Abstract:
A self-similar reduction of the Korteweg–de Vries hierarchy is considered. A linear system of equations associated with this hierarchy is presented. This Lax pair can be used to solve the Cauchy problem for the studied hierarchy. It is shown that special solutions of the self-similar reduction of the KdV hierarchy are determined via the transcendents of the first Painlevé hierarchy. Rational solutions are expressed by means of the Yablonskii–Vorob’ev polynomials. The map of the solutions for the second Painlevé hierarchy into the solutions for the self-similar reduction of the KdV hierarchy is illustrated using the Miura transformation. Lax pairs for equations of the hierarchy for the Yablonskii–Vorob’ev polynomial are discussed. Special solutions to the hierarchy for the Yablonskii–Vorob’ev polynomials are given. Some other hierarchies with properties of the Painlevé hierarchies are presented. The list of nonlinear differential equations whose general solutions are expressed in terms of nonclassical functions is extended.
\Bibitem{Kud19}
\by Nikolay A. Kudryashov
\paper Rational and Special Solutions for Some Painlevé Hierarchies
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 1
\pages 90--100
\mathnet{http://mi.mathnet.ru/rcd391}
\crossref{https://doi.org/10.1134/S1560354719010052}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85061209063}
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This publication is cited in the following 9 articles:
V. V. Tsegel'nik, “On Bäcklund transformations for some second-order nonlinear differential equations”, Theoret. and Math. Phys., 217:2 (2023), 1755–1766
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
Sh. Chen, Yu. Li, M. Jiang, B. Guan, Ya. Liu, F. Bu, “Abundant traveling wave solutions to an intrinsic fractional discrete nonlinear electrical transmission line”, Results Phys., 28 (2021), 104587
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
Nikolay A. Kudryashov, Dariya V. Safonova, Anjan Biswas, “Painlevé Analysis and a Solution to the Traveling Wave Reduction of the Radhakrishnan – Kundu – Lakshmanan Equation”, Regul. Chaotic Dyn., 24:6 (2019), 607–614
Kudryashov N.A., “General Solution of Traveling Wave Reduction For the Kundu-Mukherjee-Naskar Model”, Optik, 186 (2019), 22–27
Kudryashov N.A., “First Integrals and Solutions of the Traveling Wave Reduction For the Triki-Biswas Equation”, Optik, 185 (2019), 275–281
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