Geodesic
Limit relation between Toda chains and the~elliptic $SL(N,\mathbb C)$ top
Teoretičeskaâ i matematičeskaâ fizika, Tome 171 (2012) no. 2, pp. 179-195

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We study a limit relation between the elliptic $SL(N,\mathbb C)$ top and Toda chains. We show that in the case of the nonautonomous $SL(2,\mathbb C)$ top, whose equations of motion are related to the Painlevé VI equation, it turns out to be possible to modify the previously proposed procedure and in the limit obtain the nonautonomous Toda chain, whose equations of motion are equivalent to a particular case of the Painlevé III equation. We obtain the limit of the Lax pair for the elliptic $SL(2,\mathbb C)$ top, which allows representing the equations of motion of the nonautonomous Toda chain as the equation for isomonodromic deformations.
Keywords: integrable system, nonautonomous system, Painlevé equation, Inozemtsev limit, topological structure of phase space. 
@article{TMF_2012_171_2_a0,
     author = {G. Aminov},
     title = {Limit relation between {Toda} chains and the~elliptic $SL(N,\mathbb C)$ top},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {179--195},
     publisher = {mathdoc},
     volume = {171},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2012_171_2_a0/}
}
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G. Aminov. Limit relation between Toda chains and the~elliptic $SL(N,\mathbb C)$ top. Teoretičeskaâ i matematičeskaâ fizika, Tome 171 (2012) no. 2, pp. 179-195. http://geodesic.mathdoc.fr/item/TMF_2012_171_2_a0/