The modified Hunter–Saxton equation

Abstract

We introduce a quadratic pseudo-potential for the Hunter–Saxton equation (HS), as an application of the fact that HS describes pseudo-spherical surfaces. We use it to compute conservation laws and to obtain a full Lie algebra of nonlocal symmetries for HS which contains a semidirect sum of the loop algebra over sl(2,R) and the centerless Virasoro algebra. We also explain how to find families of solutions to HS obtained using our symmetries, and we apply them to the construction of a recursion operator. We then reason by analogy with the theory of the Korteweg–de Vries and Camassa–Holm equations and we define a “modified” Hunter–Saxton (mHS) equation connected with HS via a “Miura transform”. We observe that this new equation describes pseudo-spherical surfaces (and that therefore it is the integrability condition of an sl(2,R)-valued over-determined linear problem), we present two conservation laws, and we solve an initial value problem with Dirichlet boundary conditions. We also point out that our mHS equation plus its corresponding Miura transform are a formal Bäcklund transformation for HS. Thus, our result on existence and uniqueness of solutions really is a rigorous analytic statement on Bäcklund transformations.