How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6

IF 0.9 3区数学 Q3 MATHEMATICS, APPLIED

Mathematical Physics, Analysis and Geometry Pub Date : 2021-11-28 DOI:10.1007/s11040-021-09413-2

Misha Schmalian, Yuri B. Suris, Yuriy Tumarkin

引用次数: 3

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.

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如何修复不可积的Kahan离散化。2具有6次不变曲线的平面系统

我们在平面上找到了一种新的单参数可积二次克雷莫纳映射族，它保留了一束6次1属的曲线。它们被证明是一组新的二次向量场的kahan型离散化，该二次向量场具有6次多项式积分，其水平曲线也是1属。这些向量场是由Hitchin, Manton和Murray引入的二十面体对称磁单极子的约化纳姆系统的非齐次推广。这些新型非齐次系统的直接Kahan离散化是不可积的。然而，这个缺点是通过在离散化系数中引入阶\(O(\epsilon ^2)\)的调整来修复的，其中\(\epsilon \)是步长。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.