Technical Physics

, Volume 49, Issue 1, pp 1–7 | Cite as

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

  • A. V. Kashevarov
Theoretical and Mathematical Physics

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 ycAi(x) in the limit ν → 0, are found. The results are discussed from the standpoint of the isomonodromic deformation method.

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References

  1. 1.
    V. I. Gromak, Differentsial’nye Uravneniya 18, 753 (1982).MATHMathSciNetGoogle Scholar
  2. 2.
    H. Flaschka and A. C. Newell, Commun. Math. Phys. 76, 65 (1980).CrossRefMathSciNetGoogle Scholar
  3. 3.
    A. A. Kapaev and V. Yu. Novokshenov, Dokl. Akad. Nauk SSSR 290, 590 (1986) [Sov. Phys. Dokl. 31, 719 (1986)].MathSciNetGoogle Scholar
  4. 4.
    A. P. Its and A. A. Kapaev, Izv. Akad. Nauk SSSR, Ser. Mat. 51, 878 (1987).Google Scholar
  5. 5.
    A. A. Kapaev, Teor. Mat. Fiz. 77, 323 (1988).MATHMathSciNetGoogle Scholar
  6. 6.
    R. R. Rosales, Proc. R. Soc. London, Ser. A 361(1706), 265 (1978).ADSMATHMathSciNetGoogle Scholar
  7. 7.
    J. Miles, Proc. R. Soc. London, Ser. A 361(1706), 277 (1978).ADSMATHMathSciNetGoogle Scholar
  8. 8.
    I. M. Cohen, Phys. Fluids 6, 1492 (1963).CrossRefGoogle Scholar
  9. 9.
    A. V. Kashevarov, Zh. Vychisl. Mat. Mat. Fiz. 38, 992 (1998).MATHMathSciNetGoogle Scholar
  10. 10.
    I. Babuška, M. Práger, and E. Vitásek, Numerical Processes in Differential Equations (Wiley, New York, 1966; Mir, Moscow, 1969).Google Scholar
  11. 11.
    Handbook of Mathematical Functions, Ed. by M. Abramowitz and I. A. Stegun (Dover, New York, 1971; Nauka, Moscow, 1979).Google Scholar

Copyright information

© MAIK "Nauka/Interperiodica" 2004

Authors and Affiliations

  • A. V. Kashevarov
    • 1
  1. 1.Zhukovsky Central Institute of AerohydrodynamicsZhukovskii, Moscow OblastRussia

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