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BKP and CKP revisited: The odd KP system | Aristophanes Dimakis
; Folkert Muller-Hoissen
; | Date: |
4 Oct 2008 | Abstract: | Restricting a linear system for the KP hierarchy to those independent
variables t\_n with odd n, its compatibility (Zakharov-Shabat conditions) leads
to the "odd KP hierarchy". The latter consists of pairs of equations for two
dependent variables, taking values in a (typically noncommutative) associative
algebra. If the algebra is commutative, the odd KP hierarchy is known to admit
reductions to the BKP and the CKP hierarchy. We approach the odd KP hierarchy
and its relation to BKP and CKP in different ways, and address the question
whether noncommutative versions of the BKP and the CKP equation (and some of
their reductions) exist. In particular, we derive a functional representation
of a linear system for the odd KP hierarchy, which in the commutative case
produces functional representations of the BKP and CKP hierarchies in terms of
a tau function. Furthermore, we consider a functional representation of the KP
hierarchy that involves a second (auxiliary) dependent variable and features
the odd KP hierarchy directly as a subhierarchy. A method to generate large
classes of exact solutions to the KP hierarchy from solutions to a linear
matrix ODE system, via a hierarchy of matrix Riccati equations, then also
applies to the odd KP hierarchy, and this in turn can be exploited, in
particular, to obtain solutions to the BKP and CKP hierarchies. | Source: | arXiv, 0810.0757 | Services: | Forum | Review | PDF | Favorites |
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