Mathematical Research Letters

Volume 15 (2008)

Number 5

The family of analytic Poisson brackets for the Camassa–Holm hierarchy

Pages: 867 – 879



M. I. Gekhtman (University of Notre Dame)

K. L. Vaninsky (Michigan State University)


We consider the integrable Camassa–Holm hierarchy on the line with positive initial data rapidly decaying at infinity. It is known that flows of the hierarchy can be formulated in a Hamiltonian form using two compatible Poisson brackets. In this note we propose a new approach to Hamiltonian theory of the CH equation. In terms of associated Riemann surface and the Weyl function we write an analytic formula which produces a family of compatible Poisson brackets. The formula includes an entire function $f(z)$ as a parameter. The simplest choice $f(z)=1$ or $f(z)=z$ corresponds to the rational or trigonometric solutions of the Yang-Baxter equation and produces two original Poisson brackets. All other Poisson brackets corresponding to other choices of the function $f(z)$ are new.

Published 1 January 2008