Higher Gel'fand-Dikii structures

TL;DR

This paper constructs an infinite family of compatible Poisson structures on the KdV phase space, incorporating non-local terms and providing a generating function via Baker-Akhiezer functions, advancing the understanding of higher Gel'fand-Dikii structures.

Contribution

It extends the theory of Gel'fand-Dikii structures by explicitly constructing higher Poisson structures with non-local terms on the KdV phase space.

Findings

Generated an infinite family of compatible Poisson structures.

Expressed non-local terms using the KdV hierarchy.

Provided a generating function with Baker-Akhiezer functions.

Abstract

We apply the procedure of Magri and Weinstein to produce an infinity of compatible Poisson structures on a bihamiltonian manifold, to the case of the KdV phase space. The higher Gel'fand-Dikii structures thus obtained contain non local terms, which we express with the help of the r.h.s. of the KdV hierarchy. We also give a generating function for all these Poisson structues, in terms of the Baker-Akhiezer functions. Finally we describe the symplectic leaves of these Poisson structures.