论文标题
中心咖啡因几何形状和拉梅方程中的self-bäcklund曲线
Self-Bäcklund curves in centroaffine geometry and Lamé's equation
论文作者
论文摘要
25年前，美国Pinkall发现，Korteweg-De Vries方程可以实现为核糖植物几何形状中曲线的演变。从那时起，许多作者用核查几何形状来解释KDV的各种特性及其概括。特别是，可以将Korteweg-De Vries方程的Bäcklund变换视为质心曲线之间的关系。 我们的论文涉及Self-Bäcklund质心曲线。我们描述了这些曲线的一般特性，并根据椭圆函数提供了详细的描述。我们的工作与F. Wegner在欧几里得几何形状中类似问题的研究中进行了质心，这与ULAM描述（二维）身体的问题有关，这些物体（二维）身体在所有位置和自行车运动学方面都处于平衡状态。 我们还考虑了曲线被多边形代替的问题的离散化。这与理想多边形的KDV离散化和交叉比例动力学有关。
Twenty five years ago U. Pinkall discovered that the Korteweg-de Vries equation can be realized as an evolution of curves in centoraffine geometry. Since then, a number of authors interpreted various properties of KdV and its generalizations in terms of centoraffine geometry. In particular, the Bäcklund transformation of the Korteweg-de Vries equation can be viewed as a relation between centroaffine curves. Our paper concerns self-Bäcklund centroaffine curves. We describe general properties of these curves and provide a detailed description of them in terms of elliptic functions. Our work is a centroaffine counterpart to the study done by F. Wegner of a similar problem in Euclidean geometry, related to Ulam's problem of describing the (2-dimensional) bodies that float in equilibrium in all positions and to bicycle kinematics. We also consider a discretization of the problem where curves are replaced by polygons. This is related to discretization of KdV and the cross-ratio dynamics on ideal polygons.
