A Three-component Extension of the Camassa–Holm Hierarchy

We introduce a bi-Hamiltonian hierarchy on the loop-algebra of endowed with a suitable Poisson pair. It gives rise to the usual Camassa–Holm (CH) hierarchy by means of a bi-Hamiltonian reduction, and its first nontrivial flow provides a three-component extension of the CH equation.     

Higher Dimensional Camassa–Holm Equations

Utilizing some conservation laws of the(1+1)-dimensional Camassa–Holm(CH) equation and/or its reciprocal forms, some(n+1)-dimensional CH equations for n ≥ 1 are constructed by a modified deformation algorithm.The Lax integrability can be proven by applying the same deformation algorithm to the Lax pair of the(1+1)-dimensional CH equation. A novel type of peakon solution is implicitly given and expressed by the Lambert W function.     

A multi-symplectic numerical integrator for the two-component Camassa–Holm equation

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.     

Geometric Integrability of the Camassa–Holm Equation

It is observed that the Camassa–Holm equation describes pseudo-spherical surfaces and that therefore, its integrability properties can be studied by geometrical means. An sl(2, R)-valued linear problem whose integrability condition is the Camassa–Holm equation is presented, a Miura transform and a modified Camassa–Holm equation are introduced, and conservation laws for the Camassa–Holm equation are then directly constructed. Finally, it is pointed out that this equation possesses a nonlocal symmetry, and its flow is explicitly computed.     

Integrable generalization of the associated Camassa–Holm equation

In this paper, we study an integrable generalization of the associated Camassa–Holm equation. The generalized system is shown to be integrable in the sense of Lax pair and the bilinear Bäcklund transformations are presented through the Bell polynomial technique. Meanwhile, its infinite conservation laws are constructed, and conserved densities and fluxes are given in explicit recursion formulas. Furthermore, a Darboux transformation for the system is derived with the help of the gauge transformation between two Lax pairs. As an application, soliton and periodic wave solutions are given through the Darboux transformation.     

Dual Hierarchies of a Multi-Component Camassa–Holm System

In this paper, we derive the bi-Hamiltonian structure of a multi-component Camassa–Holm system, which associates with the multi-component AKNS hierarchy and multi-component KN hierarchy via the tri-Hamiltonian duality method. Furthermore, the spectral problems of the dual hierarchies may be obtained.     

Reciprocal Transformations of Two Camassa–Holm Type Equations

The relation between the Camassa–Holm equation and the Olver–Rosenau–Qiao equation is obtained,and we connect a new Camassa–Holm type equation proposed by Qiao etc. with the first negative flow of the Kd V hierarchy by a reciprocal transformation.     

Estimates for the KdV-Limit of the Camassa–Holm Equation

In a number of scaling limits, we prove estimates relating the solutions of the Camassa–Holm equation to the solutions of the associated KdV equation. As a consequence, suitable solutions of the water wave problem and solutions of the Camassa–Holm equation stay close together for long times.     

Wave-Breaking and Peakons for a Modified Camassa–Holm Equation

In this paper, we investigate the formation of singularities and the existence of peaked traveling-wave solutions for a modified Camassa-Holm equation with cubic nonlinearity. The equation is known to be integrable, and is shown to admit a single peaked soliton and multi-peakon solutions, of a different character than those of the Camassa-Holm equation. Singularities of the solutions can occur only in the form of wave-breaking, and a new wave-breaking mechanism for solutions with certain initial profiles is described in detail.     

Multi-soliton solutions of a two-component Camassa–Holm system:Darboux transformation approach

We propose and develop another approach to constructing multi-soliton solutions of an integrable two-component Camassa–Holm(CH2)system.With the help of a reciprocal transformation and a gauge transformation,we relate the CH2 system to a negative flow of the Broer–Kaup or twoboson hierarchy.The solutions of this negative flow are given in terms of Wronskians via Darboux transformation.Then the multi-soliton solutions of the CH2 system are recovered in parametric form by inverting the reciprocal transformation and the gauge transformation.     

Multiply Kink and Anti-Kink Solutions for a Coupled Camassa–Holm Type Equation

This article amis at revealing dynamical behavior of a coupled Camassa–Holm type equation, which was proposed by Geng and Wang based on a 4×4 matrix spectral problem with two potentials. Its kink and anti-kink solutions are presented explicitly. In particular, some exact multi-kink and anti-kink wave solutions are discussed and under some conditions, the kink and anti-kinks look like hat-shape solitons. The dynamic characters of the obtained solutions are investigated by figures. The method used in this paper can be widely applied to looking for the multi-kinks for Camassa–Holm type equations possessing cubic nonlinearity.     

An Isospectral Problem for Global Conservative Multi-Peakon Solutions of the Camassa–Holm Equation

We introduce a generalized isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation. Utilizing the solution of the indefinite moment problem given by M. G. Krein and H. Langer, we show that the conservative Camassa–Holm equation is integrable by the inverse spectral transform in the multi-peakon case.     

Dual Hierarchies of a Multi-Component Camassa——Holm System

In this paper, we derive the bi-Hamiltonian structure of a multi-component Camassa-Holm system, which associates with the multi-component AKNS hierarchy and multi-component KN hierarchy via the tri-Hamiltonian duality method. Furthermore, the spectral problems of the dual hierarchies may be obtained.     

The classification of single travelling wave solutions to the Camassa–Holm–Degasperis–Procesi equation for some values of the convective parameter

By the complete discrimination system for the polynomial, we give the classification of single travelling wave solutions to the Camassa–Holm–Degasperis–Procesi equation for some values of the convective parameter.     

On the development of a high-order compact scheme for exhibiting the switching and dissipative solution natures in the Camassa–Holm equation

In this paper a two-step iterative solution algorithm for solving the Camassa–Holm equation, which involves only the first-order derivative term, is presented. In each set of the u − P and u − m differential equations, one is governed by the inviscid nonlinear convection–reaction equation for the time-evolving fluid velocity component along the horizontal direction. The other equation is known as the inhomogeneous Helmholtz equation. The resulting reduction of differential order facilitates us to develop the flux discretization scheme in a stencil with comparatively fewer points. For accurately predicting the unidirectional propagation of the shallow water wave, the modified equation analysis for eliminating several leading discretization error terms and the Fourier analysis for minimizing a particular type of wave-like error are employed. In this study, the fifth-order spatially accurate combined compact upwind scheme is developed in a three-point stencil for approximating the first-order derivative term. For the purpose of retaining a long-term accurate Hamiltonian and multi-symplectic geometric structures in Camassa–Holm equation, the time integrator (or time-stepping scheme) chosen in this study should conserve symplecticity. Another main emphasis of conducting the present calculation of Camassa–Holm equation is to shed light on the conservation of Hamiltonians up to the time before wave breaking. We also intended to elucidate the switching scenario by virtue of the peakon–peakon interaction problem and the dissipative scenario after the time of head-on collision in the peakon–antipeakon interaction problem.     

Peakon–antipeakon interaction in the Dullin–Gottwald–Holm equation

In this Letter, we obtain a two-peakon solution to a special Dullin–Gottwald–Holm equation explicitly by direct computation, and then discuss the peakon–antipeakon interaction in this equation. Our results show that, like the Camassa–Holm equation, during the soliton absorption time, the peakon–antipeakon in this equation is essentially a single-peaked, double-cornered wave. However, the two corners may travel in opposite direction for some ω, which is different from that in the Camassa–Holm equation.