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TMF, 2007, Volume 153, Number 1, Pages 58–67 (Mi tmf6121)  

This article is cited in 9 scientific papers (total in 9 papers)

Special polynomials and rational solutions of the hierarchy of the second Painlevé equation

M. V. Demina, N. A. Kudryashov

Moscow Engineering Physics Institute (State University)

Abstract: We study special polynomials used to represent rational solutions of the hierarchy of the second Painlevé equation. We find several recursion relations satisfied by these polynomials. In particular, we obtain a differential–difference relation that allows finding any polynomial recursively. This relation is an analogue of the Toda chain equations.

Keywords: Painlevé equations, hierarchy of the second Painlevé equation, rational solution, special polynomial, Toda chain

DOI: https://doi.org/10.4213/tmf6121

Full text: PDF file (395 kB)
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English version:
Theoretical and Mathematical Physics, 2007, 153:1, 1398–1406

Bibliographic databases:

Received: 27.12.2006

Citation: M. V. Demina, N. A. Kudryashov, “Special polynomials and rational solutions of the hierarchy of the second Painlevé equation”, TMF, 153:1 (2007), 58–67; Theoret. and Math. Phys., 153:1 (2007), 1398–1406

Citation in format AMSBIB
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  • https://doi.org/10.4213/tmf6121
  • http://mi.mathnet.ru/eng/tmf/v153/i1/p58

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Kudryashov NA, Demina MV, “The generalized Yablonskii-Vorob'ev polynomials and their properties”, Physics Letters A, 372:29 (2008), 4885–4890  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Kudryashov N.A., “Soliton, rational and special solutions of the Korteweg-de Vries hierarchy”, Applied Mathematics and Computation, 217:4 (2010), 1774–1779  crossref  mathscinet  zmath  isi  scopus
    3. Demina M.V., Kudryashov N.A., “Point Vortices and Polynomials of the Sawada-Kotera and Kaup-Kupershmidt Equations”, Regular & Chaotic Dynamics, 16:6 (2011), 562–576  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Demina M.V., Kudryashov N.A., “Point Vortices and Classical Orthogonal Polynomials”, Regul. Chaotic Dyn., 17:5 (2012), 371–384  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Demina M.V., Kudryashov N.A., “Vortices and Polynomials: Non-Uniqueness of the Adler-Moser Polynomials for the Tkachenko Equation”, J. Phys. A-Math. Theor., 45:19 (2012), 195205  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Demina M.V., Kudryashov N.A., “Rotation, Collapse, and Scattering of Point Vortices”, Theor. Comput. Fluid Dyn., 28:3 (2014), 357–368  crossref  isi  scopus
    7. Maria V. Demina, Nikolai A. Kudryashov, “Multi-particle Dynamical Systems and Polynomials”, Regul. Chaotic Dyn., 21:3 (2016), 351–366  mathnet  crossref  mathscinet
    8. Balogh F., Bertola M., Bothner T., “Hankel Determinant Approach to Generalized Vorob?ev?Yablonski Polynomials and Their Roots”, Constr. Approx., 44:3 (2016), 417–453  crossref  mathscinet  zmath  isi  elib  scopus
    9. Gromak V.I., “Solutions of the Fourth-Order Equation in the Generalized Hierarchy of the Second Painleve Equation”, Differ. Equ., 55:3 (2019), 328–339  crossref  mathscinet  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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