Integrability properties of Kahan's method
Elena Celledoni, Robert I McLachlan, David I McLaren, Brynjulf Owren, and G R W Quispel
TL;DR
This paper demonstrates that Kahan's discretization method can preserve integrability in various quadratic vector fields, including notable systems like Painlevé I, and explores the role of Manin transformations in this context.
Contribution
The paper introduces new integrable examples of quadratic vector fields preserved by Kahan's method and analyzes the connection with Manin transformations.
Findings
Kahan's method preserves integrability in several new quadratic systems.
Examples include generalized Suslov, Ishii, Nambu, Riccati, and Painlevé I systems.
Manin transformations are linked to Kahan discretizations of certain vector fields.
Abstract
We present several novel examples of integrable quadratic vector fields for which Kahan's discretization method preserves integrability. Our examples include generalized Suslov and Ishii systems, Nambu systems, Riccati systems, and the first Painlev\'e equation. We also discuss how Manin transformations arise in Kahan discretizations of certain vector fields.
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