论文标题
AKNS层次结构和古典R-Matrix的多形描述
Multiform description of the AKNS hierarchy and classical r-matrix
论文作者
论文摘要
近年来，在哈密顿的形式主义中，某些可综合的古典田地理论的新属性已被发现，并通过协方差汉密尔顿领域理论的思想得以进行重新制定：从这个意义上讲，其古典$ r $ $ -matrix结构的协变性是揭开的。在这里，我们解决了将这些结果扩展到整个层次结构的公开问题。我们选择Ablowitz-Kaup-Newell-Segur（AKNS）层次结构。为此，我们首次介绍了整个AKNS层次结构的Lagrangian多形式。我们使用它来明确构建我们先前引入的必要对象：符号多形，多时间泊松支架和汉密尔顿多形式。配备了这些功能，我们证明了以下结果：$（i）$ lax形式包含整个层次结构的LAX矩阵序列，具有合理的经典$ r $ r $ -matrix结构； $（ii）$ AKNS层次结构的零曲率方程是与我们的汉密尔顿多形和多时间泊松支架相关的多形汉密尔顿方程； $（iii）$ hamiltonian Multiform提供了一种表征层次结构的无限保护定律集，让人联想到熟悉的标准$ \ {i，h \} = 0 $ for第一个积分$ i $。
In recent years, new properties of space-time duality in the Hamiltonian formalism of certain integrable classical field theories have been discovered and have led to their reformulation using ideas from covariant Hamiltonian field theory: in this sense, the covariant nature of their classical $r$-matrix structure was unraveled. Here, we solve the open question of extending these results to a whole hierarchy. We choose the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. To do so, we introduce for the first time a Lagrangian multiform for the entire AKNS hierarchy. We use it to construct explicitly the necessary objects introduced previously by us: a symplectic multiform, a multi-time Poisson bracket and a Hamiltonian multiform. Equipped with these, we prove the following results: $(i)$ the Lax form containing the whole sequence of Lax matrices of the hierarchy possesses the rational classical $r$-matrix structure; $(ii)$ The zero curvature equations of the AKNS hierarchy are multiform Hamilton equations associated to our Hamiltonian multiform and multi-time Poisson bracket; $(iii)$ The Hamiltonian multiform provides a way to characterise the infinite set of conservation laws of the hierarchy reminiscent of the familiar criterion $\{I,H\}=0$ for a first integral $I$.
