Geometric properties of the Riemann surfaces associated with the Noumi-Yamada systems with a large parameter
type: Item
instance of: scholarly article
title: Geometric properties of the Riemann surfaces associated with the Noumi-Yamada systems with a large parameter
zbMATH Open document ID: 1242.34154
author: Takashi Aoki
author: N. Honda
published in: Journal of the Mathematical Society of Japan
publication date: 2011-12-28T00:00:00Z
review text: The Noumi-Yamada systems \((NY)_l\), \(l=2,3,4,\dots\), are systems of \(l+1\) first order ODEs for \(l+1\) unknowns which enjoy the affine Weyl group symmetry of type \(A_l^{(1)}\). These systems can be understood as the generalizations of the classical Painlevé equations PII, PIV and PV. Using appropriate scaling, it is possible to introduce the large parameter \(\eta\) and to study particular \(0\)-parameter solutions to \((NY)_l\) using the so-called exact WKB analysis. The above mentioned \(0\)-parameter formal series in \(\eta^{-1}\) is completely specified by the leading order term satisfying a system of algebraic equations. Earlier, the authors proved the existence of a \(0\)-parameter solution in the case of even \(l\). In the present paper, they treat the cases of even and odd \(l\) in a unified manner proving that the relevant algebraic system determines a variety which is isomorphic to the variety determined by a hyperbolic system with an appropriate normalization. In this way, the authors find the number of the possible leading order terms (i.e.\ the number of the branches of a multi-valued holomorphic function satisfying the algebraic system) outside the set of the turning points of the first kind (zeros of the Jacobian of the algebraic system) as well as the number of such turning points.
reviewed by: Andrei A. Kapaev
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 34M60
Mathematics Subject Classification ID: 34E20
Mathematics Subject Classification ID: 34M25
zbMATH DE Number: 5992428
zbMATH Keywords: Noumi-Yamada systems
zbMATH Keywords: Painlevé hierarchy
zbMATH Keywords: formal solutions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/06341085
OpenAlex ID: W2107503505
cites work: Q5455322
cites work: Q5401195
cites work: Dual class of a subvariety
cites work: Toward the exact WKB analysis for higher-order Painlevé equations -- the case of Noumi-Yamada systems
cites work: Q4824246
DOI: 10.2969/JMSJ/06341085
description: scientific article; zbMATH DE number 5992428