Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy

Details Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy

2001-05-09

arxiv.org/abs/nlin/0105023v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

26 pages, 9 figures

Scientific paper

10.1088/0305-4470/35/29/313

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 is discussed. The dSKP and discrete Schwarzian Boussinesq equations are shown to represent discretizations of families of quasi-conformal mappings.

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