Antiplectic Structure, Diffeomorphism and Generalized KdV Family
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Antiplectic Structure, Diffeomorphism and Generalized KdV Family Partha Guha
Received: 3 August 2005 / Accepted: 27 January 2006 © Springer Science+Business Media B.V. 2006
Abstract In this paper we show that the generalized KdV, generalized Camassa–Holm equations and the corresponding Möbius invariant generalized Schwarzian KdV, Schwarzian CH equations can be realized in terms of flows induced by Vect (S1 ) on the space of differential operators and on the space of immersion curves, respectively. These are Euler–Poincaré type flows, and one of the flow takes place on an infinite-dimensional Poisson manifold and the other on a slightly degenerate infinite-dimensional Symplectic manifold. They form an Antiplectic pair. We also study Euler–Poincaré flow with respect to H1 metric, and this induces generalized Camassa–Holm equation. In the final section we discuss the Antiplectic pair in 2 + 1 dimensions. Mathematics Subject Classifications (2000) 35Q53 · 53A07 · 53B50 Key words diffeomorphism · geodesic flows · generalized (Rudykh) KdV · antiplectic structure · Schwarzian KdV · Schwarzian CH · KP family
1. Introduction It is well known that the KdV equation is the archetypal example of a scalar Lax equation [2], which is an equation defined by a Lax pair of scalar differential operators d1(n) = [P, 1(n) ] dt
with 1(n) =
dn dn−2 + un−2 n−2 + · · · + u0 , n dx dx
Dedicated to Professor George Wilson on his 65th birthday with great respect and admiration. P. Guha (B) S.N. Bose National Centre for Basic Sciences JD Block, Sector-3, Salt Lake, Calcutta 700098, India e-mail: [email protected]
98
Acta Appl Math (2006) 91: 97–118
where P is a differential operator whose coefficients are differential polynomials in the variables, essentially determined by the requirement that [P, 1(n) ] be an operator of order less than n. The space of linear differential operators on a manifold M considered as a module over the group of diffeomorphisms is a well-known classical text. This space has various algebraic structures [8], e.g. the structure of an associative algebra and of a Lie algebra. This was studied in the one-dimensional case, by Wilczynski and Cartan [5, 34]. The space of higher order linear differential operator has an interesting structures of infinite dimensional Poisson manifold with respect to the Adler–Gelfand–Dickey [2, 9], Poisson structure, which is also known as the classical W-algebras in the physics literature. These are generalization of the Virasoro algebra. Earlier we studied the KdV flow as the Euler–Poincaré flow on the space of differential operators on S1 . We identified this space as the space of projective connections on a circle. The projective connections are classified from a geometric point of view by Kuiper [19]. Lazutkin and Penkratova [21] were the first to formulate this analytically. The connection between the geodesic flow on the Bott–Virasoro group and the periodic KdV equation follows from the work of Ovsienko, Khesin, Segal and Witten [28, 31, 38]. In fact, Ovsienko and Khesin [28] g
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