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

• Autores: Michael Gekhtman, K. L. Vaninsky
• Localización: Mathematical research letters, ISSN 1073-2780, Vol. 15, Nº 5-6, 2008, págs. 867-879
• Idioma: inglés
• DOI: 10.4310/mrl.2008.v15.n5.a4
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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.