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by A Zabrodin, A Zotov

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by
Igor Rumanov
, 2014

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...to be simpler than the solution of the FP equation one started with. This gives additional justification for the approach taken here. A similar approach was put forward recently by Zabrodin and Zotov =-=[43]-=-. One difference is that these authors restrict their Lax pairs to be traceless, which we do not do. More importantly, however, we stress that the approach can be applied e.g. to β-ensembles for all 4...

by
Igor Rumanov
, 2014

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... FP equations are autonomous and therefore admit simple solutions. It turns out that the whole related nonlinear integrable structure can be derived from this type FP equations but nonautonomous ones =-=[107, 94]-=-, which is one of the main subjects of the present paper. Besides, one is more often interested in the integrals of the p.d.f. rather than the p.d.f. itself. The integrals sometimes also satisfy certa...

by
A. Levin, M. Olshanetsky, A. Zotov

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...) = [L(z),Ma(z)] , a = 1, ..., n (6.43) and ∂τL(z)− ∂zMτ = [L(z),Mτ (z)] (6.44) as well. The phenomenon is known as the Painlevé-Calogero Correspondence [43] (see also [91] for developed version and =-=[92, 93, 94, 95]-=- for the quantum versions). Notice that the time variables ta in (6.37) are replaced by the positions of marked points xa in (6.43) while Ma(z) are the same in both sets of equations. In the same mann...

by
A. M. Grundl, D. Riglioni

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...́ equation as the compatibility condition of a set of linear differential equations goes back to the work of R. Fuchs [8]. In some recent papers by Suleimanov [9] and later by Zabrodin and Zotov [10]-=-=[12]-=- such a mechanism has been used to define the Painlevé equations in the so-called ”Calogero form”. The definition of Painlevé equations as a Hamiltonian system (also known as ”Calogero form”) has a ...

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