The Hunter–Saxton Equation: A Geometric Approach

Abstract

We provide a rigorous foundation for the geometric interpretation of the Hunter–Saxton equation as the equation describing the geodesic flow of the $\dot{H}^1$ right-invariant metric on the quotient space $Rot(\mathbb{S})\backslash\mathcal{D}^k(\mathbb{S})$ of the infinite-dimensional Banach manifold $\mathcal{D}^k(\mathbb{S})$ of orientation-preserving $H^k$-diffeomorphisms of the unit circle $\mathbb{S}$ modulo the subgroup of rotations $Rot(\mathbb{S})$. Once the underlying Riemannian structure has been established, the method of characteristics is used to derive explicit formulas for the geodesics corresponding to the $\dot{H}^1$ right-invariant metric, yielding, in particular, new explicit expressions for the spatially periodic solutions of the initial-value problem for the Hunter–Saxton equation.