Hamiltonian multiform description of an integrable hierarchy

V. Caudrelier, Matteo Stoppato
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引用次数: 6

Abstract

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.
可积层次的哈密顿多形式描述
基于可积性的拉格朗日多重形式的概念,以及作者关于协变哈密顿形式在可积场论中的作用的结果,我们提出了可积1+1维场论的哈密顿多重形式的概念。它们提供了拉格朗日多形式的哈密顿形式,并将可积层次结构中的任意数量的流封装在单个对象中。对于给定的层次,以拉格朗日多重形式为出发点,基于协变哈密顿场论技术的推广,给出了哈密顿多重形式的系统构造。这也产生了另外两个重要的对象:辛多重形式和相关的多时间泊松括号。如果我们将注意力限制在层次结构中的单个流上,它们将简化为多辛形式和相关的协变泊松括号。我们的框架提供了另一种方法来定义和推导层次结构的守恒定律。我们用三个例子来说明我们的结果:潜在的Korteweg-de Vries层次结构,sin - gordon层次结构(光锥坐标)和ablowitz - kap - newwell - segur层次结构。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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