Javier Mas

A one-parameter family of hamiltonian structures for the KP hierarchy and a continuous deformation of the nonlinear WKP algebra

AbstractThe KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting W-algebra is a one-parameter deformation of WKP admitting a central extension for generic values of the parameter, reducing naturally to Wn for special values of the parameter, and contracting to the centrally extended W1+∞, W∞ and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic towKP. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation ofn

INTEGRABILITY AND BIHAMILTONIAN STRUCTURE OF THE EVEN ORDER SKdV HIERARCHIES

hat W_infty

The topography of W∞-type algebras

n which contracts to a new nonlinear algebra of the W∞-type.

Centrally extended W1+∞ and the KP hierarchy

Abstract We construct the second hamiltonian structure of the KP hierarchy as a natural extension of the Gelfand-Dickey brackets of the generalized KdV hierarchies. The first structure — which has been recently identified as W 1+∞ —is coordinated with the second structure and arises as a trivial (generalized) cocycle. The second structure gives rise to a non linear algebra, denoted W KP , with generators of weights 1, 2, … . The reduced algebra obtained by setting the weight 1 field to zero contains a centerless Virasoro subalgebra, and we argue that this is a universal W algebra from which all W n algebras are obtained through reduction.

The Topography of W(infinity) type algebras

We study reductions of the even order SKP hierarchy. We prove that these systems are integrable and bihamiltonian. We derive an infinite set of independent polynomial conservation laws, prove their nontriviality, and derive Lenard relations between them. A further reduction of the simplest such hierarchy is identified with the supersymmetric KdV hierarchy of Manin and Radul. We prove that it inherits all the bihamiltonian and integrability properties from the unreduced hierarchy.

DIFFEOMORPHISMS FROM HIGHER-DIMENSIONAL W-ALGEBRAS

Abstract Recently much attention has been paid to the restriction of KP to the submanifold of operators which can be represented as a ratio of two purely differetial operators L = AB f -1 . Whereas most of the aspects concerning this reduced hierarchy, like the Lax flows and the Hamiltonians, are by now well understood, there still lacks a clear and conclusive statement about the associated Poisson structure. We fill this gap by placing the problem in a more general framework and then showing how the required result follows from an interesting property of the second Gelfand-Dickey brackets under multiplication and inversion of Lax operators. As a byproduct we give an elegant and simple proof of the generalised Kuperschmidt-Wilson theorem.

On the two-boson picture of the KP hierarchy

We chart out the landscape of

The Topography of

Winfty