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UC hierarchy and monodromy preserving deformation

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Published/Copyright: April 11, 2012
Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2014 Issue 690

Abstract.

The UC hierarchy is an extension of the KP hierarchy, which possesses not only an infinite set of positive time evolutions but also that of negative ones. Through a similarity reduction we derive from the UC hierarchy a class of the Schlesinger systems including the Garnier system and the sixth Painlevé equation, which describes the monodromy preserving deformations of Fuchsian linear differential equations with certain spectral types. We also present a unified formulation of the above Schlesinger systems as a canonical Hamiltonian system whose Hamiltonian functions are polynomials in the canonical variables.

I would like to express my sincere gratitude to Hidetaka Sakai for his helpful comments on Hamiltonian structure of the Schlesinger systems and informing me about the article [Phys. D 1 (1980), 80–158]. I also greatly appreciate illuminating discussions with Yasuhiro Ohta and Takao Suzuki. This work was partly conducted during my stay in the Issac Newton Institute for Mathematical Sciences at the program “Discrete Integrable Systems” (2009).

Received: 2010-2-24
Revised: 2012-2-14
Published Online: 2012-4-11
Published in Print: 2014-5-1

© 2014 by Walter de Gruyter Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. UC hierarchy and monodromy preserving deformation
  3. Vector-valued p-adic Siegel modular forms
  4. K-theoretic Schubert calculus for OG(n,2n+1) and jeu de taquin for shifted increasing tableaux
  5. Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications
  6. Inverse tunneling estimates and applications to the study of spectral statistics of random operators on the real line
  7. Mean curvature flow of entire graphs in a half-space with a free boundary
  8. Ricci flow and the holonomy group
  9. Uniqueness of compact tangent flows in Mean Curvature Flow
  10. Even unimodular Lorentzian lattices and hyperbolic volume
  11. Algebraic points on Shimura curves of Γ0(p)-type
  12. Errata to Algebraic points on Shimura curves of Γ0(p)-type (J. reine angew. Math., DOI 10.1515/crelle-2012-0068)
  13. Property A and the operator norm localization property for discrete metric spaces
  14. Stable manifolds for holomorphic automorphisms
https://doi.org/10.1515/crelle-2012-0022

Articles in the same Issue

  1. Frontmatter
  2. UC hierarchy and monodromy preserving deformation
  3. Vector-valued p-adic Siegel modular forms
  4. K-theoretic Schubert calculus for OG(n,2n+1) and jeu de taquin for shifted increasing tableaux
  5. Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications
  6. Inverse tunneling estimates and applications to the study of spectral statistics of random operators on the real line
  7. Mean curvature flow of entire graphs in a half-space with a free boundary
  8. Ricci flow and the holonomy group
  9. Uniqueness of compact tangent flows in Mean Curvature Flow
  10. Even unimodular Lorentzian lattices and hyperbolic volume
  11. Algebraic points on Shimura curves of Γ0(p)-type
  12. Errata to Algebraic points on Shimura curves of Γ0(p)-type (J. reine angew. Math., DOI 10.1515/crelle-2012-0068)
  13. Property A and the operator norm localization property for discrete metric spaces
  14. Stable manifolds for holomorphic automorphisms
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