Integrability of higher pentagram maps

Details Integrability of higher pentagram maps Integrability of higher pentagram maps

2012-04-03

arxiv.org/abs/1204.0756v1

Mathematics

Dynamical Systems

40 pages, 4 figures

Scientific paper

We define higher pentagram maps on polygons in $P^d$ for any dimension $d$, which extend R.Schwartz's definition of the 2D pentagram map. We prove their integrability for both closed and twisted polygons by presenting their Lax representation. The corresponding continuous limit of the pentagram map in dimension $d$ is shown to be the $(2,d+1)$-equation of the KdV hierarchy, generalizing the Boussinesq equation in 2D. We also study in detail the 3D case, where we describe the spectral curve, first integrals, the corresponding Liouville tori and the motion along them, as well as an invariant symplectic structure.

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