Lagrangian time-discretization of the Hunter-Saxton equation

TL;DR

This paper develops Lagrangian time-discretizations of the Hunter-Saxton equation using the Moser-Veselov approach, exploring their properties on the Virasoro group and diffeomorphisms, and conjectures integrability of one discretization.

Contribution

It introduces novel Lagrangian discretizations of the Hunter-Saxton equation on important geometric groups, proposing their potential integrability.

Findings

Discretizations defined on the Virasoro group and diffeomorphisms.

Conjecture that one discretization is integrable.

Framework for future analysis of discretized Hunter-Saxton equations.

Abstract

We study Lagrangian time-discretizations of the Hunter-Saxton equation. Using the Moser-Veselov approach, we obtain such discretizations defined on the Virasoro group and on the group of orientation-preserving diffeomorphisms of the circle. We conjecture that one of these discretizations is integrable.