长对角五角星形映射
Long-diagonal pentagram maps
摘要：
R. Schwartz在1992年提出的离散可积系统是目前最流行和最经典的离散可积系统之一。本文引入并证明了$\mathbb{R}\mathrm{P}^d$中多边形上长对角五角星形映射的可积性，包括所有已知的可积情形。我们还建立了一个等价的长对角和双对角映射，并提出了一个简单的自包含建设的拉克斯形式。最后证明了所有这些映射的连续极限等价于$ (2,d+1)$ -KdV方程，推广了$d=2$的Boussinesq方程.
英文摘要：
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 $d=2$.
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