Long-diagonal pentagram maps

2022/03/15 by Izosimov, Anton, Khesin, Boris · 3 citations
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Metric Geometry (math.MG) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2203.07578

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 ℝP^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.

Cited by

• The dimer model and dynamical incidence geometry
• Degree growth of skew pentagram maps
• Collapsing in polygonal dynamics

Related

• Integrability of higher pentagram maps
• The Pentagram map in higher dimensions and KdV flows
• The Pentagram map: a discrete integrable system
• The Algebraic Dynamics of the Pentagram Map
• Pentagram maps over rings, Grassmannians, and skewers