1. How one can repair non-integrable Kahan discretizations. II. A planar system with invariant curves of degree 6
- Author
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Misha Schmalian, Yuriy Tumarkin, Yuri B. Suris, Suris, Yuri B [0000-0001-9378-0314], Apollo - University of Cambridge Repository, and Suris, Yuri B. [0000-0001-9378-0314]
- Subjects
Pure mathematics ,Polynomial ,4902 Mathematical Physics ,Integrable system ,Discretization ,FOS: Physical sciences ,Integrable discretization ,Article ,Elliptic pencil ,Mathematics - Algebraic Geometry ,Quadratic equation ,Birational maps ,Genus (mathematics) ,FOS: Mathematics ,Rational elliptic surface ,ddc:510 ,Invariant (mathematics) ,Algebraic Geometry (math.AG) ,Mathematical Physics ,Mathematics ,Degree (graph theory) ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,4904 Pure Mathematics ,510 Mathematik ,Mathematical Physics (math-ph) ,49 Mathematical Sciences ,Vector field ,Geometry and Topology ,Discrete integrable systems ,Exactly Solvable and Integrable Systems (nlin.SI) - 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 $O(\epsilon^2)$ in the coefficients of the discretization, where $\epsilon$ is the stepsize., Comment: 15 pages, 4 figures
- Published
- 2021
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