Continuous limits of generalized pentagram maps
Abstract
We provide a rigorous treatment of continuous limits for various generalizations of the pentagram map on polygons in RP^{d} by means of quantum calculus. Describing this limit in detail for the case of the shortdiagonal 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 χpentagram 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 KdVtype equation can be discretized through pentagramtype maps.
 Publication:

Journal of Geometry and Physics
 Pub Date:
 September 2021
 DOI:
 10.1016/j.geomphys.2021.104292
 arXiv:
 arXiv:2010.00723
 Bibcode:
 2021JGP...16704292N
 Keywords:

 37J35;
 Mathematics  Dynamical Systems;
 Mathematics  Differential Geometry
 EPrint:
 21 pages, 3 figures. v2: minor changes for clarity