Bg. Konopelchenko et Wk. Schief, Three-dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality, P ROY SOC A, 454(1980), 1998, pp. 3075-3104
Citations number
38
Categorie Soggetti
Multidisciplinary
Journal title
PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
It is shown that the discrete Darboux system, descriptive of conjugate latt
ices in Euclidean spaces, admits constraints on the (adjoint) eigenfunction
s which may be interpreted as discrete orthogonality conditions on the latt
ices. Thus, it turns out that the elementary quadrilaterals of orthogonal l
attices are cyclic. Orthogonal lattices on lines, planes and spheres are di
scussed and the underlying integrable systems in one, two and three dimensi
ons are derived explicitly. A discrete analogue of Bianchi's Ribaucour tran
sformation is set down and particular orthogonal lattices are given. As a b
y-product, discrete Dini surfaces are obtained.