2005/06/24 by K. L. Vaninsky · 17 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Algebraic structures and combinatorial models #Nonlinear Photonic Systems #Mathematics #Integrable system #Camassa–Holm equation #Poisson bracket #Mathematical analysis #Ellipsoid #Hamiltonian (control theory) #Phase space #String (physics) #Pure mathematics #Mathematical physics
paper · doi:10.1002/cpa.20089
published in Communications on Pure and Applied Mathematics 58(9), 1149-1187 (Wiley)
openalex publication_date 2005/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15
Abstract We consider the integrable Camassa‐Holm (CH) equation on the line with positive initial data rapidly decaying at infinity. On such a phase space we construct a one‐parameter family of integrable hierarchies that preserves the mixed spectrum of the associated string spectral problem. This family includes the CH hierarchy. We demonstrate that the constructed flows can be interpreted as Hamiltonian flows on the space of Weyl functions of the associated string spectral problem. The corresponding Poisson bracket is the Atiyah‐Hitchin bracket. Using an infinite dimensional version of the Jacobi ellipsoidal coordinates, we obtain a one‐parameter family of canonical coordinates linearizing the flows. © 2005 Wiley Periodicals, Inc.