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SIGMA 3 (2007), 020, 29 pages nlin.SI/0605027
https://doi.org/10.3842/SIGMA.2007.020
Contribution to the Vadim Kuznetsov Memorial Issue
Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows
Henrik Aratyn a and Johan van de Leur b
a) Department of Physics,
University of Illinois at Chicago, 845 W. Taylor St., Chicago, IL 60607-7059, USA
b) Mathematical Institute, University of Utrecht, P.O. Box 80010, 3508 TA Utrecht, The
Netherlands
Received October 11, 2006, in final form January 09, 2007; Published online February 06, 2007
Abstract
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
n × 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.
Key words:
Clifford algebra; tau-functions; Kac-Moody algebras; loop groups; Camassa-Holm equation; Cecotti-Vafa equations; AKNS hierarchy.
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