The Laplace sequence of conjugate nets provides geometric interpretation of the 2D (hyperbolic) Toda field system, which works also on the discrete level. In the elliptic case, formally very similar to the hyperbolic one, we have the so called harmonic sequence of harmonic maps of Riemann surfaces into complex projective spaces. The corresponding Laplace transformation of the discrete 2D elliptic Schroedinger operator (the self-adjoint 7 point scheme on the triangular lattice) was introduced ten years ago by Novikov. I would like to show that the known properties of this operator, like existence of the Laplace and Darboux-type transformations, are consequences of simple geometric properties of the B-(Moutard) quadrilateral lattice. As a by-product of this new interpretation we obtain, for example, a Darboux-type transformation for a linear problem on the honeycomb grid.
(joint work with M. Nieszporski and P. M. Santini).
Contact A.P. Fordy (email: allan@maths.leeds.ac.uk ) for further details.
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Last updated February 9th, 2007.