Transformations of Quadrilateral Lattices

Details Transformations of Quadrilateral Lattices Transformations of Quadrilateral Lattices

1997-12-20

arxiv.org/abs/solv-int/9712017v2

J. Math. Phys. 41 (2000) 944-990

Nonlinear Sciences

Exactly Solvable and Integrable Systems

50 pages, 15 figures; minor corrections, added references

Scientific paper

10.1063/1.533175

Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogues: the multidimensional quadrilateral lattices, i.e. lattices x: Z^N -> R^M, whose elementary quadrilaterals are planar. Our investigation is based on the discrete analogue of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogues 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 D-bar formalism.

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