Lie-theoretical generalization and discretization of the Pinney equation
type: Item
instance of: scholarly article
title: Lie-theoretical generalization and discretization of the Pinney equation
zbMATH Open document ID: 0910.34013
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-04-19T00:00:00Z
review text: The authors discuss two types of Pinney equations which admit nonlinear superposition principles. An introduction is given to superposition principles based on classical results of Lie for the structure of corresponding differential equations. Then the Pinney equation \[ \ddot u+w(t) \cdot u={\alpha \over u^3} \] is written as a first-order system, and the corresponding Lie algebra \(Sl(2)\) of generators is transformed to a normal form depending on \(\alpha\) and the fields \(\mathbb{R}\) or \(\mathbb{C}\). On this background suitable transformations are constructed, and formulas for the general solution using two solutions and the superposition principle. Connections to nonlinear Schrödinger equations and discretization formulas are given.
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 34L40
zbMATH DE Number: 1115533
zbMATH Keywords: Pinney equation
zbMATH Keywords: differential equations
zbMATH Keywords: superposition principle
zbMATH Keywords: discretization
author: Colin Rogers
author: W. K. Schief
author: Pavel Winternitz
reviewed by: Günter Czichowski
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5674
OpenAlex ID: W2074500712
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DOI: 10.1006/JMAA.1997.5674
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description: scientific article; zbMATH DE number 1115533