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Weakly nonassociative algebras, Riccati and KP hierarchies

Aristophanes Dimakis·Folkert Muller-Hoissen
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Abstract

It has recently been observed that certain nonassociative algebras (called "weakly nonassociative", WNA) determine, via a universal hierarchy of ordinary differential equations, solutions of the KP hierarchy with dependent variable in an associative subalgebra (the middle nucleus). We recall central results and consider a class of WNA algebras for which the hierarchy of ODEs reduces to a matrix Riccati hierarchy, which can be easily solved. The resulting solutions of a matrix KP hierarchy then determine (under a rank 1 condition) solutions of the scalar KP hierarchy. We extend these results to the discrete KP hierarchy. Moreover, we build a bridge from the WNA framework to the Gelfand-Dickey formulation of the KP hierarchy.

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Aristophanes Dimakis, Folkert Muller-Hoissen. 2008-01-16. Weakly nonassociative algebras, Riccati and KP hierarchies. https://arxiv.org/abs/nlin/0701010

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