Abstract
The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n + 1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations S (n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams M.R., Harnad J. and Hurtubise J. (1933). Darboux coordinates and Liouville–Arnold integration in loop algebras. Commun. Math. Phys. 155: 385–413
Andreev F.V. and Kitaev A.V. (2002). Transformations \(RS^{2}_{4}(3)\) Commun. Math. Phys. 228(1): 151–176
Anosov, D.V., Bolibruch, A.A.: The Riemann-Hilbert Problem. Volume E 22, Aspects of Mathematics, Braunsdiweig: Friedrich Vieweg & Sohn Verlag (1994)
Arinkin D. and Lysenko S. (1977). On the moduli of SL(2)-bundles with connections on \(P^{1}\setminus\{x_1,\dots,x_4\}\) Int. Math. Res. Not. 1997(19): 983–999
Audin, M.: Lectures on gauge theory and integrable systems. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 488, Dordrecht: Kluwer, 1997
Babelon O. and Talon M. (2003). Riemann surfaces, separation of variables and classical and quantum integrability. Phys.Lett. A 312: 71–77
Bolibruch, A.A.: The 21-st Hilbert problem for linear Fuchsian systems. In: Developments in mathematics: the Moscow school. London: Chapman and Hall, 1993
Bolibruch, A.A.: On isomonodromic deformations of Fuchsian systems. J. Dynam. Control Systems 3, 589–604 (1997)
Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York-Toronto-London: McGraw-Hill Book Company, Inc., 1955
Costin O. and Costin R.D. (2002). Asymptotic properties of a family of solutions of the Painlevé equation VI. Int. Math. Res. Not. 22: 1167–1182
Dubrovin B. (1985). Matrix finite-gap operators. J. Soviet Math. 28: 20–50
Dubrovin, B.: Geometry of 2D topological field theories. Volume 1620 of Springer Lecture Notes in Math. Integrable Systems and Quantum Groups, M. Francaviglia, S. Greco, eds. Berlin-Heidelberg- New York:Springer 1996
Dubrovin, B., Diener, P.: Algebro-geometrical Darboux coordinates in R-matrix formalism. Preprint 88/94/FM, 1994
Dubrovin B. and Mazzocco M. (2000). Monodromy of certain Painlevé-VI transcendents and reflection groups. Invent. Math. 141: 55–147
Faddeev, L.D., Takhtajan, L.A.: Hamiltonian methods in the theory of solitons. Springer Series in Soviet Mathematics, Berlin: Springer-Verlag, 1987
Flaschka H. and McLaughlin D.W. (1976). Cononically conjugate variables for the Korteweg–de Vries equation and the Toda lattice with periodic boundary conditions. Progr. Theor. Phys. 55: 438–456
Fuchs, R.: Lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math. Ann. 63, 301–321 (1907)
Garnier R. (1912). Sur des équations différentielles du troisième ordre dont l’intégrale générale est uniforme et sur une classe d’équations nouvelles d’ordre supérieur dont l’intégrale générale a ses points critiques fixes. Ann. Sci. École Norm. Sup. 29(3): 1–126
Garnier R. (1926). Solution du probleme de Riemann pour les systemes différentielles linéaires du second ordre. Ann. Sci. École Norm. Sup. 43: 239–352
Gekhtman M. (1995). Separation of variables in the classical SL(N) magnetic chains. Commun. Math. Phys. 167: 593–605
Griffiths P.A. (1984). Linearizing flows and a cohomological interpretation of Lax equations. Math. Sci. Res. Inst. Publ. 2: 36–46
Guzzetti D. (2002). The elliptic representation of the general Painlevé VI equation. Comm. Pure Appl. Math. 55(10): 1280–1363
Harnad, J.: Quantum isomonodromic deformations and the Knizhnik–Zamolodchikov equations. In: Symmetries and integrability of difference equations (Esttrel, PQ, 1994), CRM Lecture Notes 9, Providence, RI: Amer. Math.Soc., 1996
Hitchin N. (1997). Frobenius manifolds (with notes by David Calderbank). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 488: 69–112
Hitchin N. (2003). A lecture on the octahedron. Bull. London Math. Soc. 35(5): 577–600
Ince, E.L.: Ordinary differential equations. New York: Dover Publications INC., 1956
Its, A.R., Novokshenov, V.Yu.:The isomonodromic deformation method in the theory of Painlevé equations, volume 1191 of Lecture notes in mathematics. Berlin-Heidelberg-New York: Springer, 1980
Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé, a Modern Theory of Special Functions. Volume E 16, Aspects of Mathematics, Branschweig: Friedrich Vieweg # sohn, 1991
Jimbo, M.: Monodromy problem and the boundary condition for some Painlevé equations. Publ. Res. Inst. Math. Sci. 18, 1137–1161 (1982)
Jimbo, M., Miwa, T.:Monodromy preserving deformations of linear ordinary differential equations with rational coefficients II. Physica 2D, 2(3), 407–448 (1981)
Jimbo, M., Miwa, T.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients III. Physica 2D 4(1), 26–46 (1982)
Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients I. Physica 2D 2(2), 306–352 (1981)
Katz, N.: Rigid Local Systems. Volume 139 of Ann. Math. Studies. Princeton, NJ: Princeton University Press, 1996
Kimura H. and Okamoto K. (1983). On the isomonodromic deformation of linear ordinary differential equations of higher order. Funkcial. Ekvac. 26(1): 37–50
Krichever I.M. (1977). Methods of algebraic geometry in the theory of non-linear equations. Russ. Math. Surv. 32: 185–213
Krichever I.M. (2002). Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations. Mosc. Math. J. 2(4): 717–806
Levelt, A.H.M.:Hypergeometric functions. Doctoral thesis, University of Amsterdam, 1961
Malgrange, B.: Sur les déformations isomonodromiques I. Singularités régulières. Volume 37 of Mathematics and Physics. Progr. Math. Boston: Birkhauser 1983
Manin, Yu.I.: Sixth Painlevé equation, universal elliptic curve, and mirror of P 2. Amer Math Soc Transl. Ser. 2, 186, 131–151 (1998)
Manin, Yu.I.: Frobenius manifolds, quantum cohomology and moduli spaces. Colloquium Publ. Volume 47, Providence, RI: Amer. Math. Soc. (1999)
Marsden J. and Weinstein A. (1974). Reduction of symplectic manifolds with symmetry. Rep. Mathematical Phys. 5(1): 121–130
Mazzocco M. (2001). Picard and Chazy solutions to the PVI equation. Math. Ann. 321(1): 131–169
Mazzocco M. (2001). Rational solutions of the Painlevé VI equation. J. Phys. A: Math. Gen. 34: 2281–2294
Miwa T. (1981). Painlevé property of monodromy preserving equations and the analyticity of τ-functions. Publ. RIMS 17: 703–721
Moser J. (1975). Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16: 197–220
Ohtsuki M. (1982). On the number of apparent singularities of a linear differential equation. Tokyo J. Math. 5: 23–29
Okamoto, K.: Isomonodromic deformations, Painlevé equations and the Garnier system. J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math. 33, 576–618 (1986)
Okamoto K. (1987). Studies on the Painlevé equations I, sixth Painlevé equation. Ann. Mat. Pura Appl. 146: 337–381
Okamoto, K.: Painlevé equations and Dynkin diagrams. In: Painlevé Transcendents, London: Plenum, 1992, pp. 299–313
Okamoto K. and Kimura H. (1986). On Particular solutions of the Garnier system and the hypergeometric functions of several variables. Quart. J. Math. Oxford 37: 61–80
Reshetikhin N. (1992). The Knizhnik–Zamolodchikov system as a deformation of ‘the isomonodromy problem. Lett. Math. Phys. 26: 167–177
Schlesinger L. (1912). Ueber eine Klasse von Differentsial System Beliebliger Ordnung mit Festen Kritischer Punkten. J. fur Math. 141: 96–145
Scott D.R.D. (1994). Classical functional Bethe ansatz for SL(N): separation of variables for the magnetic chain. J. Math. Phys. 35: 5831–5843
Sibuya, Y.: Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation, Volume 82 of Trans. of math. Monographs. Providence, RI: Amer Math. Soc., 1990
Sklyanin E.K. (1989). Separation of variables in Gaudin model. J.Soviet Math. 47: 2473–2488
Tsuda, T.: Universal characters and integrable systems. PhD thesis, Tokyo Graduate School of Mathematics, 2003
Umemura H. (1990). Irreducibility of the first differential equation of Painlevé. Nagoya Math. J. 117: 231–252
Veselov A.P. and Novikov S.P. (1984). Poisson brackets and complex tori. Trudy Mat. Inst. Steklov. 165: 49–61
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Takhtajan
Rights and permissions
About this article
Cite this article
Dubrovin, B., Mazzocco, M. Canonical Structure and Symmetries of the Schlesinger Equations. Commun. Math. Phys. 271, 289–373 (2007). https://doi.org/10.1007/s00220-006-0165-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-006-0165-3