Application of an incidence theorem for conics: Cauchy problem and integrability of the dCKP equation

Abstract

We demonstrate that Pascal's classical theorem for hexagons inscribed in conics allows one to define in a compact manner maps which are governed algebraically by the integrable discrete CKP equation. A theorem for conics on oriented triangulated surfaces is used to construct a well-posed Cauchy problem for these dCKP maps. Moreover, the same theorem is exploited to construct in a purely geometric manner a Backlund transformation for dCKP maps. Thus, the integrability of dCKP maps and their underlying nonlinear soliton equation is shown to be encoded in an incidence theorem of projective geometry.