### 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 language | English (US) |
---|---|

Pages (from-to) | 613-633 |

Number of pages | 21 |

Journal | Communications in Mathematical Physics |

Volume | 203 |

Issue number | 3 |

State | Published - 1999 |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*Communications in Mathematical Physics*,

*203*(3), 613-633.

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

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 203, no. 3, pp. 613-633.

}

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

AN - SCOPUS:0033242958

VL - 203

SP - 613

EP - 633

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -