We show that the continuous limit of a wide natural class of the right-invariant discrete Lagrangian systems on the Virasoro group gives the family of integrable PDE’s containing Camassa-Holm, Hunter-Saxton and Korteweg-de Vries equations. This family has been recently derived by Khesin and Misio-lek as Euler equations on the Virasoro algebra for H1 α,βmetrics. Our result demonstrates a univers...

Abstract We present an overview of some contributions the author regarding Camassa–Holm type equations. show that equation unifying both and Novikov equations can be derived using invariance under certain suitable scaling, conservation Sobolev norm existence peakon solutions. Qualitative analysis two-peakon dynamics is given.

We show how the change from Eulerian to Lagrangian coordinates for the two-component Camassa–Holm system can be understood in terms of certain reparametrizations of the underlying isospectral problem. The respective coordinates correspond to different normalizations of an associated first order system. In particular, we will see that the two-component Camassa– Holm system in Lagrangian variable...

Article history: Received 4 April 2008 Revised 11 January 2009 Available online 28 February 2009

We unify a few of the best known results on wave breaking for the Camassa– Holm equation (by R. Camassa, A. Constantin, J. Escher, L. Holm, J. Hyman and others) in a single theorem: a sufficient condition for the breakdown is that u′0 + |u0| is strictly negative in at least one point x0 ∈ R. Such blowup criterion looks more natural than the previous ones, as the condition on the initial data is...

Integrable equations with second order Lax pair like KdV and CamassaHolm (CH) exhibit interesting conformal properties and can be written in terms of the so-called conformal invariants (Schwarz form). These properties for the CH hierarchy are discussed in this contribution. The squared eigenfunctions of the spectral problem, associated to the Camassa-Holm equation represent a complete basis of ...

We present a method for the classification of all weak travelling-wave solutions for some dispersive nonlinear wave equations. When applied to the Camassa-Holm or the Degasperis-Procesi equation, the approach shows the existence of not only smooth, peaked and cusped travelling-wave solutions, but also more exotic solutions with fractal-like wave profiles.

We consider higher-order Camassa–Holm equations describing exponential curves of the manifold of smooth orientation preserving diffeomorphisms of the unit circle in the plane. We establish the existence of a strongly continuous semigroup of global weak solutions. We also present some invariant spaces under the action of that semigroup. Moreover, we prove a “weak equals strong” uniqueness result.

We study the existence and stability of periodic travelling-wave solutions for generalized Benjamin-Bona-Mahony and Camassa-Holm equations. To prove orbital stability, we use the abstract results of Grillakis-Shatah-Strauss and the Floquet theory for periodic eigenvalue problems. Mathematics Subject Classification: 35B10, 35Q35, 35Q53, 35B25, 34C08, 34L40

The squared eigenfunctions of the spectral problem associated with the CamassaHolm (CH) equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the CH hierarchy as a generalized Fourier transform (GFT). All the fundamental properties of the CH equation, such as the integrals of motion, the description of the equations of the whole hierarchy...