Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2006-11-09
J. Phys. A: Math. Theor. 40 (2007) 12539-12561
Nonlinear Sciences
Exactly Solvable and Integrable Systems
19 pages, 11 figures; v2. some typos corrected; v3. new references added, higlighted similarities and differences with recent
Scientific paper
10.1088/1751-8113/40/42/S03
We study multidimensional quadrilateral lattices satisfying simultaneously two integrable constraints: a quadratic constraint and the projective Moutard constraint. When the lattice is two dimensional and the quadric under consideration is the Moebius sphere one obtains, after the stereographic projection, the discrete isothermic surfaces defined by Bobenko and Pinkall by an algebraic constraint imposed on the (complex) cross-ratio of the circular lattice. We derive the analogous condition for our generalized isthermic lattices using Steiner's projective structure of conics and we present basic geometric constructions which encode integrability of the lattice. In particular, we introduce the Darboux transformation of the generalized isothermic lattice and we derive the corresponding Bianchi permutability principle. Finally, we study two dimensional generalized isothermic lattices, in particular geometry of their initial boundary value problem.
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