On the Darboux transformation of the second order differential operator of Fuchsian type on the Riemann sphere
type: Item
instance of: scholarly article
title: On the Darboux transformation of the second order differential operator of Fuchsian type on the Riemann sphere
zbMATH Open document ID: 0711.34008
author: Mayumi Ohmiya
published in: Osaka Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The main purpose of the present paper is to clarify some analytical properties of the Darboux transformation of the second order ordinary differential operator \(L(P)=D^ 2-P(x)\), \(D=d/dx\), of Fuchsian type on the Riemann sphere \({\mathbb{P}}_ 1\). For brevity it is assumed that P(x) is of the form \((*)\quad P(x)=\sum^{n}_{j=1}\alpha_ j(x-a_ j)^{- 2}.\) The contents of the paper are as follows. In section 1 preliminary considerations about the Darboux transformation are made. Thus, if \({\mathbb{F}}\) is the set of all L(P) such that P(x) is of the form (*) and the ratio \(y_ 2(x)/y_ 1(x)\) of the fundamental system of solutions of \(L(P)y=y''-P(x)y=0\) is a rational function, then it is shown that the Darboux transformation L(P;\(\zeta\),Y) of L(P) is of Fuchsian type on \({\mathbb{P}}_ 1\) for any \(\zeta \in {\mathbb{P}}_ 1\) if and only if L(P)\(\in {\mathbb{F}}\). In section 2, the Darboux transformation is investigated in connection with the isomonodromic deformation. In section 3, the relation between Darboux transformation and the Lenard relation is discussed. Section 4 is devoted to the construction of the space of rational function solutions of the higher order KdV equation \(\partial u(x,\xi)/\partial \xi =X_ n(u(x;\xi)),\) where \(\xi\) is a complex variable and \(X_ n\) is given by the Lenard relation.
reviewed by: Pavel Talpalaru
Mathematics Subject Classification ID: 34A25
zbMATH DE Number: 4169077
zbMATH Keywords: Darboux transformation
zbMATH Keywords: second order ordinary differential operator
zbMATH Keywords: isomonodromic deformation
zbMATH Keywords: Lenard relation
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4169077