Abstract Camassa-Holm type equations arise as models for the unidirectional propagation of shallow water waves over a flat bottom. They also describe finite length, small amplitude radial deformation in cylindrical compressible hyperelastic rods. Under appropriate assumption on initial data, time T T , and coefficients...

We derive the precise stability criterion for smooth solitary waves in b-family of Camassa–Holm equations. The exist on constant background. In integrable cases b=2 and b=3, we show analytically that is satisfied are orbitally stable with respect to perturbations H3(R). non-integrable cases, numerically asymptotically every b>1. orbital theory relies a Hamiltonian formulation which different fr...

We consider a three-parameter family of non-linear equations with (p+1)-order non-linearities. Such includes as particular member the well-known b-equation, which encloses famous Camassa–Holm equation. For certain choices parameters, we establish global existence result and show scenario that prevents wave breaking solutions. Also, explore unique continuation properties for some values paramete...

We consider a new partial differential equation, of a similar form to the Camassa-Holm shallow water wave equation, which was recently obtained by Degasperis and Procesi using the method of asymptotic integrability. We prove the exact integrability of the new equation by constructing its Lax pair, and we explain its connection with a negative flow in the Kaup-Kupershmidt hierarchy via a recipro...

on the interval [0, 2π]. Spectral discretizations of this equation have been in use ever since the work of Camassa and Holm [2] and Camassa, Holm and Hyman [3]. However, to the knowledge of the authors, no proof that such a discretization actually converges has appeared heretofore. Therefore, this issue is taken up here. Our method of proof is related to the work of Maday and Quarteroni on the ...

is an extensively studied nonlinear equation. It first appeared as an abstract biHamiltonian partial differential equation in an article of Fuchssteiner and Fokas [39] but did not receive much attention until Camassa and Holm [15] (see also [16]) derived it as a nonlinear wave equation that models unidirectional wave propagation on shallow water and discovered its rich mathematical structure. I...

The Degasperis-Procesi equation can be derived as a member of a oneparameter family of asymptotic shallow water approximations to the Euler equations with the same asymptotic accuracy as that of the CamassaHolm equation. It is noted that the Degasperis-Procesi equation, unlike the Camassa-Holm equation, has not only peakon solitons but also shock peakons. In this paper, we study the orbital sta...

It is shown that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time. In particular, a strong solution of the Cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution. This work has...

A recently proposed three-component Camassa-Holm equation is considered. It shown that this system a bi-Hamiltonian system.