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Classical solutions of the degenerate Garnier system and their coalescence structures  [PDF]
Takao Suzuki
Mathematics , 2004,
Abstract: We study the degenerate Garnier system which generalizes the fifth Painlev\'{e} equation. We present two classes of particular solutions, classical transcendental and algebraic ones. Their coalescence structure is also investigated.
Bi-orthogonal systems on the unit circle, regular semi-classical weights and the discrete Garnier equations  [PDF]
N. S. Witte
Mathematics , 2008,
Abstract: We demonstrate that a system of bi-orthogonal polynomials and their associated functions corresponding to a regular semi-classical weight on the unit circle constitute a class of general classical solutions to the Garnier systems by explicitly constructing its Hamiltonian formulation and showing that it coincides with that of a Garnier system. Such systems can also be characterised by recurrence relations of the discrete Painlev\'e type, for example in the case with one free deformation variable the system was found to be characterised by a solution to the discrete fifth Painlev\'e equation. Here we derive the canonical forms of the multi-variable generalisation of the discrete fifth Painlev\'e equation to the Garnier systems, i.e. for arbitrary numbers of deformation variables.
On movable singularities of Garnier systems  [PDF]
R. R. Gontsov
Mathematics , 2009,
Abstract: We study movable singularities of Garnier systems using the connection of the latter with Schlesinger isomonodromic deformations of Fuchsian systems
Apparent singularities of Fuchsian equations, and the Painlevé VI equation and Garnier systems  [PDF]
R. R. Gontsov,I. V. Vyugin
Mathematics , 2009, DOI: 10.1016/j.geomphys.2011.08.002
Abstract: We study movable singularities of Garnier systems using the connection of the latter with isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions for some inverse monodromy problems are also considered.
Geometry and classification of solutions of the Classical Dynamical Yang-Baxter Equation  [PDF]
Pavel Etingof,Alexander Varchenko
Mathematics , 1997, DOI: 10.1007/s002200050292
Abstract: The classical Yang-Baxter equation (CYBE) is an algebraic equation central in the theory of integrable systems. Its solutions were classified by Belavin and Drinfeld. Quantization of CYBE led to the theory of quantum groups. A geometric interpretation of CDYB was given by Drinfeld and gave rise to the theory of Poisson-Lie groups. The classical dynamical Yang-Baxter equation (CDYBE) is an important differential equation analagous to CYBE and introduced by Felder as the consistency condition for the Knizhnik-Zamolodchikov-Bernard equations for correlation functions in conformal field theory on tori. Quantization of CDYBE allowed Felder to introduce an interesting elliptic analog of quantum groups. It becomes clear that numerous important notions and results connected with CYBE have dynamical analogs. In this paper we classify solutions to CDYBE and give geometric interpretation to CDYBE. The classification and interpretation are remarkably analogous to the Belavin-Drinfeld picture.
Classical differential geometry and integrability of systems of hydrodynamic type  [PDF]
S. P. Tsarev
Physics , 1993,
Abstract: Remarkable parallelism between the theory of integrable systems of first-order quasilinear PDE and some old results in projective and affine differential geometry of conjugate nets, Laplace equations, their Bianchi-Baecklund transformations is exposed. These results were recently applied by I.M.Krichever and B.A.Dubrovin to prove integrability of some models in topological field theories. Within the geometric framework we derive some new integrable (in a sense to be discussed) generalizations describing N-wave resonant interactions.
Explicit solutions of the classical Calogero & Sutherland systems for any root system  [PDF]
R. Sasaki,K. Takasaki
Physics , 2005, DOI: 10.1063/1.2162334
Abstract: Explicit solutions of the classical Calogero (rational with/without harmonic confining potential) and Sutherland (trigonometric potential) systems is obtained by diagonalisation of certain matrices of simple time evolution. The method works for Calogero & Sutherland systems based on any root system. It generalises the well-known results by Olshanetsky and Perelomov for the A type root systems. Explicit solutions of the (rational and trigonometric) higher Hamiltonian flows of the integrable hierarchy can be readily obtained in a similar way for those based on the classical root systems.
Irregular isomonodromic deformations for Garnier systems and Okamoto's canonical transformations  [PDF]
M. Mazzocco
Physics , 2003,
Abstract: In this paper we describe the Garnier systems as isomonodromic deformation equations of a linear system with a simple pole at zero and a Poincar\'e rank two singularity at infinity. We discuss the extension of Okamoto's birational canonical transformations to the Garnier systems in more than one variable and to the Schlesinger systems.
Vacuum Solutions of Classical Gravity on Cyclic Groups from Noncommutative Geometry  [PDF]
Jian Dai,Xing-Chang Song
Physics , 2001,
Abstract: Based on the observation that the moduli of a link variable on a cyclic group modify Connes' distance on this group, we construct several action functionals for this link variable within the framework of noncommutative geometry. After solving the equations of motion, we find that one type of action gives nontrivial vacuum solution for gravity on this cyclic group in a broad range of coupling constants and that such solutions can be expressed with Chebyshev's polynomials.
The Discrete and Continuous Painleve VI Hierarchy and the Garnier Systems  [PDF]
F. W. Nijhoff,A. J. Walker
Physics , 2000,
Abstract: We present a general scheme to derive higher-order members of the Painleve VI (PVI) hierarchy of ODE's as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation and that consists of a system of partial difference equations on a multidimensional lattice. The connection with the isomonodromic Garnier systems is discussed.
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