Abstract:
It is well known that the self-similar solutions of the Korteweg–de Vries equation and the modified Korteweg–de 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 Korteveg–de Vries, modified Korteweg–de Vries, Kaup–Kupershmidt, Caudrey–Dodd–Gibbon and Fordy–Gibbons 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.
This research was partially supported by the Federal Target Program “Scientific and ScientificPedagogical
Personnel of Innovative Russia” for 2009–2013, by RFBR grant 11–01–00798–a and
“Researches and Developments in Priority Directions of Development of the Scientific Technological
Complex of Russia” for 2007–2013.
Citation:
Nikolay A. Kudryashov, “Higher Painlevé Transcendents as Special Solutions of Some Nonlinear Integrable Hierarchies”, Regul. Chaotic Dyn., 19:1 (2014), 48–63
\Bibitem{Kud14}
\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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\crossref{https://doi.org/10.1134/S1560354714010043}
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Linking options:
https://www.mathnet.ru/eng/rcd140
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This publication is cited in the following 21 articles:
P.R. Gordoa, A. Pickering, J.A.D. Wattis, “Solution classes of the matrix second Painlevé hierarchy”, Physica D: Nonlinear Phenomena, 435 (2022), 133295
Nikolay A. Kudryashov, “Lax Pairs and Rational Solutions of Similarity Reductions for
Kupershmidt and Sawada – Kotera Hierarchies”, Regul. Chaotic Dyn., 26:3 (2021), 271–292
Kudryashov N.A., “Lax Pairs For One of Hierarchies Similar to the First Painleve Hierarchy”, Appl. Math. Lett., 116 (2021), 107003
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
Nikolay A. Kudryashov, “Rational Solutions of Equations Associated with the Second Painlevé Equation”, Regul. Chaotic Dyn., 25:3 (2020), 273–280
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
N. A. Kudryashov, Mechanisms and Machine Science, 80, Advanced Technologies in Robotics and Intelligent Systems, 2020, 25
Nikolay A. Kudryashov, “Rational and Special Solutions for Some Painlevé Hierarchies”, Regul. Chaotic Dyn., 24:1 (2019), 90–100
N. A. Kudryashov, “On Integrability of the FitzHugh – Rinzel Model”, Rus. J. Nonlin. Dyn., 15:1 (2019), 13–19
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
N. A. Kudryashov, “Lax pair and first integrals of the traveling wave reduction for the KdV hierarchy”, Appl. Math. Comput., 350 (2019), 323–330
N. A. Kudryashov, “Nonlinear differential equations associated with the first Painlevé hierarchy”, Appl. Math. Lett., 90 (2019), 223–228
P. R. Gordoa, A. Pickering, “Backlund transformations for a new extended Painlevé hierarchy”, Commun. Nonlinear Sci. Numer. Simul., 69 (2019), 78–97
N. A. Kudryashov, “The Painlevé approach for finding solitary wave solutions of nonlinear nonintegrable differential equations”, Optik, 183 (2019), 642–649
Nikolay A. Kudryashov, “Exact Solutions and Integrability of the Duffing–Van der Pol Equation”, Regul. Chaotic Dyn., 23:4 (2018), 471–479
P. R. Gordoa, A. Pickering, “On an extended second Painlevé hierarchy”, J. Differ. Equ., 263:7 (2017), 4070–4125
P. R. Gordoa, A. Pickering, Z. N. Zhu, “On matrix Painlevé hierarchies”, J. Differ. Equ., 261:2 (2016), 1128–1175
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
Nikolay A. Kudryashov, “Analytical Solutions of the Lorenz System”, Regul. Chaotic Dyn., 20:2 (2015), 123–133
Alexey V. Borisov, Nikolay A. Kudryashov, “Paul Painlevé and His Contribution to Science”, Regul. Chaotic Dyn., 19:1 (2014), 1–19
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