经典 Yang–Baxter 方程、Lagrange 多重形式与超局部可积层级
Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies
AI总结：
本文利用依赖经典 r-矩阵的生成性 Lagrange 多重形式，将 CYBE 纳入变分框架，并据此构造和耦合一大类超局部可积层级。
摘要：
我们首次将经典 Yang–Baxter 方程（CYBE）置于变分背景中，方法是把它与 Lagrange 多重形式理论联系起来；该理论是一个旨在以变分方式刻画可积性的框架。这在 Lagrange 多重形式与 CYBE——可积系统最基本的概念之一——之间建立了重要联系。这一目标通过引入一个生成性 Lagrange 多重形式来实现，该多重形式依赖于一个带谱参数的斜对称经典 r-矩阵。多重形式 Euler–Lagrange 方程产生一个生成性 Lax 方程，而后者又给出一个生成性零曲率方程。CYBE 在三个层面发挥作用：1）它保证生成性 Lax 方程各流之间的交换性；2）它保证生成性零曲率方程成立；3）它蕴含生成性 Lagrange 多重形式的闭包关系。一个可积层级通过固定若干数据来确定：有限集合 S∈CP¹、一个 Lie 代数 g、一个取值于 g 且极点位于 S 中的有理函数，以及一个 r-矩阵。我们通过给出若干属于有理性或三角类的已知及新例子，展示了该框架如何能够生成一大类超局部可积层级。这些例子包括 Ablowitz–Kaup–Newell–Segur 层级、sine-Gordon（sG）层级，以及与 Zakharov–Mikhailov 型模型相关的各种层级；后者把 Faddeev–Reshetikhin（FR）模型以及最近提出的变形 sigma/Gross–Neveu 模型作为特例包含在内。我们还通过展示如何把多个可积层级耦合起来以构造可积场论及其层级的新例子，说明了方法的多功能性。文中给出两个例子：非线性 Schrödinger 系统与 FR 模型的耦合，以及 sG 与各向异性 FR 模型的耦合。
英文摘要：
We cast the classical Yang-Baxter equation (CYBE) in a variational context for the first time, by relating it to the theory of Lagrangian multiforms, a framework designed to capture integrability in a variational fashion. This provides a significant connection between Lagrangian multiforms and the CYBE, one of the most fundamental concepts of integrable systems. This is achieved by introducing a generating Lagrangian multiform which depends on a skew-symmetric classical $r$-matrix with spectral parameters. The multiform Euler-Lagrange equations produce a generating Lax equation which yields a generating zero curvature equation. The CYBE plays a role at three levels: 1) It ensures the commutativity of the flows of the generating Lax equation; 2) It ensures that the generating zero curvature equation holds; 3) It implies the closure relation for the generating Lagrangian multiform. The specification of an integrable hierarchy is achieved by fixing certain data: a finite set $S\in CP^1$, a Lie algebra $\mathfrak{g}$, a $\mathfrak{g}$-valued rational function with poles in $S$ and an $r$-matrix. We show how our framework is able to generate a large class of ultralocal integrable hierarchies by providing several known and new examples pertaining to the rational or trigonometric class. These include the Ablowitz-Kaup-Newell-Segur hierarchy, the sine-Gordon (sG) hierarchy and various hierachies related to Zakharov-Mikhailov type models which contain the Faddeev-Reshetikhin (FR) model and recently introduced deformed sigma/Gross-Neveu models as particular cases. The versatility of our method is illustrated by showing how to couple integrable hierarchies together to create new examples of integrable field theories and their hierarchies. We provide two examples: the coupling of the nonlinear Schrödinger system to the FR model and the coupling of sG with the anisotropic FR model.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
