Structure and duality of D-modules related to KP hierarchy

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].

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“…. , of the shift operator), see also [42]. Later it reappeared in [43,44] as the 'Sato system' of the (ordinary) KP hierarchy 10 .…”

Section: Further Remarksmentioning

confidence: 98%