Binary Darboux transformation for a negative-order AKNS equation

We study a binary Darboux transformation for a negative-order AKNS equation. Iterating the transformation, we obtain $N$-fold quasi-Grammian solutions expressed in terms of quasideterminants. In some simple cases, we construct explicit solutions of the studied equation with nonvanishing and vanishing backgrounds including bright and dark breathers, a soliton, and solutions with one and two humps.