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Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy | | B.G. Konopelchenko
; W.K. Schief
; | | Date: |
9 May 2001 | | Subject: | Exactly Solvable and Integrable Systems; Complex Variables | nlin.SI math.CV | | Abstract: | 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. | | Source: | arXiv, nlin.SI/0105023 | | Services: | Forum | Review | PDF | Favorites | |
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