Transformations of quadrilateral lattices



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Doliwa, Adam and Santin, Maria Santin and Mañas Baena, Manuel (2000) Transformations of quadrilateral lattices. Journal of mathematical physics, 41 (2). pp. 944-990. ISSN 0022-2488

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Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogs: the multidimensional quadrilateral lattices, i.e., lattices x:Z(N)--> R-M, N less than or equal to M, whose elementary quadrilaterals are planar. Our investigation is based on the discrete analog of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogs of the Laplace, Combescure, Levy, radial, and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between "transformations" and "discretizations" is also investigated for quadrilateral lattices. We finally interpret these results within the <(partial derivative)over bar> formalism.

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©2000 American Institute of Physics.
A.D. would like to thank A. Sym for pointing out (see also Ref. 47) the important role of the rectilinear congruences in the theory of integrable geometries (soliton surfaces). He also acknowledges partial support from KBN Grant No. 2P03 B 18509. M.M. acknowledges partial support from CICYT Proyect No. PB95-0401 and from the exchange agreement between Università La Sapienza of Rome and Universidad Complutense of Madrid

Uncontrolled Keywords:Integrable hierarchies; Coordinate systems; Hydrodynamic-type; Equation; Surfaces; Discretization
Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:32473
Deposited On:23 Jul 2015 10:32
Last Modified:10 Dec 2018 15:10

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