On Bäcklund and Ribaucour transformations for surfaces with constant negative curvature

Abstract

Given a surface in the Euclidean 3-space with constant negative Gaussian curvature, the composition of Bäcklund transformations generates a 4-parameter family of surfaces with the same curvature. Since Ribaucour transformations of such a surface also provides a 4-parameter family of surfaces of the same type, one has the following natural question. Are these two methods equivalent? The answer is negative in general. Necessary and sufficient conditions for the surfaces given by the two procedures to be congruent are obtained. An explicit example is given of a composition of Bäcklund transformations which is not a Ribaucour transformation. The analytic interpretation of this result provides a method of producing solutions of the sine-Gordon equation, distinct from those obtained by Bäcklund transformations. New exact solutions are given and some of the corresponding surfaces are visualized.

Polynomial conserved quantities of Lie applicable surfaces

“…In [2, §273] and [1, §398], Bianchi constructs a 3-parameter family of Ribaucour transformations of pseudospherical (K = −1) surfaces in Euclidean space into surfaces of the same kind by performing two successive complex conjugate Bäcklund transformations. This has been investigated recently in [21]. This transformation preserves I + III.…”

“…21.ŝ is a null rank 1 parallel subbundle of d + mη if and only ifŝ = T −1 (m)L for some constantL ∈ P(L).…”

On Backlund transformations of surfaces by extended Harry-Dym flow

“…with the choicess b = r = 0 and f = τ -1/2 in the space curve flow (5). We note that the curve flow with constant curvature κ is considered here.…”

“…Due to the aforementioned features, numerous studies have been carried out on Backlund transformations from past to present. For example, Palmer studied Backlund transformations for surfaces in [3], Schief gave analog of Darboux's Backlund transformation for isothermic surfaces in [4] and Goulart obtained Backlund and Ribaucour transformations for some special surfaces in [5]. Korpinar gave numerical solutions of heat-like equation in [6] and Ijaz studied Heat Transfer Analysis in MHD flow in [7].…”

On Bäcklund and Ribaucour transformations for hyperbolic linear Weingarten surfaces

“…In this paper, we will show that in general the families obtained by these procedures are distinct, in contrast with what happens in the case of constant positive Gaussian curvature (see for example Tenenblat [18]) and surfaces of nonzero constant mean curvature (Jeromin-Pedit [12]). In the particular case K = −1, necessary and sufficient conditions were established by Goulart-Tenenblat [11], for a composition of Bäcklund transformations to be congruent to a Ribaucour transformation.…”