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Teoreticheskaya i Matematicheskaya Fizika, 2000, Volume 122, Number 1, Pages 5–22
DOI: https://doi.org/10.4213/tmf551 (Mi tmf551) 		 
This article is cited in 13 scientific papers (total in 13 papers)

On the relation between the continuous and discrete Painlevé equations

P. A. Clarkson, E. L. Mansfield, H. N. Webster

University of Kent
Full-text PDF (273 kB) Citations (13)
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DOI: https://doi.org/10.4213/tmf551
Abstract: A method for deriving difference equations (the discrete Painlevé equations in particular) from the Bäcklund transformations of the continuous Painlevé equations is discussed. This technique can be used to derive several of the known discrete Painlevé equations (in particular, the first and second discrete Painlevé equations and some of their alternative versions). The Painlevé equations possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions for special values of the parameters. Hence, the aforementioned relations can be used to generate hierarchies of exact solutions for the associated discrete Painlevé equations. Exact solutions of the Painlevé equations simultaneously satisfy both a differential equation and a difference equation, analogously to the special functions.
English version:
Theoretical and Mathematical Physics, 2000, Volume 122, Issue 1, Pages 1–16
DOI: https://doi.org/10.1007/BF02551165
Bibliographic databases:
Language: Russian
Citation: P. A. Clarkson, E. L. Mansfield, H. N. Webster, “On the relation between the continuous and discrete Painlevé equations”, TMF, 122:1 (2000), 5–22; Theoret. and Math. Phys., 122:1 (2000), 1–16
Citation in format AMSBIB
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  • https://doi.org/10.4213/tmf551
  • https://www.mathnet.ru/eng/tmf/v122/i1/p5
    • This publication is cited in the following 13 articles:
    1. A. Baddour, M. Malykh, L. Sevastianov, “On Periodic Approximate Solutions of Dynamical Systems with Quadratic Right-Hand Side”, J Math Sci, 261:5 (2022), 698  crossref
    2. A. Baddur, M. D. Malykh, L. A. Sevastyanov, “O periodicheskikh priblizhennykh resheniyakh dinamicheskikh sistem s kvadratichnoi pravoi chastyu”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XXXIII, Zap. nauchn. sem. POMI, 507, POMI, SPb., 2021, 157–172  mathnet
    3. Artyom V. Yurov, Valerian A. Yurov, “On the Question of the Bäcklund Transformations and Jordan Generalizations of the Second Painlevé Equation”, Symmetry, 13:11 (2021), 2095  crossref
    4. Van Assche W., Filipuk G., Zhang L., “Multiple Orthogonal Polynomials Associated With An Exponential Cubic Weight”, J. Approx. Theory, 190:SI (2015), 1–25  crossref  mathscinet  zmath  isi  scopus  scopus
    5. A. Mohammadi, E. Hesameddini, “On some aspects of Bäcklund transformations”, Applied Mathematics and Computation, 215:11 (2010), 3985  crossref
    6. Clarkson P.A., “Painlevé Equations - Nonlinear Special Functions”, Orthogonal Polynomials and Special Functions: Computation and Applications, Lecture Notes in Mathematics, 1883, 2006, 331–411  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    7. Peter A. Clarkson, “Special Polynomials Associated with Rational Solutions of the Painlevé Equations and Applications to Soliton Equations”, Comput. Methods Funct. Theory, 6:2 (2006), 329  crossref
    8. Clarkson P.A., “On rational solutions of the fourth Painlevé equation and its Hamiltonian”, Group Theory and Numerical Analysis, CRM Proceedings & Lecture Notes, 39, 2005, 103–118  crossref  mathscinet  zmath  isi
    9. Gromak V.I., Zenchenko A.S., “On the theory of higher-order Painlevé equations”, Differential Equations, 40:5 (2004), 625–633  mathnet  mathnet  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    10. Clarkson, PA, “Remarks on the Yablonskii-Vorob'ev polynomials”, Physics Letters A, 319:1–2 (2003), 137  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    11. Clarkson, PA, “Hierarchies of difference equations and Backlund transformations”, Journal of Nonlinear Mathematical Physics, 10 (2003), 13  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    12. Clarkson, PA, “The third Painlevé equation and associated special polynomials”, Journal of Physics A-Mathematical and General, 36:36 (2003), 9507  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    13. Kudryashov, NA, “Discrete equations corresponding to fourth-order differential equations of the P-2 and K-2 hierarchies”, Anziam Journal, 44 (2002), 149  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
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    		Теоретическая и математическая физика Theoretical and Mathematical Physics
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