On the Camassa-Holm and Hunter-Saxton equations

We survey recent results for the Camassa-Holm equation u(t) - u(xx)t + 2 kappa u(x) + 3uu(x) - 2u(x)u(xx) - uu(xxx) = 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 u(t) -u(xxt) + 1/2 g(u)(x) = gamma(2u(x)u(xx) + uu(xxx)). Finally, we discuss convergence of finite difference schemes for the Hunter-Saxton equation (u(t) + uu(x))(x) = 1/2 (u(x))(2) and describe semi-discrete, implicit as well as explicit upwind schemes that converge to diffusive solutions of the Hunter-Saxton equation.