论文标题
广义五角星地图的连续限制
Continuous limits of generalized pentagram maps
论文作者
论文摘要
我们通过量子计算，为$ \ mathbb {rp}^d $ pentagram图的各种概括提供了对连续限制的严格处理。在短期五角星地图的情况下详细描述了此限制，我们验证该构建产生了$（2，d+1）$ -KDV方程，此外，限制中Pentagram地图的LAX形式被证明已被证明成为相应KDV系统的LAX表示。更一般而言，我们引入了$χ$ -pentagram Map，这是一种通过指定点直接将子空间的交叉点进行的几何结构定义曲线演变。我们表明，其不同的配置产生了某些其他KDV方程，并提供了一个论点，以证明可以通过Pentagram型映射离散任何KDV型方程。
We provide a rigorous treatment of continuous limits for various generalizations of the pentagram map on polygons in $\mathbb{RP}^d$ by means of quantum calculus. Describing this limit in detail for the case of the short-diagonal pentagram map, we verify that this construction yields the $(2,d+1)$-KdV equation, and moreover, the Lax form of the pentagram map in the limit is proved to become the Lax representation of the corresponding KdV system. More generally, we introduce the $χ$-pentagram map, a geometric construction defining curve evolutions by directly taking intersections of subspaces through specified points. We show that its different configurations yield certain other KdV equations and provide an argument towards disproving the conjecture that any KdV-type equation can be discretized through pentagram-type maps.
