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 TMF, 2001, Volume 127, Number 2, Pages 284–303 (Mi tmf458)

Coulomb Gas Representation for Rational Solutions of the Painlevé Equations

V. G. Marikhin

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: We consider rational solutions for a number of dynamic systems of the type of the nonlinear Schrödinger equation, in particular, the Levi system. We derive the equations for the dynamics of poles and Bäcklund transformations for these solutions. We show that these solutions can be reduced to rational solutions of the Painlevé IV equation, with the equations for the pole dynamics becoming the stationary equations for the two-dimensional Coulomb gas in a parabolic potential. The corresponding Coulomb systems are derived for the Painlevé II-VI equations. Using the Hamiltonian formalism, we construct the spin representation of the Painlevé equations.

DOI: https://doi.org/10.4213/tmf458

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English version:
Theoretical and Mathematical Physics, 2001, 127:2, 646–663

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Revised: 04.01.2001

Citation: V. G. Marikhin, “Coulomb Gas Representation for Rational Solutions of the Painlevé Equations”, TMF, 127:2 (2001), 284–303; Theoret. and Math. Phys., 127:2 (2001), 646–663

Citation in format AMSBIB
\Bibitem{Mar01} \by V.~G.~Marikhin \paper Coulomb Gas Representation for Rational Solutions of the Painlev\'e Equations \jour TMF \yr 2001 \vol 127 \issue 2 \pages 284--303 \mathnet{http://mi.mathnet.ru/tmf458} \crossref{https://doi.org/10.4213/tmf458} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1863564} \zmath{https://zbmath.org/?q=an:1017.33010} \transl \jour Theoret. and Math. Phys. \yr 2001 \vol 127 \issue 2 \pages 646--663 \crossref{https://doi.org/10.1023/A:1010449603754} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000170547800004} 

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This publication is cited in the following articles:
1. Clarkson, PA, “The fourth Painlevé equation and associated special polynomials”, Journal of Mathematical Physics, 44:11 (2003), 5350
2. Filipuk, GV, “The symmetric fourth Painlevé hierarchy and associated special polynomials”, Studies in Applied Mathematics, 121:2 (2008), 157
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