2001/05/31 by B. G. Konopelchenko, W. K. Schief
Chemistry · Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons #math.CV #nlin.SI
paper · pdf · doi:10.1088/0305-4470/35/29/313
26 pages, 9 figures
arxiv created 2001/07/02 · openalex publication_date 2002/07/12 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/04
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.