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

Abstract

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.