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

TL;DR

This paper introduces a novel analytic approach to the Hamiltonian structure of the Camassa--Holm hierarchy, generating a family of compatible Poisson brackets parameterized by an entire function, expanding the known integrable structures.

Contribution

It presents a new method to derive a family of compatible Poisson brackets for the Camassa--Holm hierarchy using Riemann surface and Weyl function techniques.

Findings

Derives explicit formulas for a family of Poisson brackets

Identifies special cases corresponding to rational and trigonometric solutions

Introduces new Poisson brackets beyond known structures

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.