We establish the local well-posedness in H(S) with any s > 72 for a modified Camassa-Holm equation derived as the EPDiff equation with respect to the H(S) metric, and obtain the global existence of the weak solution in H(S) under some sign assumption on the initial values and prove the convergence of the corresponding finite particle approximation method.

The Camassa-Holm equation possesses well-known peaked solitary waves that are called peakons. Their orbital stability has been established by Constantin and Strauss in [6]. We prove here the stability of ordered trains of peakons. We also establish a result on the stability of multipeakons.

We show that the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym hierarchies can be obtained by applying a reduction process to a simple Poisson pair defined on the loop algebra of sl(2,R). The reduction process is a bi-Hamiltonian reduction, that can be canonically performed on every bi-Hamiltonian manifold.

In this paper, a generalized Camassa–Holm equation is studied by using the integral bifurcation method. Many travelling waves such as peaked compacton, compacton, peaked solitary wave, solitary wave and kink-like wave are found. In some parameter conditions, exact parametric representations of these travelling waves in explicit form and implicit form are obtained. 2008 Elsevier Inc. All rights ...

We derive conditions on the initial data, including cases where the initial momentum density is not of one sign, that produce blow-up of the induced solution to the modified integrable Camassa-Holm equation with cubic nonlinearity. The blow-up conditions are formulated in terms of the initial momentum density and the average initial energy.

We investigate the nonhomogeneous initial boundary value problem for the Camassa-Holm equation on an interval. We provide a local in time existence theorem and a weak strong uniqueness result. Next we establish a result on the global asymptotic stabilization problem by means of a boundary feedback law.

We consider a generalized hyperelastic-rod wave equation (or generalized Camassa– Holm equation) describing nonlinear dispersive waves in compressible hyperelastic rods. We establish existence of a strongly continuous semigroup of global weak solutions for any initial data from H1(R). We also present a “weak equals strong”uniqueness result.

We study an equation lying ‘mid-way’ between the periodic HunterSaxton and Camassa-Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and ...