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

Abstract

Abstract. We study an equation lying 'mid-way' between the periodic HunterSaxton 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.

Nonlocal symmetries and conservation laws of nonlocal Camassa-Holm type equations

“…Interestingly, it also can be derived by the tri-Hamiltonian duality approach due to Olver and Rosenau [17] by using the bi-Hamiltonian structures of the KdV equation. If we set m = −u xx in (1.1), it reduces to the Hunter-Saxton equation [15] , which is also integrable. If we set m = μ(u) − u xx , where μ(u) = 1 0 u(t, x) dx, equation (1.1) becomes the so-called μ-CH equation, which was introduced by Khesin, Lenels and Misiolek [1] , being regraded as a nonlocal version of the CH equation.…”

Conservative weak solutions of the periodic Cauchy problem for μHS equation

“…10 Note that the presence of the nonlocal expression µ(u) in (1) does not prevent it from being a well-defined partial differential equation (PDE) rather than a mixed integro-differential equation since µ(u) is constant in time and is completely determined by the initial condition.…”

“…(1) is "integrable" in the following sense: it has a Lax pair representation, a bi-Hamiltonian structure, and peaked soliton (peakon) solutions. The Lax pair for (1) 10 is…”

A sign-changing Liouville equation

“…This has made both versions of (1.2) a frequent topic of investigation. In particular cases, the equation can be interpreted as a model for a selfinteracting scalar field in two-dimensional space-time, whose properties have been the subject of extensive study (see for instance [2][3][4][5][6][7][8][9][10][11]). Equation (1.1)i) is obtained from (1.2) on changing to characteristic coordinates τ = α + t and x = α − t and…”