On solutions of the Chazy equation

Authors

  • Kiryl G. Atrokhau Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus
  • Elena V. Gromak Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus https://orcid.org/0000-0003-3646-6227

Keywords:

Chazy equation, Chazy system, Painlevé property, elliptic functions

Abstract

The Chazy system determines the necessary and sufficient conditions for the absence of movable critical points of solutions of the particular third order differential equation that was considered by Chazy in one of the first papers on the classification of higher-order ordinary differential equations with respect to the Painlevé property. The solution of the complete Chazy system in the case of constant poles has been already obtained. However, the question of integrating the Chazy equation remained open until now. In this paper, we prove that in the case of constant poles, under some additional conditions, this equation is integrated in elliptic functions.

Author Biographies

  • Kiryl G. Atrokhau, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

    senior lecturer at the department of differential equations and systems analysis, faculty of mechanics and mathematics

  • Elena V. Gromak, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

    PhD (physics and mathematics); associate professor at the department of theory of functions, faculty of mechanics and mathematics

References

  1. Ince EL. Ordinary Differential Equations. New York, Dover, 1956. 558 p.
  2. Iwasaki K, Kimura H, Shimomura S, Yoshida M. From Gauss to Painlevé: a Modern Theory of Special Functions. Braunschweig: Vieweg; 1991. 347 p. (Aspects of mathematics; volume 16).
  3. Chazy J. Sur les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes. Acta Mathematica. 1911;34(1):317–385.
  4. Bureau FJ. Differential equations with fixed critical points. Annali di Matematica Pura ed Applicata. 1964;64(1):229–364. DOI: 10.1007/BF02410054.
  5. Bureau FJ. Differential equations with fixed critical points. Annali di Matematica Pura ed Applicata. 1964;66(1):1–116. DOI: 10.1007/BF02412437.
  6. Martynov IP. Third-order equations without moving critical singularities. Differential Equations. 1985;21(6):937–946.
  7. Exton H. Nonlinear ordinary differential equations with fixed critical points. Rendiconti di Matematica. 1973;6(2):419–462.
  8. Cosgrove CM. Higher-order Painlevé equations in the polynomial class I. Bureau symbol P2. Studies in Applied Mathematics. 2000;104(1):1–65. DOI: 10.1111/1467-9590.00130.
  9. Mŭgan U, Jrad F. Painlevé test and higher order differential equations. Journal of Nonlinear Mathematical Physics. 2002;9(3): 282–310. DOI: 10.2991/jnmp.2002.9.3.4.
  10. Cosgrove CM. Higher-order Painlevé equations in the polynomial class II. Bureau symbol P1. Studies in Applied Mathematics. 2006;116(4):321–413. DOI: 10.1111/j.1467-9590.2006.00346.x.
  11. Kudryashov NA. Fourth-order analogies to the Painlevé equations. Journal of Physics A: Mathematical and General. 2002; 35(21):4617–4632. DOI: 10.1088/0305-4470/35/21/310.
  12. Sobolevsky S. Painlevé classification of binomial type ordinary differential equations of the arbitrary order. Studies in Applied Mathematics. 2006;117(3):215–237. DOI: 10.1111/j.1467-9590.2006.00353.x.
  13. Gromak VI. On solutions of the Chazy system. Differential Equations. 2007;43(5):631–635. DOI: 10.1134/S0012266107050060.
  14. Atrokhov KG, Gromak VI. Solution of the Chazy system. Differential Equations. 2010;46(6):783–797. DOI: 10.1134/ S0012266110060030.
  15. Gromak EV. [On integration of the Chazy equation with constant poles in elliptic functions]. In: Rogozin SV, editor. Tezisy dokladov mezhdunarodnoi nauchnoi konferentsii «Analiticheskie metody analiza i differentsial’nykh uravnenii»; 11–14 sentyabrya 2012 g.; Minsk, Belarus’ [Abstracts of the International scientific conference «Analytical methods of analysis and differential equations»; 2012 September 11–14; Minsk, Belarus]. Minsk: Institute of Mathematics, Natioanl Academy of Sciences of Belarus; 2012. p. 28. Russian.
  16. Gromak EV. [On the third-order P-type equations]. In: Demenchuk AK, Krasovskii SG, Makarov EK, editors. XVI Mezhdunarodnaya nauchnaya konferentsiya po differentsial’nym uravneniyam (Eruginskie chteniya – 2014); 20–22 maya 2014 g.; Novopolotsk, Belarus’. Chast’ 1 [XVI International scientific conference on differential equations (Erugin readings – 2014); 2014 May 20–22; Novopolotsk, Belarus. Part 1]. Minsk: Institute of Mathematics, National Academy of Sciences of Belarus; 2014. p. 11–12. Russian.

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Published

2021-08-05

Issue

No. 2 (2021): Journal of the Belarusian State University. Mathematics and Informatics

Section

Differential Equations and Optimal Control

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How to Cite

[1]
Atrokhau, K.G. and Gromak, E.V. 2021. On solutions of the Chazy equation. Journal of the Belarusian State University. Mathematics and Informatics. 2 (Aug. 2021), 51–59. DOI:https://doi.org/10.33581/2520-6508-2021-2-51-59.