论文标题
汉密尔顿的多形描述可集成的层次结构
Hamiltonian multiform description of an integrable hierarchy
论文作者
论文摘要
由拉格朗日多形式的概念的概念（提供了可集成性的拉格朗日表述），以及作者对协变量的汉密尔顿形式形式对综合场理论的作用的结果，我们提出了汉密尔顿多形式的概念。他们提供了拉格朗日多形式的汉密尔顿对应物，并将其封装在单个对象中，一个任意数量的集成层次结构中的流量。对于给定的层次结构，以拉格朗日的多形式为起点，我们基于协方差汉密尔顿现场理论的技术的概括，为汉密尔顿多形式提供了系统的结构。这还产生了另外两个重要对象：一个符号多形和相关的多时间泊松支架。如果我们将注意力限制在层次结构中的单个流量上，它们将减少到多核形式和相关的协变量泊松支架。我们的框架提供了定义和得出层次结构的保护法的另一种方法。我们在三个例子上说明了结果：潜在的Korteweg-de Vries层次结构，正弦式层次结构（在光锥坐标中）和Ablowitz-Kaup-Newell-Newell-Segur层次结构。
Motivated by the notion of Lagrangian multiforms, which provide a Lagrangian formulation of integrability, and by results of the authors on the role of covariant Hamiltonian formalism for integrable field theories, we propose the notion of Hamiltonian multiforms for integrable $1+1$-dimensional field theories. They provide the Hamiltonian counterpart of Lagrangian multiforms and encapsulate in a single object an arbitrary number of flows within an integrable hierarchy. For a given hierarchy, taking a Lagrangian multiform as starting point, we provide a systematic construction of a Hamiltonian multiform based on a generalisation of techniques of covariant Hamiltonian field theory. This also produces two other important objects: a symplectic multiform and the related multi-time Poisson bracket. They reduce to a multisymplectic form and the related covariant Poisson bracket if we restrict our attention to a single flow in the hierarchy. Our framework offers an alternative approach to define and derive conservation laws for a hierarchy. We illustrate our results on three examples: the potential Korteweg-de Vries hierarchy, the sine-Gordon hierarchy (in light cone coordinates) and the Ablowitz-Kaup-Newell-Segur hierarchy.
