Continuous limits of generalized pentagram maps

TL;DR

This paper rigorously analyzes the continuous limits of generalized pentagram maps in projective space, revealing their connection to KdV equations and introducing the $hi$-pentagram map, which relates geometric constructions to integrable systems.

Contribution

It provides a detailed continuous limit analysis of generalized pentagram maps, linking them to KdV equations and introducing the $hi$-pentagram map with implications for discretization of integrable systems.

Findings

The short-diagonal pentagram map's limit yields the (2,d+1)-KdV equation.

The Lax form of the pentagram map converges to the Lax representation of the KdV system.

Different configurations of the $hi$-pentagram map produce various KdV equations.

Abstract

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 $\chi$-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.