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GENERALIZED ISOTHERMIC LATTICES
, 2007
"... Abstract. 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 Möbius sphere one obtains, after the stereogra ..."
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Abstract. 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 Möbius sphere one obtains, after the stereographic projection, the discrete isothermic surfaces defined by Bobenko and Pinkall by an algebraic constraint imposed on the (complex) crossratio 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.
Desargues maps and their reductions
 Proceedings of the 2nd Workshop on Nonlinear and Modern Mathematical Physics (March 2013, Tampa FL), AIP
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Darboux transformations for 6point scheme
 J. Phys. A: Math. Theor
, 2007
"... We introduce (binary) Darboux transformation for general differential equation of the second order in two independent variables. We present a discrete version of the transformation for a 6point difference scheme. The scheme is appropriate to solving a hyperbolic type initialboundary value problem. ..."
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We introduce (binary) Darboux transformation for general differential equation of the second order in two independent variables. We present a discrete version of the transformation for a 6point difference scheme. The scheme is appropriate to solving a hyperbolic type initialboundary value problem. We discuss several reductions and specifications of the transformations as well as construction of other Darboux covariant schemes by means of existing ones. In particular we introduce a 10point scheme which can be regarded as the discretization of selfadjoint hyperbolic equation. Integrable systems, Jonas transformations, Moutard transformations 1
DARBOUX TRANSFORMATIONS FOR LINEAR OPERATORS ON TWO DIMENSIONAL REGULAR LATTICES
"... Abstract. Darboux transformations for (systems of) linear operators on regular two dimensional lattices are reviewed. 1. ..."
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Abstract. Darboux transformations for (systems of) linear operators on regular two dimensional lattices are reviewed. 1.
Darboux transformations for qdiscretizations of 2D second order differential equations
"... This article is a part of the special issue titled “Symmetries and Integrability of Difference ..."
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This article is a part of the special issue titled “Symmetries and Integrability of Difference