In this paper we discuss pseudo-peakons, a new class of weak solutions found in the study of higher order equations of Camassa-Holm (CH) type. A pseudo-peakon is a weak bounded solution with differentiable first derivative and continuous and bounded second derivative, but such that any higher order derivative blows up. We recall that pseudo-peakons appear if we change the momentum m appearing in the Camassa-Holm equation from m=(1−∂x2)u to m=(1−α∂x2)(1−β∂x2)u (α and β are two real parameters). Here we note that they also appear if we look for “geometrically integrable” fifth order equations, that is, for equations describing one-parametric families of pseudo-spherical surfaces in a sense explained in Section 1.
