论文标题
半混凝土Lagrangian 2形和Toda层次结构
Semi-discrete Lagrangian 2-forms and the Toda hierarchy
论文作者
论文摘要
我们提出了一个集成差分方程（半污染集成系统）的变分理论。这是以“ Lagrangian Multiforms”和“ Pluri-Lagrangian Systems”的名称所知的思想的自然扩展，这些系统以前在完全离散和完全连续的情况下已经建立。这些想法的主要特征是捕获单个变异原理的通勤方程的层次结构。我们的主要示例是说明拉格朗日多种制服的新半分化理论是Toda晶格。该ODE描述了与邻次相互作用的颗粒的一维晶格的连续时间的演变。它是ODE的可集成层次结构的一部分，每个ODE都涉及连续变量和许多晶格移动的导数。我们将使用拉格朗日多形理论来得出TODA层次结构的连续变量中的PDE，这不涉及任何晶格变化。作为第二个例子，我们简要讨论了与Volterra晶格有关的半分化电位KDV方程。
We present a variational theory of integrable differential-difference equations (semi-discrete integrable systems). This is a natural extension of the ideas known by the names "Lagrangian multiforms" and "Pluri-Lagrangian systems", which have previously been established in both the fully discrete and fully continuous cases. The main feature of these ideas is to capture a hierarchy of commuting equations in a single variational principle. Our main example to illustrate the new semi-discrete theory of Lagrangian multiforms is the Toda lattice. This ODE describes the evolution in continuous time of a 1-dimensional lattice of particles with nearest-neighbour interaction. It is part of an integrable hierarchy of ODEs, each of which involves a derivative with respect to a continuous variable and a number of lattice shifts. We will use the Lagrangian multiform theory to derive PDEs in the continuous variables of the Toda hierarchy, which do not involve any lattice shifts. As a second example, we briefly discuss the semi-discrete potential KdV equation, which is related to the Volterra lattice.
