On equivalence of singularities of second order linear differential equations by point transformations
type: Item
instance of: scholarly article
title: On equivalence of singularities of second order linear differential equations by point transformations
zbMATH Open document ID: 1486.34170
author: Martin Klimeš
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2021-11-29T00:00:00Z
review text: The author considers a second-order linear ordinary differential equation \[ \dfrac{d^2y}{dx^2}+a_1(x)\dfrac{dy}{dx}+a_0(x)y=0, \quad (x,y)\in\Omega\times\mathbb C, \tag{1}\] on a complex domain \(\Omega\subset\mathbb C\), the coefficients \(a_0(x)\) and \(a_1(x)\) are meromorphic on \(\Omega\), together with its companion system \[\dfrac{dv}{dx}=\begin{pmatrix}0&1\\ -a_0(x)&-a_1(x)\end{pmatrix}v, \quad v=\left(y,\dfrac{dy}{dx}\right) ^t, \quad (x,v)\in\Omega\times\mathbb C^2.\tag{2}\] Given a linear point transformation of both the independent variable \(x\) and of the unknown function \(y\) \[\widetilde{x}=\phi(x),\quad\widetilde{y}=t(x)y \] with \(\phi:\Omega\rightarrow\Omega\) an analytic diffeomorphism and \(t(x)\) either a non-vanishing analytic, or a non-zero meromorphic, function, the corresponding equations for \(y\) and \(\widetilde{y}\) are called analytically (resp. meromorphically) equivalent. With respect to the transformation (2), the question of the local classification of the singularities of Eq. (1) is asked. The author proves in particular that, under a convenient non-degeneracy condition, the equations for \(y\) and \(\widetilde{y}\) are analytically equivalent if and only if the associated companion systems are analytically equivalent as systems. He also provides the Lie algebras of the analytic linear infinitesimal symmetries of the singularities.
reviewed by: Pascal Remy
Mathematics Subject Classification ID: 34M30
Mathematics Subject Classification ID: 34M03
Mathematics Subject Classification ID: 34M40
zbMATH DE Number: 7435284
zbMATH Keywords: linear ordinary differential equations
zbMATH Keywords: local analytic classification
zbMATH Keywords: normal forms
zbMATH Keywords: regular singularity
zbMATH Keywords: irregular singularity
zbMATH Keywords: Stokes phenomenon
zbMATH Keywords: analytic Lie symmetries
MaRDI profile type: Publication
full work available at URL: afst.1684
OpenAlex ID: W3209850683
Wikidata QID: Q115159084
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DOI: 10.5802/AFST.1684
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description: scientific article; zbMATH DE number 7435284