Integrable Multidimensional Discrete Geometry

Darboux transformations for linear operators on two-dimensional regular lattices

“…In this section we present the Darboux transformations for the four-point scheme (the discrete Laplace equation) from the point of view of systems of such equations, and the corresponding permutability theorems. To keep the paper of reasonable size and in order to present the results from a simple algebraic perspective we do not discuss important relations of the subject to incidence and difference geometry [7, 13, 20, 21, 25-27, 31, 32, 35, 38, 39, 54, 55, 57, 58, 100, 101] (see also [8,33] and earlier works [97,98]), application of analytic [10, 29, 32, 34-37, 70, 106-108] and algebro-geometric [4,5,22,23,26,27,39,47,59,60] techniques of the integrable systems theory to construct large classes of solutions of the linear systems in question and solutions of the corresponding nonlinear discrete equations.…”

Pseudospherical surfaces on time scales: a geometric definition and the spectral approach

“…In the discrete case (e.g., T 1 = T 2 = Z) we have σ j (u) = T j u and ρ j (u) = T −1 j u, where T 1 , T 2 are usual shift operators. Therefore delta and nabla differentiation can be associated with forward and backward data, respectively [10]. Definition 4 ( [5]).…”

“…The difference geometry [22] is a discrete analogue of the differential geometry. In the past years one can observe a fast development of the integrable difference geometry (see, for instance, [3,4,[8][9][10][11]23]) closely related to the classical differential geometry [12,13]. It is interesting that in the discrete case one recovers explicit constructions and transformations known in the continuous case (e.g., Darboux, Bäcklund, Ribaucour, Laplace and Jonas transformations, soliton and finite-gap solutions etc).…”

Field-Induced Phase Transition between Tetragonal and Rhombohedral Phases in Pb(Zn1/3Nb2/3)O3–8%PbTiO3

“…Finally we will see how each approximant is associated with a highest weight vector. Relations between these highest weight vectors are of the 'Lozenge' type which occur in the standard theory [5,7] and in present treatments of integrable discrete systems [6,8].…”