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# Mathematical Research Letters

## Volume 15 (2008)

### Number 5

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

Pages: 867 – 879

DOI: http://dx.doi.org/10.4310/MRL.2008.v15.n5.a4

#### Authors

#### Abstract

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.