Nonassociativity and Integrable Hierarchies

Details Nonassociativity and Integrable Hierarchies Nonassociativity and Integrable Hierarchies

2005-12-31

arxiv.org/abs/nlin/0601001v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

36 pages, second version substantially revised and rewritten

Scientific paper

Let A be a nonassociative algebra such that the associator (A,A^2,A) vanishes. If A is freely generated by an element f, there are commuting derivations delta_n, n=1,2,..., such that delta_n(f) is a nonlinear homogeneous polynomial in f of degree n+1. We prove that the expressions delta_{n_1} ... delta_{n_k}(f) satisfy identities which are in correspondence with the equations of the Kadomtsev-Petviashvili (KP) hierarchy. As a consequence, solutions of the `nonassociative hierarchy' partial_{t_n}(f) = delta_n(f), n=1,2,..., of ordinary differential equations lead to solutions of the KP hierarchy. The framework is extended by introducing the notion of an A-module and constructing, with the help of the derivations delta_n, zero curvature connections and linear systems.

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