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Regul. Chaotic Dyn., 2014,  19,  1,  48–63 (Mi rcd140)  

16 ( 17 )

Higher Painlevé Transcendents as Special Solutions of Some Nonlinear Integrable Hierarchies

Nikolay A. Kudryashov

Department of Applied Mathematics, National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409 Russia

: It is well known that the self-similar solutions of the Kortewegde Vries equation and the modified Kortewegde Vries equation are expressed via the solutions of the first and second Painlevé equations. In this paper we solve this problem for all equations from the Kortevegde Vries, modified Kortewegde Vries, KaupKupershmidt, CaudreyDoddGibbon and FordyGibbons hierarchies. We show that the self-similar solutions of equations corresponding to hierarchies mentioned above can be found by means of the general solutions of higher-order Painlevé hierarchies introduced more than ten years ago.

 : Painlevé equation, Painlevé transcendent, Kortewegde Vries hierarchy, modified Kortevegde Vries hierarchy, KaupKupershmidt hierarchy, CaudreyDoddCibbon hierarchy

11-01-00798-a
This research was partially supported by the Federal Target Program Scientific and ScientificPedagogical Personnel of Innovative Russia for 20092013, by RFBR grant 110100798a and Researches and Developments in Priority Directions of Development of the Scientific Technological Complex of Russia for 20072013.


DOI: https://doi.org/10.1134/S1560354714010043

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MSC: 35Q51, 35Q53, 37K15
: 02.12.2013
:22.12.2013
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: Nikolay A. Kudryashov, “Higher Painlevé Transcendents as Special Solutions of Some Nonlinear Integrable Hierarchies”, Regul. Chaotic Dyn., 19:1 (2014), 48–63

AMSBIB
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\by Nikolay~A.~Kudryashov
\paper Higher Painlevé Transcendents as Special Solutions of Some Nonlinear Integrable Hierarchies
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 1
\pages 48--63
\mathnet{http://mi.mathnet.ru/rcd140}
\crossref{https://doi.org/10.1134/S1560354714010043}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3181036}
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  • http://mi.mathnet.ru/rus/rcd/v19/i1/p48

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

    x:
    1. Alexey V. Borisov, Nikolay A. Kudryashov, “Paul Painlevé and His Contribution to Science”, Regul. Chaotic Dyn., 19:1 (2014), 1–19  mathnet  crossref
    2. Nikolay A. Kudryashov, Dmitry I. Sinelshchikov, “Special Solutions of a High-order Equation for Waves in a Liquid with Gas Bubbles”, Regul. Chaotic Dyn., 19:5 (2014), 576–585  mathnet  crossref  mathscinet  zmath
    3. Nikolay A. Kudryashov, Dmitry I. Sinelshchikov, “On the Connection of the Quadratic Lienard Equation with an Equation for the Elliptic Functions”, Regul. Chaotic Dyn., 20:4 (2015), 486–496  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    4. Nikolay A. Kudryashov, “Analytical Solutions of the Lorenz System”, Regul. Chaotic Dyn., 20:2 (2015), 123–133  mathnet  crossref  mathscinet  zmath  adsnasa
    5. P. R. Gordoa, A. Pickering, Z. N. Zhu, “On matrix Painlevé hierarchies”, J. Differ. Equ., 261:2 (2016), 1128–1175  crossref  mathscinet  zmath  isi  scopus
    6. P. R. Gordoa, A. Pickering, “On an extended second Painlevé hierarchy”, J. Differ. Equ., 263:7 (2017), 4070–4125  crossref  mathscinet  zmath  isi  scopus
    7. Nikolay A. Kudryashov, “Exact Solutions and Integrability of the DuffingVan der Pol Equation”, Regul. Chaotic Dyn., 23:4 (2018), 471–479  mathnet  crossref  mathscinet
    8. Nikolay A. Kudryashov, “Rational and Special Solutions for Some Painlevé Hierarchies”, Regul. Chaotic Dyn., 24:1 (2019), 90–100  mathnet  crossref
    9. N. A. Kudryashov, “On Integrability of the FitzHugh Rinzel Model”, ., 15:1 (2019), 13–19  mathnet  crossref  elib
    10. N. A. Kudryashov, “Traveling wave reduction of the modified KdV hierarchy: the Lax pair and the first integrals”, Commun. Nonlinear Sci. Numer. Simul., 73 (2019), 472–480  crossref  mathscinet  isi  scopus
    11. N. A. Kudryashov, “Lax pair and first integrals of the traveling wave reduction for the KdV hierarchy”, Appl. Math. Comput., 350 (2019), 323–330  crossref  mathscinet  zmath  isi  scopus
    12. N. A. Kudryashov, “Nonlinear differential equations associated with the first Painlevé hierarchy”, Appl. Math. Lett., 90 (2019), 223–228  crossref  mathscinet  zmath  isi  scopus
    13. P. R. Gordoa, A. Pickering, “Backlund transformations for a new extended Painlevé hierarchy”, Commun. Nonlinear Sci. Numer. Simul., 69 (2019), 78–97  crossref  mathscinet  isi  scopus
    14. N. A. Kudryashov, “The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations”, Optik, 183 (2019), 642–649  crossref  isi  scopus
    15. Nikolay A. Kudryashov, “Lax Pairs and Special Polynomials Associated with Self-similar Reductions of Sawada Kotera and Kupershmidt Equations”, Regul. Chaotic Dyn., 25:1 (2020), 59–77  mathnet  crossref
    16. Nikolay A. Kudryashov, “Rational Solutions of Equations Associated with the Second Painlevé Equation”, Regul. Chaotic Dyn., 25:3 (2020), 273–280  mathnet  crossref
    17. Oswaldo González-Gaxiola, Anjan Biswas, Mir Asma, Abdullah Kamis Alzahrani, “Optical Dromions and Domain Walls with the Kundu Mukherjee Naskar Equation by the Laplace Adomian Decomposition Scheme”, Regul. Chaotic Dyn., 25:4 (2020), 338–348  mathnet  crossref
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