Related papers: Studies on the Garnier system in two variables
We present {\it symmetric Hamiltonians} for the degenerate Garnier systems in two variables. For these symmetric Hamiltonians, we make the symmetry and holomorphy conditions, and we also make a generalization of these systems involving…
We construct and characterize quantum Garnier systems in two variables including degenerate cases by certain holomorphic properties under the quantum canonical transformations.
In this note, we will compare the Garnier system in two variables with four-dimensional partial differential system in two variables with $W(D_6^{(1)})$-symmetry. Both systems are different in each compactification in the variables…
We review non-autonomous Hamiltonian systems, polynomial in two dependent variables, with the property that all of their solutions are meromorphic functions in the complex plane. These are related to known Hamiltonian systems with the…
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…
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…
The sixth Painlev\'e equation is hiding extremely rich geometric structures behind its outward appearance. This article tries to give as a total picture as possible of its dynamical natures, based on the Riemann-Hilbert approach recently…
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.
We find and study a two-parameter family of coupled Painlev\'e II systems in dimension four with affine Weyl group symmetry of several types. Moreover, we find a three-parameter family of polynomial Hamiltonian systems in two variables…
We find a one-parameter family of polynomial Hamiltonian system in two variables with $W({A}^{(1)}_1)$-symmetry. We also show that this system can be obtained by the compatibility conditions for the linear differential equations in three…
In this paper, we study the isomonodromy systems associated with the Garnier systems of type 9/2 and type 5/2+3/2. We show that the both of isomonodromy systems admit the singularity reduction (restriction to a movable pole), and the…
We construct solutions of analogues of the nonstationary Schr\"odinger equation corresponding to the polynomial isomonodromic Hamiltonian Garnier system with two degrees of freedom. This solutions are obtained from solutions of systems of…
The diagonal spin-spin correlations $ \langle \sigma_{0,0}\sigma_{N,N} \rangle $ of the Ising model on a triangular lattice with general couplings in the three directions are evaluated in terms of a solution to a three-variable extension of…
We study several variants of q-Garnier system corresponding to various directions of discrete time evolutions. We also investigate a relation between the $q$-Garnier system and Suzuki's higher order $q$-Painlev/'e system by using a duality…
We construct Poisson structures for Ermakov systems, using the Ermakov invariant as the Hamiltonian. Two classes of Poisson structures are obtained, one of them degenerate, in which case we derive the Casimir functions. In some situations,…
Bureau proposed a classification of systems of quadratic differential equations in two variables which are free of movable critical points, which was recently revised by Guillot. We revisit the quadratic Bureau-Guillot systems with the…
In this paper we study different Hamiltonian systems with polynomial and rational Hamiltonians associated with the generic third Painlev\'e equation and present explicit birational transformations relating them.
In our previous work, a unified description as polynomial Hamiltonian systems was established for a broad class of the Schlesinger systems including the sixth Painleve equation and Garnier systems. The main purpose of this paper is to…
Recently, a quantum version of Painleve equations from the point of view of their symmetries was proposed by H. Nagoya. These quantum Painleve equations can be written as Hamiltonian systems with a (noncommutative) polynomial Hamiltonian.…
In this paper, we study the second member of the second Painlev\'e hierarchy $P_{II}^{(2)}$. We show that the birational transformations take this equation to the polynomial Hamiltonian system in dimension four, and this Hamiltonian system…