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

Details The family of analytic Poisson brackets for the Camassa--Holm hierarchy The family of analytic Poisson brackets for the Camassa--Holm hierarchy

2007-08-27

arxiv.org/abs/0708.3681v1

Math. Res. Lett. 15 (2008), no. 5, 867-879

Physics

Mathematical Physics

15 pages, 3 figures

Scientific paper

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.

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