How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6
Publication Date
2021-12Journal Title
Mathematical Physics, Analysis and Geometry
ISSN
1385-0172
Publisher
Springer Science and Business Media LLC
Language
en
Type
Article
This Version
VoR
Metadata
Show full item recordCitation
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. Mathematical Physics, Analysis and Geometry https://doi.org/10.1007/s11040-021-09413-2
Abstract
<jats:title>Abstract</jats:title><jats:p>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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(\epsilon ^2)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>O</mml:mi>
<mml:mo>(</mml:mo>
<mml:msup>
<mml:mi>ϵ</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> in the coefficients of the discretization, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\epsilon $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>ϵ</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> is the stepsize.</jats:p>
Keywords
Article, Birational maps, Discrete integrable systems, Elliptic pencil, Rational elliptic surface, Integrable discretization
Sponsorship
deutsche forschungsgemeinschaft (TRR 109)
Identifiers
s11040-021-09413-2, 9413
External DOI: https://doi.org/10.1007/s11040-021-09413-2
This record's URL: https://www.repository.cam.ac.uk/handle/1810/331500
Rights
Licence:
http://creativecommons.org/licenses/by/4.0/
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