Solutions of the Betchov-Da Rios soliton equation:: a Lorentzian approach

The purpose of this paper is to find out explicit solutions of the Betchov-Da Rios soliton equation in three-dimensional Lorentzian space forms. We start with non-null curves and obtain solutions living in certain flat ruled surfaces in L-3 and H-1(3), as well as in R-3 and S-3. Next we take a null curve and have got solutions lying in the associated B-scrolls in L-3 S-1(3) and H-1(3). It should be pointed out that we extend previous results already obtained, and as far as we know, this is the first time that solutions in the De Sitter 3-space appear in the literature. Soliton solutions are characterized as null geodesics in B-scrolls. (C) 1999 Elsevier Science B.V. All rights reserved.