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Abstract:
We present an overview of recent results (nlin.SI/0501003, J.Phys.A: Math.Gen. 38 (2005) 5453).
An old (``trace") method of Okhuma and Wadati to generate soliton solutions of the KP equation via a formal power series ansatz leads to a correspondence between algebraic partial sum identities and the equations of the classical KP hierarchy (where the fields are allowed to have values in an arbitrary associative algebra). Moreover, such a correspondence can be achieved by abstracting from the partial sum calculus to an algebraic scheme, which in particular involves a (mixable) shuffle product and makes contact with Rota-Baxter algebras. Furthermore, larger sets of identities in this algebra correspond to extensions of the KP hierarchy in the case where the associative product is a Moyal-product or a certain generalization of it.
