Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy
Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy
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DOI：
10.1088/0305-4470/35/29/313
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发表时间：
2001-05
期刊：
影响因子：
--
通讯作者：
B. Konopelchenko;W. Schief
中科院分区：
文献类型：
--
作者：
B. Konopelchenko;W. Schief
它表明，可积离散Schwarzian KP（dSKP）方程，构成一个代数叠加公式，例如，Schwarzian KP族，经典的达布变换和拟共形映射封装，但古希腊几何的一个基本定理。因此，它表明，与Menelaus定理的连接，更一般地说，Clifford配置呈现dSKP方程的一个自然的对象的反演几何的平面上。讨论了dSKP格的几何可积性和代数可积性及其到Menelaus-Darboux型格、SchwarzianKdV型格、SchwarzianBoussinesq型格和Schramm型格的约化。dSKP和离散Schwarzian Boussinesq方程表示拟共形映射族的离散化。
It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, more generally, Clifford configurations renders the dSKP equation a natural object of inversive geometry on the plane. The geometric and algebraic integrability of dSKP lattices and their reductions to lattices of Menelaus–Darboux, Schwarzian KdV, Schwarzian Boussinesq and Schramm type are discussed. The dSKP and discrete Schwarzian Boussinesq equations are shown to represent discretizations of families of quasi-conformal mappings.