (1986) [Sov. Phys. Dokl. 31, 719 (1986)].

    MathSciNet  Google Scholar 

  4. A. P. Its and A. A. Kapaev, Izv. Akad. Nauk SSSR, Ser. Mat. 51, 878 (1987).

    Google Scholar 

  5. A. A. Kapaev, Teor. Mat. Fiz. 77, 323 (1988).

    MATH  MathSciNet  Google Scholar 

  6. R. R. Rosales, Proc. R. Soc. London, Ser. A 361(1706), 265 (1978).

    ADS  MATH  MathSciNet  Google Scholar 

  7. J. Miles, Proc. R. Soc. London, Ser. A 361(1706), 277 (1978).

    ADS  MATH  MathSciNet  Google Scholar 

  8. I. M. Cohen, Phys. Fluids 6, 1492 (1963).

    Article  Google Scholar 

  9. A. V. Kashevarov, Zh. Vychisl. Mat. Mat. Fiz. 38, 992 (1998).

    MATH  MathSciNet  Google Scholar 

  10. I. Babuška, M. Práger, and E. Vitásek, Numerical Processes in Differential Equations (Wiley, New York, 1966; Mir, Moscow, 1969).

    Google Scholar 

  11. Handbook of Mathematical Functions, Ed. by M. Abramowitz and I. A. Stegun (Dover, New York, 1971; Nauka, Moscow, 1979).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Zhukovsky Central Institute of Aerohydrodynamics, Zhukovskii, Moscow Oblast, 140180, Russia

    A. V. Kashevarov

Authors
  1. A. V. Kashevarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Zhurnal Tekhnichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) Fiziki, Vol. 74, No. 1, 2004, pp. 3–9.

Original Russian Text Copyright © 2004 by Kashevarov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kashevarov, A.V. The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface. Tech. Phys. 49, 1–7 (2004). https://doi.org/10.1134/1.1642671

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1134/1.1642671

Keywords

Search

Navigation