论文标题
重新审视静止流量
Stationary Flows Revisited
论文作者
论文摘要
在本文中，我们重新审视了耦合KDV类的宽松层次结构的固定流程。我们在标准KDV案例中解释了主要思想，然后以2和3汉密尔顿的代表分别考虑分散水波（DWW）案例。每种哈密顿的表示都为我们提供了不同形式的固定流动。比较这些，我们构建了泊松图，该图是非典型的，引起了固定流动的双汉米尔顿表示。另一种方法是使用Miura地图，在DWW层次结构的情况下，它具有两个“修改”。这种结构为我们提供了3个相关的静置流量的序列。我们使用泊松地图来构建三个固定层次结构中每个阶层的三 - 哈米尔顿代表。汉密尔顿（Hamiltonian）的代表性之一允许多组分平方本本特征功能扩展，该扩展为$ n $ n $ n $ n $的自由度汉密尔顿人提供了第一积分。每个固定流的LAX表示来自耦合的KDV矩阵。在3个自由度的情况下，我们对LAX矩阵和哈密顿函数进行概括，这允许与理性的Calogero-Moser（CM）系统建立联系。这使CM系统与其他电势以及LAX表示相结合。我们介绍了将可集成的Hénon-Heiles系统与CM耦合的特殊情况。
In this paper we revisit the subject of stationary flows of Lax hierarchies of a coupled KdV class. We explain the main ideas in the standard KdV case and then consider the dispersive water waves (DWW) case, with respectively 2 and 3 Hamiltonian representations. Each Hamiltonian representation gives us a different form of stationary flow. Comparing these, we construct Poisson maps, which, being non-canonical, give rise to bi-Hamiltonian representations of the stationary flows. An alternative approach is to use the Miura maps, which we do in the case of the DWW hierarchy, which has two ''modifications''. This structure gives us 3 sequences of Poisson related stationary flows. We use the Poisson maps to build a tri-Hamiltonian representation of each of the three stationary hierarchies. One of the Hamiltonian representations allows a multi-component squared eigenfunction expansion, which gives $N$ degrees of freedom Hamiltonians, with first integrals. A Lax representation for each of the stationary flows is derived from the coupled KdV matrices. In the case of 3 degrees of freedom, we give a generalisation of our Lax matrices and Hamiltonian functions, which allows a connection with the rational Calogero-Moser (CM) system. This gives a coupling of the CM system with other potentials, along with a Lax representation. We present the particular case of coupling one of the integrable Hénon-Heiles systems to CM.
