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Equations of Camassa‐Holm type and Jacobi ellipsoidal coordinates
Author(s) -
Vaninsky K. L.
Publication year - 2005
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20089
Subject(s) - mathematics , integrable system , camassa–holm equation , poisson bracket , mathematical analysis , ellipsoid , hamiltonian (control theory) , phase space , string (physics) , pure mathematics , mathematical physics , thermodynamics , mathematical optimization , physics , astronomy , lie algebra
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.