Geodesic
Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

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A multi-Poisson structure on a Lie algebra $\mathfrak{g}$ provides a systematic way to construct completely integrable Hamiltonian systems on $\mathfrak{g}$ expressed in Lax form $\partial X_\lambda /\partial t = [X_\lambda , A_\lambda ]$ in the sense of the isospectral deformation, where $X_\lambda , A_\lambda \in \mathfrak{g}$ depend rationally on the indeterminate $\lambda $ called the spectral parameter. In this paper, a method for modifying the isospectral deformation equation to the Lax equation $\partial X_\lambda /\partial t = [X_\lambda , A_\lambda ] + \partial A_\lambda /\partial \lambda $ in the sense of the isomonodromic deformation, which exhibits the Painlevé property, is proposed. This method gives a few new Painlevé systems of dimension four.
Mots-clés : Painlevé equations; Lax equations; multi-Poisson structure. 
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     author = {Hayato Chiba},
     title = {Multi-Poisson {Approach} to the {Painlev\'e} {Equations:} from the {Isospectral} {Deformation} to the {Isomonodromic} {Deformation}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a24/}
}
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Hayato Chiba. Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a24/

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