Canonical structure and symmetries of the Schlesinger equations

DOI10.1007/s00220-006-0165-3zbMath1146.32005arXivmath/0311261OpenAlexW2099806108MaRDI QIDQ2472451

Marta Mazzocco, B. A. Dubrovin

Publication date: 22 February 2008

Published in: Communications in Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0311261

zbMATH Keywords

Hamiltonian structureDarboux coordinatesPainlevé equationscanonical transformationsspectral coordinatesapparent singularitiesFuchsian systemsSchlesinger equations

Mathematics Subject Classification ID

Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies (34M55) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Relationships between algebraic curves and integrable systems (14H70) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions (37K20) Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation) (32G34) Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain (34M50)

Related Items

Classical conformal blocks and Painlevé VI ⋮ Rational \(so(3)\) Gaudin model with general boundary terms ⋮ On separation of variables for homogeneous \(sl(r)\) Gaudin systems ⋮ Hamiltonian structures of isomonodromic deformations on moduli spaces of parabolic connections ⋮ Birational Darboux coordinates on (co)adjoint orbits of \(\mathrm{GL}(N,\mathbb{C})\) ⋮ Surface operators and separation of variables ⋮ The \(2\times 2\) matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system ⋮ Normal forms for rank two linear irregular differential equations and moduli spaces ⋮ Construction and classification of certain algebraic solutions of Garnier systems ⋮ Riemann-Hilbert correspondence and blown up surface defects ⋮ Generalization of Okamoto's equation to arbitrary \(2\times 2\) Schlesinger system ⋮ From quantum curves to topological string partition functions ⋮ Pole distribution of PVI transcendents close to a critical point ⋮ Classification of algebraic solutions of irregular Garnier systems ⋮ The locus of the representation of logarithmic connections by Fuchsian equations ⋮ Algebro-geometric approach to an Okamoto transformation, the Painlevé VI and Schlesinger equations ⋮ Quantization of Calogero-Painlevé system and multi-particle quantum Painlevé equations II-VI ⋮ On rational symplectic parametrization of the coadjoint orbit of $\mathrm{GL}(N)$. Diagonalizable case ⋮ Cubic and Quartic Transformations of the Sixth Painlevé Equation in Terms of Riemann–Hilbert Correspondence ⋮ Explicit description of jumping phenomena on moduli spaces of parabolic connections and Hilbert schemes of points on surfaces ⋮ Integral transformation of solutions for a Fuchsian-class equation corresponding to the Okamoto transformation of the Painlevé VI equation ⋮ Quantum Painlevé-Calogero correspondence ⋮ Isomonodromic deformations and Painlevé equations ⋮ A review of the sixth Painlevé equation

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Separation of variables in the Gaudin model
• On the number of apparent singularities of a linear differential equation
• Monodromy problem and the boundary condition for some Painlevé equations
• Hamiltonian methods in the theory of solitons. Transl. from the Russian by A. G. Reyman
• Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I: General theory and \(\tau \)-function
• Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II
• Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III
• Painlevé property of monodromy preserving deformation equations and the analyticity of \(\tau\) functions
• Studies of the Painlevé equations. I: Sixth Painlevé equation \(P_{VI}\)
• The Knizhnik-Zamolodchikov system as a deformation of the isomonodromy problem
• Three integrable Hamiltonian systems connected with isospectral deformations
• Reduction of symplectic manifolds with symmetry
• Darboux coordinates and Liouville-Arnold integration in loop algebras
• Riemann surfaces, separation of variables and classical and quantum integrability
• Monodromy of certain Painlevé-VI transcendents and reflection groups
• Transformations \(RS_4^2(3)\) of the ranks \(\leq 4\) and algebraic solutions of the sixth Painlevé equation
• Separation of variables in the classical \(\mathrm{SL}(N)\) magnetic chain
• On isomonodromic deformations of Fuchsian systems
• Matrix finite-zone operators
• Rational solutions of the Painlevé VI equation
• Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions
• ON PARTICULAR SOLUTIONS OF THE GARNIER SYSTEMS AND THE HYPERGEOMETRIC FUNCTIONS OF SEVERAL VARIABLES
• METHODS OF ALGEBRAIC GEOMETRY IN THE THEORY OF NON-LINEAR EQUATIONS
• Sixth Painlev\'e Equation, Universal Elliptic Curve, and Mirror of $\bold{P}^2$
• Classical functional Bethe ansatz for SL(N): Separation of variables for the magnetic chain
• A LECTURE ON THE OCTAHEDRON
• The elliptic representation of the general Painlevé VI equation
• Rigid Local Systems. (AM-139)
• Picard and Chazy solutions to the Painlevé VI equation