BKP and CKP revisited: the odd KP system

Abstract

By restricting a linear system for the KP hierarchy to those independent variables tn 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 an (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 of 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.