Geodesic
Amalgamations of the Painlev\'e Equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 137 (2003) no. 3, pp. 408-423

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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. 
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     author = {N. A. Kudryashov},
     title = {Amalgamations of the {Painlev\'e} {Equations}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {408--423},
     publisher = {mathdoc},
     volume = {137},
     number = {3},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2003_137_3_a7/}
}
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N. A. Kudryashov. Amalgamations of the Painlev\'e Equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 137 (2003) no. 3, pp. 408-423. http://geodesic.mathdoc.fr/item/TMF_2003_137_3_a7/