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Regul. Chaotic Dyn.:


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

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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

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.




MSC: 35Q51, 35Q53, 37K15
: 02.12.2013

: Nikolay A. Kudryashov, “Higher Painlevé Transcendents as Special Solutions of Some Nonlinear Integrable Hierarchies”, Regul. Chaotic Dyn., 19:1 (2014), 48–63

\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


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    1. Alexey V. Borisov, Nikolay A. Kudryashov, “Paul painlevé and his contribution to science”, Regul. Chaotic Dyn., 19:1 (2014), 1–19  mathnet  crossref  mathscinet  zmath  scopus
    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  scopus
    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  scopus
    4. A. Nikolay , Kudryashov, “Analytical solutions of the lorenz system”, Regul. Chaotic Dyn., 20:2 (2015), 123–133  mathnet  crossref  mathscinet  zmath  adsnasa  scopus
    5. P. R. Gordoa, A. Pickering, Z. N. Zhu, “On matrix painleve hierarchies”, J. Differ. Equ., 261:2 (2016), 1128–1175  crossref  mathscinet  zmath  isi  scopus
    6. P. R. Gordoa, A. Pickering, “On an extended second painleve hierarchy”, J. Differ. Equ., 263:7 (2017), 4070–4125  crossref  mathscinet  zmath  isi  scopus
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