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

TL;DR

This paper demonstrates that many well-known integrable PDEs, including Camassa-Holm and Korteweg-de Vries, emerge as continuous limits of discrete Lagrangian systems on the Virasoro group, highlighting their universal nature.

Contribution

It shows that a broad class of discrete Lagrangian systems on the Virasoro group converges to integrable PDEs, unifying their geometric origins.

Findings

Derivation of integrable PDEs as limits of discrete systems

Identification of universal geometric structures underlying these equations

Connection to recent work on Euler equations on the Virasoro algebra

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