On a Camassa-Holm type equation with two dependent variables

We consider a generalization of the Camassa Holm (CH) equation with two dependent variables, called CH2, introduced by Liu and Zhang. We briefly provide an alternative derivation of it based on the theory of Hamiltonian structures on (the dual of) a Lie Algebra. The Lie Algebra here involved is the same algebra underlying the NLS hierarchy. We study the structural properties of the CH2 hierarchy within the bihamiltonian theory of integrable PDEs, and provide its Lax representation. Then we explicitly discuss how to construct classes of solutions, both of peakon and of algebro-geometrical type. We finally sketch the construction of a class of singular solutions, defined by setting to zero one of the two dependent variables.Comment: 22 pages, 2 figures. A few typos correcte

Non-Analytic Solutions of Nonlinear Wave Models

This paper surveys recent work of the coauthors on nonanalytic solutions to nonlinear wave models. We demonstrate the connection between nonlinear dispersion and the existence of a remarkable variety of nonclassical solutions, including peakons, compactons, cuspons, and others. Nonanalyticity can only occur at points of genuine nonlinearity, where the symbol of the partial differential equation degenerates, and thereby provide singularities in the associated dynamical system for traveling waves. We propose the term "pseudo-classical" to characterize such solutions, and indicate how they are recovered as limits of classical analytic solutions