Reciprocal Figures, Graphical Statics, and Inversive Geometry of the Schwarzian BKP Hierarchy
Reciprocal Figures, Graphical Statics, and Inversive Geometry of the Schwarzian BKP Hierarchy
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DOI：
10.1111/1467-9590.00402
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发表时间：
2001-07
影响因子：
2.7
通讯作者：
B. Konopelchenko;W. Schief
中科院分区：
文献类型：
--
作者：
B. Konopelchenko;W. Schief
揭示了孤子理论与麦克斯韦及其同时代人发展的图解静力学理论的一个重要而美丽的分支之间的显著联系。因此，证明了构成最简单的一对代表框架和自应力的倒易图形的倒易三角形封装了可积离散BKP方程及其Schwarzian版本。固有的莫比乌斯不变性质的Schwarzian BKP方程，然后利用定义倒易的几何设置。讨论了由倒易三角形构成的非平凡组合学格的可积对及其自然推广。这些BKP晶格的特定约化与达布（2+1）维sine-Gordon方程和仿射几何的经典齐泽卡方程的可积离散版本有关。此外，它表明，八面体图形和六面体的倒数所考虑的麦克斯韦同样产生离散可积系统和相关的可积格。
A remarkable connection between soliton theory and an important and beautiful branch of the theory of graphical statics developed by Maxwell and his contemporaries is revealed. Thus, it is demonstrated that reciprocal triangles that constitute the simplest pair of reciprocal figures representing both a framework and a self‐stress encapsulate the integrable discrete BKP equation and its Schwarzian version. The inherent Möbius invariant nature of the Schwarzian BKP equation is then exploited to define reciprocity in an inversive geometric setting. Integrable pairs of lattices of nontrivial combinatorics consisting of reciprocal triangles and their natural generalizations are discussed. Particular reductions of these BKP lattices are related to the integrable discrete versions of Darboux's (2+1)‐dimensional sine‐Gordon equation and the classical Tzitzéica equation of affine geometry. Furthermore, it is shown that octahedral figures and their hexahedral reciprocals as considered by Maxwell likewise give rise to discrete integrable systems and associated integrable lattices.