  Novel soliton structures are constructed for the Fokas-Lenells equation. In
so doing, and after discussing the stability of continuous waves, a multiple
scales perturbation theory is used to reduce the equation to a Korteweg-de
Vries system whose relative soliton solution gives rise to intricate (and
rather unexpected) solutions to the original system. Both the focusing and
defocusing equations are considered and it is found that dark solitons may
exist in both cases while in the focusing case antidark solitons are also
possible. These findings are quite surprising as the relative nonlinear
Schr\"odinger equation does not exhibit these solutions.
