Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms

Abstract

We study an equation lying `mid-way' between the periodic Hunter-Saxton and Camassa-Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped, as well as smooth, traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three.

arXiv comment30 pages, 2 figures

References in corpus (2)

• An integrable shallow water equation with peaked solitons
• A singular integrable equation from short capillary-gravity waves

Cited by in corpus (5)

• Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle
• Geometric hydrodynamics and infinite-dimensional Newton's equations
• On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations
• The curvature of semidirect product groups associated with two-component Hunter-Saxton systems
• The periodic $\mu$-$b$-equation and Euler equations on the circle