With a Cole–Hopf transformation to solutions of the noncommutative KP hierarchy in terms of Wronski matrices

Abstract

In case of the KP hierarchy where the dependent variable takes values in an (arbitrary) associative algebra R, it is known that there are solutions which can be expressed in terms of quasideterminants of a Wronski matrix which solves the linear heat hierarchy. We obtain these solutions without the help of quasideterminants in a simple way via solutions of matrix KP hierarchies (over R) and by use of a Cole-Hopf transformation. For this class of exact solutions we work out a correspondence with 'weakly nonassociative' algebras. * c 2007 by A. Dimakis and F. Müller-Hoissen 1 In several publications on 'Moyal-deformed' soliton equations the Moyal-product can be replaced almost completely by any associative (noncommutative) product, since the specific properties of the Moyal-product are not actually used. The algebraic properties of such equations are then simply those of (previously studied) matrix versions of these equations. Exceptions are in particular [17][18][19] where enlarged hierarchies are considered which appear specifically in the Moyal-deformed case. Multi-soliton solutions of the (enlarged) potential KP hierarchy with Moyal-deformed product were obtained in [19] using a method which, in the commutative case, corresponds to the well-known 'trace method' [21] (see also [5], appendix A6, and [22]).2 One link between integrable systems and quasideterminants is given by the fact that 'noncommutative' Darboux transformations [34] can be compactly expressed in the form of a quasideterminant [25,31,35,36].

Explicit Solution Formulas for the Matrix-Kp

“…for example [2,8,9,13,14]. Particularly close in spirit to the present paper are [5][6][7]10]). Our result on the general operator level is the following: THEOREM 1.…”

On a direct approach to quasideterminant solutions of a noncommutative KP equation

“…Derivatives of quasideterminants have been considered in [3,10]. Let Θ = θ (i−1) j i,j=1,...,n be the n × n wronskian matrix of θ 1 , .…”

“…The origin of these solutions were explained by Darboux and binary Darboux transformations. The quasideterminant solutions were then verified directly using formulae for derivatives of quasideterminants (see also [3]). In this approach, the nature of the noncommutativity is not specified so that the results presented were valid for, for example, the noncommutative Moyal star product and the matrix or quaternion versions of the KP equation.…”

Weakly Non-Associative Algebras and the Kadomtsev–petviashvili Hierarchy

“…We refer to [6,7] for exact solutions of the matrix KP hierarchy obtained with the help of this theorem. The following proposition provides us with a formal solution of the initial value problem for (22) for a subclass of WNA algebras.…”

“…Further examples and some applications of WNA algebras in the context of KP hierarchies appeared in [3,6,7]. There is a WNA algebra such that (22) reproduces the Gelfand-Dickey-Sato formulation of the (potential) KP hierarchy [6].…”