How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6
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Published:2021-11-28
Issue:4
Volume:24
Page:
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ISSN:1385-0172
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Container-title:Mathematical Physics, Analysis and Geometry
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language:en
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Short-container-title:Math Phys Anal Geom
Author:
Schmalian Misha, Suris Yuri B.ORCID, Tumarkin Yuriy
Abstract
AbstractWe 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 $$O(\epsilon ^2)$$
O
(
ϵ
2
)
in the coefficients of the discretization, where $$\epsilon $$
ϵ
is the stepsize.
Funder
deutsche forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
Geometry and Topology,Mathematical Physics
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