Geodesic
Coulomb Gas Representation for Rational Solutions of the Painlev\'e Equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 127 (2001) no. 2, pp. 284-303

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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. 
@article{TMF_2001_127_2_a3,
     author = {V. G. Marikhin},
     title = {Coulomb {Gas} {Representation} for {Rational} {Solutions} of the {Painlev\'e} {Equations}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {284--303},
     publisher = {mathdoc},
     volume = {127},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2001_127_2_a3/}
}
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V. G. Marikhin. Coulomb Gas Representation for Rational Solutions of the Painlev\'e Equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 127 (2001) no. 2, pp. 284-303. http://geodesic.mathdoc.fr/item/TMF_2001_127_2_a3/