### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 025, 27 pages      arXiv:1604.07847      https://doi.org/10.3842/SIGMA.2017.025
Contribution to the Special Issue on Symmetries and Integrability of Difference Equations

### Multi-Poisson Approach to the Painlevé Equations: from the Isospectral Deformation to the Isomonodromic Deformation

Hayato Chiba
Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395, Japan

Received October 11, 2016, in final form April 11, 2017; Published online April 15, 2017

Abstract
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.

Key words: Painlevé equations; Lax equations; multi-Poisson structure.

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References

1. Adler M., van Moerbeke P., Vanhaecke P., Algebraic integrability, Painlevé geometry and Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 47, Springer-Verlag, Berlin, 2004.
2. Chiba H., The first, second and fourth Painlevé equations on weighted projective spaces, J. Differential Equations 260 (2016), 1263-1313, arXiv:1311.1877.
3. Chiba H., The third, fifth and sixth Painlevé equations on weighted projective spaces, SIGMA 12 (2016), 019, 22 pages, arXiv:1506.00444.
4. Chiba H., Painlevé equations and weights, submitted.
5. Clarkson P.A., Joshi N., Mazzocco M., The Lax pair for the mKdV hierarchy, in Théories asymptotiques et équations de Painlevé, Sémin. Congr., Vol. 14, Soc. Math. France, Paris, 2006, 53-64.
6. Clarkson P.A., Joshi N., Pickering A., Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach, Inverse Problems 15 (1999), 175-187, solv-int/9811014.
7. Cosgrove C.M., Chazy classes IX-XI of third-order differential equations, Stud. Appl. Math. 104 (2000), 171-228.
8. Drinfel'd V.G., Sokolov V.V., Lie algebras and equations of Korteweg-de Vries type, J. Math. Sci. 30 (1985), 1975-2036.
9. Falqui G., Magri F., Pedroni M., Zubelli J.P., A bi-Hamiltonian theory for stationary KDV flows and their separability, Regul. Chaotic Dyn. 5 (2000), 33-52, nlin.SI/0003020.
10. Gordoa P.R., Joshi N., Pickering A., On a generalized $2+1$ dispersive water wave hierarchy, Publ. Res. Inst. Math. Sci. 37 (2001), 327-347.
11. Koike T., On new expressions of the Painlevé hierarchies, in Algebraic Analysis and the Exact WKB Analysis for Systems of Differential Equations, RIMS Kôky^uroku Bessatsu, Vol. B5, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, 153-198.
12. Kudryashov N.A., The first and second Painlevé equations of higher order and some relations between them, Phys. Lett. A 224 (1997), 353-360.
13. Levin A.M., Olshanetsky M.A., Painlevé-Calogero correspondence, in Calogero-Moser-Sutherland Models (Montréal, QC, 1997), CRM Ser. Math. Phys., Springer, New York, 2000, 313-332.
14. Magnano G., Magri F., Poisson-Nijenhuis structures and Sato hierarchy, Rev. Math. Phys. 3 (1991), 403-466.
15. Magri F., Casati P., Falqui G., Pedroni M., Eight lectures on integrable systems, in Integrability of Nonlinear Systems (Pondicherry, 1996), Lecture Notes in Phys., Vol. 495, Springer, Berlin, 1997, 256-296.
16. Magri F., Falqui G., Pedroni M., The method of Poisson pairs in the theory of nonlinear PDEs, in Direct and inverse methods in nonlinear evolution equations, Lecture Notes in Phys., Vol. 632, Springer, Berlin, 2003, 85-136, nlin.SI/0002009.
17. Nakamura A., Autonomous limit of 4-dimensional Painlevé-type equations and degeneration of curves of genus two, arXiv:1505.00885.
18. Shimomura S., A certain expression of the first Painlevé hierarchy, Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), 105-109.