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We consider a periodic 2-component Camassa–Holm equation with vorticity in the paper. We first give the local well-posedness and the blow-up criterion for strong solutions to the equation in the Sobolev space Hs, . We then present a global existence result for strong solutions to the equation. We finally obtain several blow-up results and the blow-up rate of strong solutions to the equation. We finally examine the propagation behaviour of compactly supported solutions to the equation. The obtained results cover and improve the earlier results for a periodic 2-component Camssa–Holm equation without vorticity.

A new multi-symplectic formulation of the two-component Camassa-Holm equation (2CH) is presented, and the associated local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. Furthermore, this scheme preserves exactly two discrete versions of the Casimir functions of 2CH. Numerical experiments show that the proposed numerical scheme has good conservation properties.

In this Letter we propose that for Lax integrable nonlinear partial differential equations the natural concept of weak solutions is implied by the compatibility condition for the respective distributional Lax pairs. We illustrate our proposal by comparing two concepts of weak solutions of the modified Camassa-Holm equation pointing out that in the peakon sector (a family of non-smooth solitons) only one of them, namely the one obtained from the distributional compatibility condition, supports the time invariance of the Sobolev H1 norm.

In this paper, we study a (2 + 1)-dimensional generalized Camassa-Holm (2dgCH) equation with both quadratic and cubic nonlinearity. We derive a peaked soliton (peakon) solution, double-peakon solutions, and kink-peakon solutions. In particular, weak kink - peakon solution is the first time to address in the 2 + 1-dimensional integrable system.

Here, we give the existence of analytical Cartesian solutions of the multi-component Camassa-Holm (MCCH) equations. Such solutions can be explicitly expressed, in which the velocity function is giv...

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

In this paper, we study the optimal control problem for the viscous modified Camassa–Holm equation. We first prove the existence and uniqueness of a weak solution to this equation in a short interval by using the Galerkin method. Furthermore, the existence of an optimal solution to the viscous modified Camassa–Holm equation is proved.

The μ-Camassa–Holm (μCH) equation is a nonlinear integrable partial differential equation closely related to the Camassa–Holm and the Hunter–Saxton equations. This equation admits quadratic pseudo-potentials which allow us to compute some first-order nonlocal symmetries. The found symmetries preserve the mean of solutions. Finally, we discuss also the associated μCH equation.

A new approach to the analysis of wave-breaking solutions to the generalized CamassaHolm equation is presented in this paper. Introduction of a set of variables allows for solving the singularities. A continuous semigroup of dissipative solutions is also built. The solutions have non-increasing H 1 energy and thus energy loss occurs only through wave breaking.

Second heavenly equation of Plebanski, presented in a two-component form, is known to be a 3 + 1-dimensional multi-Hamiltonian integrable system. We show that one symmetry reduction of this equation yields a two component 2 + 1-dimensionalmulti-Hamiltonian integrable system. For this system, we present Hamiltonian and recursion operators, point symmetries and integrals of motion. For another symmetry reduction, the reduced system is ”almost bi-Hamiltonian”, with two known Hamiltonian operators but the second Hamiltonian density missing.

We give a simple proof that for any non-zero initial data, the solution of the CamassaHolm equation loses instantly the property of being compactly supported.

We survey recent results on well-posedness, blow-up phenomena, lifespan and global existence for the Camassa-Holm equation. Results on weak solutions are also considered.

We give a basic uniqueness theorem in the inverse spectral theory for a Sturm-Liouville equation with a weight which is not of one sign. It is shown that the theorem may be applied to the spectral problem associated with the Camassa-Holm integrable system which models shallow water waves.

We consider the direct/inverse spectral problem for the periodic Camassa-Holm equation. In fact, we survey the direct/inverse spectral problem for the periodic weighted operator Ly = m −1 (−y �� + 1 y) acting in the space L 2 (R ,m (x)dx), where m = uxx− u> 0 is a 1-periodic positive function and u is the solution of the Camassa-Holm equation ut − uxxt +3 uux =2 uxuxx + uuxxx. For the operator L we describe the complete solution of the inverse spectral problem: i) uniqueness, prove that the spectral data uniquely determines the potential, ii) characterization, give conditions for some data to be the spectral data of some potential, iii) reconstruction, give an algorithm for recovering the potential from the spectral data, iv) a priori estimates, obtain two-sided a priori estimates of u, m in terms of gap lengths. Consider the well known Camassa-Holm equation [2]: ut − uxxt +3 uux =2 uxuxx + uuxxx. (1)

We present an approach proving the integrability of the Camassa-Holm equation for initial data of small amplitude.