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Diffeomorphisms on S1, projective structures and integrable systems

Published online by Cambridge University Press:  17 February 2009

Partha Guha
Partha Guha
Affiliation:
S.N. Bose National Centre for Basic Sciences, JD Block, Sector-3, Salt Lake, Calcutta-700091, India; e-mail: guha@BOSON.bose.res.in. Institut des Hautes Etudes Scientifiques, 35, Route de Chartres, 91440-Bures-sur-Yvette, France.
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Abstract

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In this paper we consider a projective connection as defined by the nth-order Adler-Gelfand-Dikii (AGD) operator on the circle. It is well-known that the Korteweg-de Vries (KdV) equation is the archetypal example of a scalar Lax equation defined by a Lax pair of scalar nth-order differential (AGD) operators. In this paper we derive (formally) the KdV equation as an evolution equation of the AGD operator (at least for n ≤ 4) under the action of Vect(S1). The solutions of the AGD operator define an immersion R → RPn−1 in homogeneous coordinates. In this paper we derive the Schwarzian KdV equation as an evolution of the solution curve associated with Δ(n), for n ≤ 4.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 169 - 180
Copyright
Copyright © Australian Mathematical Society 2002

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