移动标架：差分与微分-差分拉格朗日量
Moving Frames: Difference and Differential-Difference Lagrangians
AI总结：
本文建立偏微分差分方程和微分-差分方程的移动标架理论，提出可投影移动标架以保证不变导数与离散位移可交换，并将其应用于不变变分法和诺特守恒律的等变表述。
摘要：
本文发展了用于偏微分差分方程以及具有一个连续自变量的微分-差分方程的移动标架理论。在每种情形下，该理论都被应用于不变变分法和由诺特定理产生的守恒律的等变表述。微分-差分理论不仅仅是微分理论与差分理论的简单合并，还具有额外特征，这些特征反映了群作用必须保持延拓结构的需求。本文引入了可投影移动标架；这些标架使得不变导数算子与离散变量中的位移可交换。示例包括一个Toda型方程以及非线性薛定谔方程的半离散化直线法。
英文摘要：
This paper develops moving frame theory for partial difference equations and for differential-difference equations with one continuous independent variable. In each case, the theory is applied to the invariant calculus of variations and the equivariant formulation of the conservation laws arising from Noether's theorem. The differential-difference theory is not merely an amalgam of the differential and difference theories, but has additional features that reflect the need for the group action to preserve the prolongation structure. Projectable moving frames are introduced; these cause the invariant derivative operator to commute with shifts in the discrete variables. Examples include a Toda-type equation and a method of lines semi-discretization of the nonlinear Schrödinger equation.
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