Discrete Lagrangian systems on the Virasoro group and Camassa-Holm family

We show that the continuous limit of a wide natural class of the right-invariant discrete Lagrangian systems on the Virasoro group gives the family of integrable PDE's containing Camassa-Holm, Hunter-Saxton and Korteweg-de Vries equations. This family has been recently derived by Khesin and Misiolek as Euler equations on the Virasoro algebra for $H^1_{\alpha,\beta}$-metrics. Our result demonstrates a universal nature of these equations.Comment: 6 pages, no figures, AMS-LaTeX. Version 2: minor changes. Version 3: minor change

Two-dimensional algebro-geometric difference operators

A generalized inverse problem for a two-dimensional difference operator is introduced. A new construction of the algebro-geometric difference operators of two types first considered by I.M.Krichever and S.P.Novikov is proposedComment: 11 pages; added references, enlarged introduction, rewritten abstrac

Canonically conjugate variables for the periodic Camassa-Holm equation

The Camassa-Holm shallow water equation is known to be Hamiltonian with respect to two compatible Poisson brackets. A set of conjugate variables is constructed for both brackets using spectral theory.Comment: 10 pages, no figures, LaTeX; v. 2,3: references updated, minor change