Darboux Transformations Between Δα=sinh α and Δα=sin α and the Application to Pseudo-Spherical Congruences in R 2,1

Abstract

In this Letter, it is proved that by using Darboux transformation, from a known solution of the sinh-Laplace equation (resp. sine-Laplace equation), new solutions of sine-Laplace equation (resp. sinh-Laplace equation) are obtained explicitly. The corresponding geometrical configuration is the space-like pseudo-spherical line congruence with two focal surfaces of negative constant curvature being space-like and time-like, respectively.