Integrability properties of Kahan's method
type: Item
instance of: scholarly article
zbMATH Open document ID: 1336.37045
DOI: 10.1088/1751-8113/47/36/365202
author: Elena Celledoni
author: David I. McLaren
author: Brynjulf Owren
author: Robert I. McLachlan
author: G. R. W. Quispel
published in: Journal of Physics A: Mathematical and Theoretical
publication date: 2014-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1405.3740
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 65D30
zbMATH DE Number: 6353526
zbMATH Keywords: discrete integrable systems
zbMATH Keywords: geometric numerical integration
zbMATH Keywords: Hamiltonian systems
MaRDI profile type: Publication
OpenAlex ID: W3106439850
arXiv ID: 1405.3740
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title: Integrability properties of Kahan's method
review text: In the paper under review, several novel examples of integrable quadratic vector fields for which Kahan's discretization method [\textit{W. Kahan} and \textit{R.-C. Li}, J. Comput. Phys. 134, No. 2, 316--331, Art. No. CP975710 (1997; Zbl 0883.65061)] preserves integrability are presented. Some examples include generalized Suslov and Ishii [\textit{M. Ishii}, Prog. Theor. Phys. 84, No. 3, 386--391 (1990; Zbl 1100.34535)] systems, Nambu systems, Riccati systems, and the first Painlevé equation. The paper also discusses how Manin transformations arise in Kahan discretizations of certain vector fields.NEWLINENEWLINEThe paper is organized as follows. Section 2 is on the application of Kahan's method to autonomous quadratic ODEs. Subsection 2.1 treats the discretization of quadratic Hamiltonian ODEs in \(\mathbb{R}^2\), and its connection to Manin transformations. Subsection 2.2 discusses the discretization of quadratic ODEs in \(\mathbb{R}^3\) possessing two quadratic integrals. Subsection 2.3 is about the discretization of generalized Suslov systems in \(\mathbb{R}^n\), including the integrable cases \(n = 2\) and \(n = 3\). Section 3 is on the application of Kahan's method to nonautonomous quadratic ODEs. Subsection 3.1 treats the discretization of certain Riccati equations. Last but not least, Section 3.2 covers the (non-autonomous) first Painlevé equation, whose Kahan discretization is also shown to be integrable. In this striking example the integrals are not rational functions and the mapping is not integrable in terms of elliptic or hyperelliptic functions.
reviewed by: Vladislav Nikolaevich Dumachev
description: scientific article; zbMATH DE number 6353526