What is integrability of discrete variational systems?
What is integrability of discrete variational systems?
复制标题
什么是离散变分系统的可积性？
DOI：
--
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Y. Suris
中科院分区：
文献类型：
--
作者：
R. Boll;M. Petrera;Y. Suris
我们提出了一个复数拉格朗日问题的概念，它应该被理解为变分系统的多维一致性的类似物。这是Lobb和Nijhoff在2009年发起的离散可积拉格朗日系统研究方向的一个发展，然而，在多谐函数理论中，在统计力学的z不变模型及其准经典极限中，以及在可追溯到诺特的变分对称性理论中，它有着更遥远的根源。d维多元拉格朗日问题可以描述如下：给定m维空间（称为多时间，m>d）上的d形式，其系数依赖于m个自变量（称为场）的一个受欢迎的函数x，找到这些场x，这些场x为任意d维流形Σ在多时间中的作用泛函提供临界点。我们导出了d=2的离散复数拉格朗日问题的多时间欧拉-拉格朗日方程的主要组成部分，即角方程，并讨论了角方程组的相合性的概念。我们分析了一类特殊的三点二型方程的角方程组，它对应于ABS表的可积四方程。这使我们能够通过表明相应的两种形式不仅在（非变分）四方程的解上是封闭的，而且在相应的角方程的一般解上也是封闭的，从而缩小Lobb和Nijhoff工作的概念差距。我们还发现了一个不是来自多维一致四方程系统的复数拉格朗日系统的例子。
We propose a notion of a pluri-Lagrangian problem, which should be understood as an analogue of multi-dimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems initiated in 2009 by Lobb and Nijhoff, however, having its more remote roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics and their quasiclassical limit, as well as in the theory of variational symmetries going back to Noether. A d-dimensional pluri-Lagrangian problem can be described as follows: given a d-form on an m-dimensional space (called multi-time, m>d), whose coefficients depend on a sought-after function x of m independent variables (called field), find those fields x which deliver critical points to the action functionals for any d-dimensional manifold Σ in the multi-time. We derive the main building blocks of the multi-time Euler–Lagrange equations for a discrete pluri-Lagrangian problem with d=2, the so-called corner equations, and discuss the notion of consistency of the system of corner equations. We analyse the system of corner equations for a special class of three-point two-forms, corresponding to integrable quad-equations of the ABS list. This allows us to close a conceptual gap of the work by Lobb and Nijhoff by showing that the corresponding two-forms are closed not only on solutions of (non-variational) quad-equations, but also on general solutions of the corresponding corner equations. We also find an example of a pluri-Lagrangian system not coming from a multi-dimensionally consistent system of quad-equations.