Weak geodesic flow on a semi-direct product and global solutions to the periodic Hunter-Saxton system

TL;DR

This paper provides explicit periodic solutions to the two-component Hunter-Saxton system by interpreting it as a geodesic flow on a semi-direct product, enabling the construction of global weak solutions.

Contribution

It introduces a geometric approach to construct global weak solutions for the Hunter-Saxton system using semi-direct product structures.

Findings

Explicit periodic solutions to the Hunter-Saxton system.

Construction of global weak solutions via geometric interpretation.

Solutions are conservative and spatially/temporally periodic.

Abstract

We give explicit solutions to the two-component Hunter-Saxton system on the unit circle. Moreover, we show how global weak solutions can be naturally constructed using the geometric interpretation of this system as a re-expression of the geodesic flow on the semi-direct product of a suitable subgroup of the diffeomorphism group of the circle with the space of smooth functions on the circle. These spatially and temporally periodic solutions turn out to be conservative.