Differential Calculi on Associative Algebras and Integrable Systems

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

“…This is the first "negative flow" of the AKNS hierarchy (see, e.g., system (12) in [4] and also [5,10], for example). Soliton solutions of it have, apparently, first been found in [9], using Hirota's bilinear method.…”

“…We refer the reader to [12] for an introduction to this structure and an extensive list of references.…”

“…The result in Corollary 4.2 can be regarded as a reduction of that in Theorem 4.1. Corollary 4.2 has been used in many previous applications of bidifferential calculus, see in particular [12,20,21].…”

“…(1.1) is among the simplest completely integrable PDEs and it has the peculiar property mentioned above. In this work, we derive a binary Darboux transformation (see, e.g., [11]), by using a general result of bidifferential calculus, which is recalled in Section 4.1 in a self-contained way (see, e.g., [12] for an introduction to bidifferential calculus and references), and demonstrate its use for finding soliton solutions. Here we proceed beyond the class of "simple solitons", which are rational expressions built from trigonometric and hyperbolic functions of linear combinations of the independent variables.…”

“…Here Γ −1 multiplies both, η1 and η2 6. Nevertheless, the classical iterative (forward, backward and binary) Darboux transformations can also be formulated in bidifferential calculus, see[12].…”