On the algebro-geometric integration of the Schlesinger equations

Percy Deift, A. Its, A. Kapaev, X. Zhou

Research output: Contribution to journalArticle

Abstract

A new approach to the construction of isomonodromy deformations of 2 × 2 Fuchsian systems is presented. The method is based on a combination of the algebro-geometric scheme and Riemann-Hilbert approach of the theory of integrable systems. For a given number 2g + 1, g ≥ 1, of finite (regular) singularities, the method produces a 2g-parameter submanifold of the Fuchsian monodromy data for which the relevant Riemann-Hilbert problem can be solved in closed form via the Baker-Akhiezer function technique. This in turn leads to a 2g-parameter family of solutions of the corresponding Schlesinger equations, explicitly described in terms of Riemann theta functions of genus g. In the case g = 1 the solution found coincides with the general elliptic solution of the particular case of the Painlevé VI equation first obtained by N. J. Hitchin [H1].

Original languageEnglish (US)
Pages (from-to)613-633
Number of pages21
JournalCommunications in Mathematical Physics
Volume203
Issue number3
StatePublished - 1999

Fingerprint

Geometric Integration
Riemann Function
Riemann-Hilbert Problem
Theta Functions
Monodromy
Integrable Systems
Submanifolds
Hilbert
Genus
Closed-form
Singularity

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

On the algebro-geometric integration of the Schlesinger equations. / Deift, Percy; Its, A.; Kapaev, A.; Zhou, X.

In: Communications in Mathematical Physics, Vol. 203, No. 3, 1999, p. 613-633.

Research output: Contribution to journalArticle

Deift, Percy ; Its, A. ; Kapaev, A. ; Zhou, X. / On the algebro-geometric integration of the Schlesinger equations. In: Communications in Mathematical Physics. 1999 ; Vol. 203, No. 3. pp. 613-633.
@article{f2de92ae74fd4b98a2017f1c3d2f8160,
title = "On the algebro-geometric integration of the Schlesinger equations",
abstract = "A new approach to the construction of isomonodromy deformations of 2 × 2 Fuchsian systems is presented. The method is based on a combination of the algebro-geometric scheme and Riemann-Hilbert approach of the theory of integrable systems. For a given number 2g + 1, g ≥ 1, of finite (regular) singularities, the method produces a 2g-parameter submanifold of the Fuchsian monodromy data for which the relevant Riemann-Hilbert problem can be solved in closed form via the Baker-Akhiezer function technique. This in turn leads to a 2g-parameter family of solutions of the corresponding Schlesinger equations, explicitly described in terms of Riemann theta functions of genus g. In the case g = 1 the solution found coincides with the general elliptic solution of the particular case of the Painlev{\'e} VI equation first obtained by N. J. Hitchin [H1].",
author = "Percy Deift and A. Its and A. Kapaev and X. Zhou",
year = "1999",
language = "English (US)",
volume = "203",
pages = "613--633",
journal = "Communications in Mathematical Physics",
issn = "0010-3616",
publisher = "Springer New York",
number = "3",

}

TY - JOUR

T1 - On the algebro-geometric integration of the Schlesinger equations

AU - Deift, Percy

AU - Its, A.

AU - Kapaev, A.

AU - Zhou, X.

PY - 1999

Y1 - 1999

N2 - A new approach to the construction of isomonodromy deformations of 2 × 2 Fuchsian systems is presented. The method is based on a combination of the algebro-geometric scheme and Riemann-Hilbert approach of the theory of integrable systems. For a given number 2g + 1, g ≥ 1, of finite (regular) singularities, the method produces a 2g-parameter submanifold of the Fuchsian monodromy data for which the relevant Riemann-Hilbert problem can be solved in closed form via the Baker-Akhiezer function technique. This in turn leads to a 2g-parameter family of solutions of the corresponding Schlesinger equations, explicitly described in terms of Riemann theta functions of genus g. In the case g = 1 the solution found coincides with the general elliptic solution of the particular case of the Painlevé VI equation first obtained by N. J. Hitchin [H1].

AB - A new approach to the construction of isomonodromy deformations of 2 × 2 Fuchsian systems is presented. The method is based on a combination of the algebro-geometric scheme and Riemann-Hilbert approach of the theory of integrable systems. For a given number 2g + 1, g ≥ 1, of finite (regular) singularities, the method produces a 2g-parameter submanifold of the Fuchsian monodromy data for which the relevant Riemann-Hilbert problem can be solved in closed form via the Baker-Akhiezer function technique. This in turn leads to a 2g-parameter family of solutions of the corresponding Schlesinger equations, explicitly described in terms of Riemann theta functions of genus g. In the case g = 1 the solution found coincides with the general elliptic solution of the particular case of the Painlevé VI equation first obtained by N. J. Hitchin [H1].

UR - http://www.scopus.com/inward/record.url?scp=0033242958&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033242958&partnerID=8YFLogxK

M3 - Article

VL - 203

SP - 613

EP - 633

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -