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

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 a rather simple nonlinear PDE in two dimensions with a leading mixed third derivative. This PDE may be regarded as describing 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 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.

On integrability of a third-order complex nonlinear wave equation

“…where w(x) is a real function of one real variable. It was pointed out in [1] that the nonlinear equation ( 1) is invariant under the transformation…”

“…The following new third-order complex nonlinear wave equation was introduced recently by Müller-Hoissen [1]:…”

“…where f (x, t) is a complex function of two real variables, subscripts denote respective derivatives, and the asterisk stands for the complex conjugate. This nonlinear wave equation (1) was called a completely integrable partial differential equation in [1], and its multi-soliton solutions were obtained there. Let us note, however, that this new nonlinear equation was studied in [1] not in its original form (1) but in the form of the following system of two equations:…”

“…Being the first "negative flow" of the nonlinear Schrödinger equation's hierarchy, this nonlinear system (2) is integrable and possesses multi-soliton solutions. The integrable system (2) is obviously a reduction of the new nonlinear wave equation (1), in the sense that all solutions of (2) (except for f = 0, of course) are solutions of (1) as well, because (1) follows from (2) through elimination of the dependent variable a. In other words, the new equation (1) possesses multi-soliton solutions because it possesses an integrable reduction.…”

“…In Section 2, we show that the nonlinear equation (1) does not pass the Painlevé test for integrability. In Section 3, we find two reductions of the nonlinear equation (1), one integrable and one nonintegrable, whose solutions jointly cover all solutions of the original equation. Section 4 contains concluding remarks.…”