On the Camassa–Holm and Hunter–Saxton equations

Abstract

We survey recent results for the Camassa–Holm equation ut − uxxt + 2κux + 3uux − 2uxuxx − uuxxx = 0, in particular convergence of a carefully selected finite difference scheme in the case of periodic initial data, and a detailed description of algebro-geometric solutions of the Camassa–Holm hierarchy. Furthermore, we present results for the generalized hyperelastic-rod wave equation ut − uxxt + ½ g(u)x = γ(2uxuxx + uuxxx). Finally, we discuss convergence of finite difference schemes for the Hunter–Saxton equation (ut + uux)x = ½ (ux)2 and describe semi-discrete, implicit as well as explicit upwind schemes that converge to diffusive solutions of the Hunter–Saxton equation.