标题： 离散拉格朗日多形式用于ABS方程 I：四边形方程 显示英文标题

标题： Discrete Lagrangian Multiforms for ABS Equations I: Quad Equations

摘要： 离散拉格朗日多重形式理论是从变分角度研究在多维一致性意义上可积的格子方程的一种方法。 ABS分类方程的拉格朗日多重形式是该理论的起点，但通常在此背景下考虑的拉格朗日多重形式所产生的方程比ABS方程稍弱。 在本工作中，我们提出了替代的拉格朗日多重形式，其欧拉-拉格朗日方程与ABS方程等价。 此外，现有文献中对ABS拉格朗日多重形式的处理未能承认其定义中的复函数具有分支切割。 分支的选择会影响ABS方程的加法三腿形式的存在性以及拉格朗日多重形式的闭合性质。 我们给出了这两个性质的反例，但通过在作用量求和中包含与分支选择相关的整数值场，我们恢复了这些性质。 显示英文摘要

摘要： Discrete Lagrangian multiform theory is a variational perspective on lattice equations that are integrable in the sense of multidimensional consistency. The Lagrangian multiforms for the equations of the ABS classification formed the start of this theory, but the Lagrangian multiforms that are usually considered in this context produce equations that are slightly weaker than the ABS equations. In this work, we present alternative Lagrangian multiforms that have Euler-Lagrange equations equivalent to the ABS equations. In addition, the treatment of the ABS Lagrangian multiforms in the existing literature fails to acknowledge that the complex functions in their definitions have branch cuts. The choice of branch affects both the existence of an additive three-leg form for the ABS equations and the closure property of the Lagrangian multiforms. We give counterexamples for both these properties, but we recover them by including integer-valued fields, related to the branch choices, in the action sums.