Darboux transformations between Delta alpha = sinh alpha and Delta alpha =sin alpha and the application to pseudo-spherical congruences in R-2,R-1

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