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

Abstract

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 $H_{\alpha ,\beta }^1$-metrics. Our result demonstrates a universal nature of these equations.

6 pages, no figures, AMS-LaTeX. Version 2: minor changes. Version 3: minor changes

References in corpus (2)

• An integrable shallow water equation with peaked solitons
• Lagrangian time-discretization of the Hunter-Saxton equation