The purpose of this paper is to study the dynamics of the interaction among a special class of solutions of the one-dimensional Camassa-Holm equation. The equation yields soliton solutions whose identity is preserved through nonlinear interactions. These solutions are characterized by a discontinuity at the peak in the wave shape and are thus called peakon solutions. We apply a particle method ...

Using Grozman’s formalism of invariant differential operators we demonstrate the derivation of N = 2 Camassa-Holm equation from the action of V ect(S1|2) on the space of pseudo-differential symbols. We also use generalized logarithmic 2-cocycles to derive N = 2 super KdV equations. We show this method is equally effective to derive Camassa-Holm family of equations and these system of equations ...

There is a deep connection between solutions of nonlinear equations and both geodesic flows and billiards on Riemannian manifolds. For example, in Alber, Camassa, Holm and Marsden [1995], a link between umbilic geodesics and billiards on n-dimensional quadrics and new soliton-like solutions of nonlinear equations in the Dym hierarchy was investigated. The geodesic flows provide the spatial x-fl...

We derive the modulation equations or Whitham equations for the Camassa– Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir...

The generalized Camassa-Holm equation ut + 2kux − uxxt + auux = 2uxuxx + uuxxx + γuxxx is considered in this paper. Under traveling wave variable substitution, the equation is related to a planar singular system. By making a transformation this singular system becomes a regular system. Through discussing the dynamical behavior of the regular system, the explicit periodic blow-up solutions and s...

Considered herein are the generalized Camassa-Holm and Degasperis-Procesi equations in the spatially periodic setting. The precise blow-up scenarios of strong solutions are derived for both of equations. Several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established. The exact blow-up rates are also...

A method for symbolically computing conservation laws of nonlinear partial differential equations (PDEs) in multiple space dimensions is presented in the language of variational calculus and linear algebra. The steps of the method are illustrated using the ZakharovKuznetsov and Kadomtsev-Petviashvili equations as examples. The method is algorithmic and has been implemented in Mathematica. The s...