An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions

Details An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions

2005-01-02

arxiv.org/abs/nlin/0501003v3

J.Phys. A38 (2005) 5453-5506

Nonlinear Sciences

Exactly Solvable and Integrable Systems

59 pages, relative to the second version a few minor corrections, but quite a lot of amendments, to appear in J. Phys. A

Scientific paper

10.1088/0305-4470/38/24/005

A well-known ansatz (`trace method') for soliton solutions turns the equations of the (noncommutative) KP hierarchy, and those of certain extensions, into families of algebraic sum identities. We develop an algebraic formalism, in particular involving a (mixable) shuffle product, to explore their structure. More precisely, we show that the equations of the noncommutative KP hierarchy and its extension (xncKP) in the case of a Moyal-deformed product, as derived in previous work, correspond to identities in this algebra. Furthermore, the Moyal product is replaced by a more general associative product. This leads to a new even more general extension of the noncommutative KP hierarchy. Relations with Rota-Baxter algebras are established.

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