Geometric discretization of the Koenigs nets

Abstract

We introduce the Koenigs lattice, which is a new integrable reduction of the quadrilateral lattice (discrete conjugate net) and provides natural integrable discrete analogue of the Koenigs net. We construct the Darboux-type transformations of the Koenigs lattice and we show permutability of superpositions of such transformations, thus proving integrability of the Koenigs lattice. We also investigate the geometry of the discrete Koenigs transformation. In particular we characterize the Koenigs transformation in terms of an involution determined by a congruence conjugate to the lattice.

17 pages, 2 figures; some spelling and typing errors corrected

References in corpus (3)

• Asymptotic lattices and their integrable reductions I: the Bianchi and the Fubini-Ragazzi lattices
• Geometric discretization of the Bianchi system
• The integrable discretization of the Bianchi-Ernst system

Cited by in corpus (9)

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• Darboux transformations for linear operators on two dimensional regular lattices
• Hirota equation and the quantum plane
• Discrete Koenigs nets and discrete isothermic surfaces
• The normal dual congruences and the dual Bianchi lattice