How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6
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Schmalian, M., Suris, Y. B., & Tumarkin, Y. (2021). How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6. https://doi.org/10.1007/s11040-021-09413-2
Abstract: We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order O(ϵ2) in the coefficients of the discretization, where ϵ is the stepsize.
Article, Birational maps, Discrete integrable systems, Elliptic pencil, Rational elliptic surface, Integrable discretization
deutsche forschungsgemeinschaft (TRR 109)
External DOI: https://doi.org/10.1007/s11040-021-09413-2
This record's URL: https://www.repository.cam.ac.uk/handle/1810/331457