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Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows

Author(s): Henrik Aratyn | Johan van de Leur

Journal: Symmetry, Integrability and Geometry: Methods and Applications
ISSN 1815-0659

Volume: 3;
Start page: 020;
Date: 2007;
Original page

Keywords: Clifford algebra | tau-functions | Kac-Moody algebras | loop groups | Camassa-Holm equation | Cecotti-Vafa equations | AKNS hierarchy

We use a Grassmannian framework to define multi-component tau functions as expectation values of certain multi-component Fermi operators satisfying simple bilinear commutation relations on Clifford algebra. The tau functions contain both positive and negative flows and are shown to satisfy the $2n$-component KP hierarchy. The hierarchy equations can be formulated in terms of pseudo-differential equations for $nimes n$ matrix wave functions derived in terms of tau functions. These equations are cast in form of Sato-Wilson relations. A reduction process leads to the AKNS, two-component Camassa-Holm and Cecotti-Vafa models and the formalism provides simple formulas for their solutions.
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