In this paper we take up the question of a small dispersion limit for the Camassa–Holm equation. The particular limit we study involves a modification of the Camassa–Holm equation, seen in the recent theoretical developments by Himonas and Misiołek, as well as the first author, where well-posedness is proved in weak Sobolev spaces. This work led naturally to the question of how solutions actual...

Geometric integrators are presented for a class of nonlinear dispersive equations which includes the Camassa-Holm equation, the BBM equation and the hyperelasticrod wave equation. One group of schemes is designed to preserve a global property of the equations: the conservation of energy; while the other one preserves a more local feature of the equations: the multi-symplecticity.

In this paper we study finite deformations in a pre-stressed, hyperelastic compressible plate. For small-amplitude nonlinear waves, we obtain equations that involve an amplitude parameter ε. Using an asymptotic perturbation technique, we derive a new family of two-dimensional nonlinear dispersive equations. This family includes the KdV, Kadomtsev-Petviashvili and Camassa-Holm equations.

Zero curvature formulations, pseudo-potentials, modified versions, “Miura transformations”, and nonlocal symmetries of the Korteweg–de Vries, Camassa–Holm and Hunter–Saxton equations are investigated from an unified point of view: these three equations belong to a two–parameters family of equations “describing pseudo-spherical surfaces”, and therefore their basic integrability properties can be...

Peaked periodic waves in the Camassa--Holm equation are revisited. Linearized evolution equations derived for perturbations to peaked waves, and linearized instability is proven in...

where α, γ, ω are given real constants. Equation (1) was first introduced as a model describing propagation of unidirectional gravitational waves in a shallow water approximation over a flat bottom, with u representing the fluid velocity [DGH01]. For α = 0 and for α = 1, γ = 0 we obtain the Korteweg–de Vries and the Camassa–Holm [CH93, J02] equations, respectively. Both of them describe unidire...

The μ-Camassa–Holm (μCH) equation is a nonlinear integrable partial differential equation closely related to the Camassa–Holm equation. We prove that the periodic peaked traveling wave solutions (peakons) of the μCH equation are orbitally stable.

This paper mainly proves the generic properties of the Camassa-Holm equation and the two-component Camassa-Holm equation by Thom’s transversality Lemma. We reveal their differences in generic regularity and singular behavior.

It was recently proven by De Lellis, Kappeler, and Topalov in [17] that the periodic Cauchy problem for the Camassa-Holm equations is locally well-posed in the space Lip(T) endowed with the topology of H(T). We prove here that the Lagrangian flow of these solutions are analytic with respect to time and smooth with respect to the initial data. These results can be adapted to the higher-order Cam...

We consider the viscous n-dimensional Camassa-Holm equations, with n = 2, 3, 4 in the whole space. We establish existence and regularity of the solutions and study the large time behavior of the solutions in several Sobolev spaces. We first show that if the data is only in L then the solution decays without a rate and that this is the best that can be expected for data in L. For solutions with ...