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Elliptic discrete integrable systems and bi-elliptic addition formulae

椭圆离散可积系统和双椭圆加法公式

基本信息

批准号：
EP/W007290/1
负责人：
Frank Nijhoff
金额：
$ 8.39万
依托单位：
University of Leeds
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW007290%2F1
关键词：
Elliptic discrete integrable systems bi

项目摘要

In mathematics, when classifying mathematical objects, one often finds in the class a 'master' object with a number of free parameters from which the other objects in the class can be obtained by taking certain limiting values for those parameters. In 2003 a classification was given of quadrilateral lattice equations, which are partial difference equations for a function depending on discrete variables (e.g. variables that label the sites of a space-time lattice) where the equation only involves the values of the function at the four vertices of an elementary quadrilateral. This is the simplest, (and so far the only) situation of a full classification of integrable partial difference equations of a single function subject to some additional conditions. The equations are nonlinear, but only in an affine-linear way (i.e. each vertex value appears only once in the equation) but the main property is that the equations considered obey the property of 'multidimensional consistency' (abbreviated by MDC), i.e. one can extend the number of dimensions of the lattice from 2 to any dimension, and impose the equation in each pair of directions on that lattice and the equation will still have large classes of nontrivial solutions. This remarkable property is considered to be an indication of the 'integrability' of the equation in question, which means that the MDC guarantees that you can actually construct exact solutions of the equation through a procedure called B"acklund transformation, by which one can obtain a new solution from an already given (possibly trivial) solution. In the classification of quad equations there was a master equation that appeared, discovered by V. Adler in 1998, and which is referred to as Q4. All other quad equations in the class are special parameter cases of this master equation. Because of the pivotal role of the Q4 equation, it is important to have a good insight into the structure of solutions. However, this turned out to be particularly challenging, since the natural parameters of the equation are subject to an algebraic condition which tells us that the parameters are points on an elliptic curve. These elliptic curves have been widely studied since the early 19th century, and are themselves parametrised in terms of a class of functions called 'elliptic functions' (like the circle is parametrised by trigonometric functions). The situation with regard to Q4 is, however, even more complicated, as to obtain even the simplest non-trivial solution of Q4 one needs a combination of two different types of elliptic functions associated with two essentially different elliptic curves. On a single elliptic curve there is a natural group law, that connects three intersection points on a straight line intersecting the curve, which is called an addition formula. It turns out that for solving Q4 there emerges a novel type of addition formulae that mixes the elliptic functions belonging to different elliptic curves. While the theory of elliptic functions and their addition rules is a classic subject, these new rules seem never to have been considered in the vast literature on elliptic functions and curves, so they merit a study in their own right. In the project I will endeavour to attain understanding of these novel 'bi-elliptic' addition formulae which govern the dynamics of the integrable systems that are defined by the Q4 equation. The project has an even more ambitious aim based on the hypothesis that behind the whole parameter-family of Q4 equations lurks a (possibly novel) algebraic object that governs the symmetries of the Q4 equation in terms of both the movable variables on the curve as well as of the parameters that fixes the curve itself, in addition to the independent and dependent variables that describe the complex dynamics encoded in this fascinating but still mysterious equation.

在数学中，当对数学对象进行分类时，人们经常会在类中找到一个带有许多自由参数的“主”对象，通过对这些参数取某些限制值，可以从中获得类中的其他对象。在2003年，四边形格点方程被分类，这是一个依赖于离散变量（例如标记时空格点的变量）的函数的偏差分方程，其中方程只涉及基本四边形的四个顶点的函数值。这是最简单的，（到目前为止唯一）的情况下，一个完整的分类可积偏差分方程的一个单一的功能，一些额外的条件。方程是非线性的，但仅以仿射线性方式（即每个顶点值在方程中只出现一次），但主要属性是所考虑的方程服从“多维一致性”属性（缩写为MDC），即可以将晶格的维数从2扩展到任何维数，并且在每对方向上将方程施加到该格上，并且方程仍将具有大类的非平凡解。这个显着的性质被认为是所讨论方程的“可积性”的指示，这意味着MDC保证您实际上可以通过称为B 'acklund变换的过程构造方程的精确解，通过该过程可以获得新的解从已经给定的（可能是微不足道的）解。在四元方程的分类中，出现了一个主方程，由V. Adler于1998年发现，称为Q4。类中的所有其他四元方程都是这个主方程的特殊参数情况。由于Q4方程的关键作用，对解的结构有一个很好的了解是很重要的。然而，这被证明是特别具有挑战性的，因为方程的自然参数受到代数条件的影响，该条件告诉我们参数是椭圆曲线上的点。这些椭圆曲线自世纪早期以来就被广泛研究，并且它们本身被称为“椭圆函数”的一类函数参数化（就像圆被三角函数参数化一样）。然而，关于Q4的情况甚至更复杂，因为为了获得Q4的甚至最简单的非平凡解，需要与两条本质上不同的椭圆曲线相关联的两种不同类型的椭圆函数的组合。在一条椭圆曲线上有一个自然的群律，它连接与曲线相交的直线上的三个交点，称为加法公式。结果表明，为了求解Q4，出现了一种新的加法公式，该公式混合了属于不同椭圆曲线的椭圆函数。虽然椭圆函数及其加法规则的理论是一个经典的主题，但这些新规则似乎从未在关于椭圆函数和曲线的大量文献中被考虑过，因此它们本身值得研究。在该项目中，我将努力达到这些新的“双椭圆”除了管理由Q4方程定义的可积系统的动力学公式的理解。该项目有一个更雄心勃勃的目标，其基础是假设在整个参数族Q4方程背后隐藏着一个（可能是新颖的）代数对象，它根据曲线上的可移动变量以及固定曲线本身的参数来控制Q4方程的对称性，除了描述这个迷人但仍然神秘的方程中编码的复杂动力学的自变量和因变量之外。