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

TL;DR

This paper develops a vectorial binary Darboux transformation for a specific part of the AKNS hierarchy, leading to new solutions including rogue waves for a complex PDE with geometric interpretation.

Contribution

It introduces a novel vectorial binary Darboux transformation for the negative AKNS hierarchy and derives multi-soliton and rogue wave solutions for related PDEs.

Findings

Derived a vectorial binary Darboux transformation for the negative AKNS hierarchy

Generated multi-soliton and rogue wave solutions for a complex PDE

Connected the PDE to geometric dynamics of a scalar field

Abstract

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.