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.

Equations of Camassa-Holm type and Jacobi ellipsoidal coordinates

A number of the equations in classical mechanics is integrable in Jacobi elliptic coordinates. In recent years a generalization of elliptic coordinates to the infinite case has been offered. We consider this generalization and indicate the connection of infinite-dimensional elliptic coordinates with some inverse spectral problems for infinite Jacobi matrices and sturm-Liouville operators (two spectra inverse problems). Also a link with the rank one perturbation theory is shown.

On Some Properties of Infinite-dimensional Elliptic Coordinates

In 1857 George Boole found an identity for a class of rational functions, which for a given function, connects the sum of its residues at finite points with the difference between the sums of its zeros and poles. We consider generalizations of this identity to the case of Nevanlinna functions. We apply these results to obtain some new trace formulas for infinite Jacobi matrices and for differential operators of the second order.

On a G. Boole’s Identity for Rational Functions and Some Trace Formulas

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A. G. Kostyuchenko

Papers

Infinite-dimensional elliptic coordinates