A vectorial binary Darboux transformation for the first member of the negative part of the AKNS hierarchy

Using bidifferential calculus, we derive a vectorial binary Darboux transformation for the first member of the ‘negative’ part of the AKNS hierarchy. A reduction leads to the first ‘negative flow’ of the NLS hierarchy, which in turn is a reduction of a rather simple nonlinear complex PDE in two dimensions, with a leading mixed third derivative. This PDE may be regarded as describing geometric dynamics of a complex scalar field in one dimension, since it is invariant under coordinate transformations in one of the two independent variables. We exploit the correspondingly reduced vectorial binary Darboux transformation to generate multi-soliton solutions of the PDE, also with additional rational dependence on the independent variables, and on a plane wave background. This includes rogue waves.