Spaces of initial conditions of Garnier system and its degenerate systems in two variables
type: Item
instance of: scholarly article
title: Spaces of initial conditions of Garnier system and its degenerate systems in two variables
zbMATH Open document ID: 1110.37046
author: Masaki Suzuki
published in: Journal of the Mathematical Society of Japan
publication date: 2007-01-30T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jmsj/1179759538
review text: Here, the spaces of initial conditions of Garnier system \(\mathcal H_J\), \(J=11111\) and its degenerate systems \(\mathcal H_J\), \(J=1112\), 113, 122, 14, 23, 5, in two variables [see \textit{H. Kimura}, Ann. Mat. Pura Appl. IV. Ser. 155, 25--74 (1989; Zbl 0693.34043)] are constructed. These systems are completely integrable Hamiltonian systems of degree 2 of the form \(dq_k=\sum_{i=1,2}\frac{\partial H_{Ji}}{\partial p_k}ds_i\), \(dp_k=-\sum_{i=1,2}\frac{\partial H_{Ji}}{\partial q_k}ds_i\), \(k=1,2\). The Hamiltonians for all \(J\) are certain polynomials of \(q_1\), \(q_2\), \(p_1\), \(p_2\) whose coefficients are rational functions of \(s=(s_1,s_2)\) holomorphic in a domain \(B_{J}\subset \mathbb C^2\). These systems are obtained as monodromy preserving deformations with regular or irregular singular points and apparent singular points. The author describes these systems as symplectic manifolds and expresses them as polynomial Hamiltonian systems on all affine charts.
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 33E17
Mathematics Subject Classification ID: 34M55
zbMATH DE Number: 5119782
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: Garnier systems in two variables
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
OpenAlex ID: W2059085330
cites work: The degeneration of the two dimensional Garnier system and the polynomial Hamiltonian structure
cites work: Q4033104
cites work: Initial value spaces of degenerate Garnier systems
cites work: Discrete decomposability of the restriction of \(A_q(\lambda)\) with respect to reductive subgroups. II: Micro-local analysis and asymptotic \(K\)-support
cites work: On some Hamiltonian structures of Painlevé systems. II
cites work: Q3862133
cites work: Q4366336
cites work: AN AUGMENTATION OF THE PHASE SPACE OF THE SYSTEM OF TYPE A4(1)
DOI: 10.2969/JMSJ/1179759538
Recommended article: The degeneration of the two dimensional Garnier system and the polynomial Hamiltonian structure
Recommended article: Q3437305
Recommended article: On the Garnier System of Half-Integer Type in Two Variables
Recommended article: Q3545618
Recommended article: Pole loci of solutions of a degenerate Garnier system
Recommended article: Special solutions and linear monodromy for the two-dimensional degenerate Garnier system G(1112)
Recommended article: Q3796046
Recommended article: Q5368120
Recommended article: Q3693904
Recommended article: Q4278505
description: scientific article; zbMATH DE number 5119782