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TMF, 2003, Volume 137, Number 3, Pages 408–423 (Mi tmf281)  

Amalgamations of the Painlevé Equations

N. A. Kudryashov

Moscow Engineering Physics Institute (State University)

Abstract: We present new hierarchies of nonlinear ordinary differential equations (ODEs) that are generalizations of the Painlevé equations. These hierarchies contain the Painlevé equations as special cases. We emphasize the sixth-order ODEs. Special solutions for one of them are expressed via the general solutions of the $P_1$ and $P_2$ equations and special cases of the $P_3$ and $P_5$ equations. Four of the six Painlevé equations can be considered special cases of these sixth-order ODEs. We give linear representations for solving the Cauchy problems for the hierarchy equations using the inverse monodromy transform.

Keywords: Painlevé equations, Painlevé transcendents, higher analogues, isomonodromic linear problem

DOI: https://doi.org/10.4213/tmf281

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English version:
Theoretical and Mathematical Physics, 2003, 137:3, 1703–1715

Bibliographic databases:


Citation: N. A. Kudryashov, “Amalgamations of the Painlevé Equations”, TMF, 137:3 (2003), 408–423; Theoret. and Math. Phys., 137:3 (2003), 1703–1715

Citation in format AMSBIB
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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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