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

• Published in: Mathematische annalen Vol. 342; no. 3; pp. 617 - 656
• Main Authors: Khesin, Boris, Lenells, Jonatan, Misiołek, Gerard
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-11-2008
• Subjects: Mathematics; Mathematics and Statistics; Soliton; Solitary Wave; Sectional Curvature; Bihamiltonian Structure; Shallow Water Equation
• ISSN: 0025-5831; 1432-1807; 1432-1807
• DOI: 10.1007/s00208-008-0250-3

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 and 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.