Long-diagonal pentagram maps

TL;DR

This paper introduces and proves the integrability of long-diagonal pentagram maps in higher-dimensional projective spaces, unifying known cases and linking them to the $(2,d+1)$-KdV equation as a continuous limit.

Contribution

It extends the pentagram map to long-diagonal cases in $ ext{RP}^d$, proves their integrability, and establishes their connection to the $(2,d+1)$-KdV equation.

Findings

Proved integrability of long-diagonal pentagram maps in $ ext{RP}^d$.

Established equivalence between long-diagonal and bi-diagonal maps.

Connected the continuous limit to the $(2,d+1)$-KdV equation.

Abstract

The pentagram map on polygons in the projective plane was introduced by R. Schwartz in 1992 and is by now one of the most popular and classical discrete integrable systems. In the present paper we introduce and prove integrability of long-diagonal pentagram maps on polygons in $\mathbb{R}\mathrm{P}^d$, encompassing all known integrable cases. We also establish an equivalence of long-diagonal and bi-diagonal maps and present a simple self-contained construction of the Lax form for both. Finally, we prove the continuous limit of all these maps is equivalent to the $ (2,d+1)$-KdV equation, generalizing the Boussinesq equation for $d=2$.