On the summability of formal solutions in Liouville-Green theory
type: Item
instance of: scholarly article
title: On the summability of formal solutions in Liouville-Green theory
zbMATH Open document ID: 1017.34089
DOI: 10.1023/A:1016382404980
published in: Journal of Dynamical and Control Systems
publication date: 2002-12-15T00:00:00Z
review text: The authors consider the second-order linear differential equation \[ \varepsilon ^2y''=(1+\varepsilon ^2\psi (x,\varepsilon))y\tag{*} \] where \(\varepsilon\) is a small complex parameter and \(\psi\) is complex-analytic w.r.t. \((x,\varepsilon)\) and even w.r.t. \(\varepsilon\). It is known [see \textit{F. W. J. Olver}, Asymptotics and special functions. New York-London: Academic Press (1974; Zbl 0303.41035), Chapter 10], that equation \((*)\) has two formal solutions \(y^{\pm x/\varepsilon }h^{\pm}(x,\varepsilon)\), where \(h^{\pm}(x,\varepsilon)\) is a formal power series of \(\varepsilon\) whose coefficients are functions analytic in \(x\). It has been shown in \textit{T. M. Dunster, D. A. Lutz} and \textit{R. Schäfke} [Proc. R. Soc. Lond., Ser. A 440, 37-54 (1993; Zbl 0798.34083)] that one or both of these solutions are \(1\)-summable in certain directions if the \(x\)-domain of \(\psi\) satisfies certain conditions. The authors of the present paper give necessary (and sufficient) conditions for the \(1\)-summability of one or both of the above formal solutions in terms of \(\psi\). The method of proof relies on the construction of a differential equation of the kind of \((*)\), with \(\psi\) of the form \(\varepsilon ^2\varphi (x)\), which exhibits a prescribed Stokes phenomenon w.r.t. \(\varepsilon\).
Mathematics Subject Classification ID: 34M40
Mathematics Subject Classification ID: 34M60
zbMATH DE Number: 1845282
zbMATH Keywords: Liouville-Green theory
zbMATH Keywords: formal series
zbMATH Keywords: summability
zbMATH Keywords: Stokes phenomenon
zbMATH Keywords: singular perturbation
zbMATH Keywords: differential equation
author: Sigrun Bodine
author: Reinhard Schäfke
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
full work available at URL: a:1016382404980
OpenAlex ID: W1536531842
Recommended article: Q4953935
Recommended article: On the summability of formal solutions of linear partial differential equations
Recommended article: Q3451878
Recommended article: On a generalization of the Liouville formula
Recommended article: On the summability of the formal solutions for some PDEs with irregular singularity
Recommended article: Integrability of Liouville theory: proof of the DOZZ formula
Recommended article: Asymptotic formulae of Liouville--Green type for higher even-order equations
Recommended article: On the summability of formal solutions for doubly singular nonlinear partial differential equations
Recommended article: Q5034900
Recommended article: Summability of formal solutions for some generalized moment partial differential equations
description: scientific article; zbMATH DE number 1845282