WELL-POSEDNESS AND PEAKON SOLUTIONS FOR A HIGHER ORDER CAMASSA-HOLM TYPE EQUATION

In this paper,we delve into a generalized higher order Camassa-Holm type equation,(or,an ghmCH equation for short).We establish local well-posedness for this equation under the condition that the initial data uo belongs to the Sobolev space H'(R)for some s>2.In addition,we obtain the weak formulation of this equation and prove the existence of both single peakon solution and a multi-peakon dynamic system.

The classification of travelling wave solutions and superposition of multi-solutions to Camassa-Holm equation with dispersion

Under the travelling wave transformation, the Camassa-Holm equation with dispersion is reduced to an integrable ordinary differential equation （ODE）, whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa-Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa-Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa-Holm equation with dispersion.

ALMOST PERIODIC SOLUTION OF ONE DIMENSIONAL VISCOUS CAMASSA-HOLM EQUATION

This article studies one dimensional viscous Camassa-Holm equation with a periodic boundary condition. The existence of the almost periodic solution is investigated by using the Galerkin method.

GLOBAL ATTRACTOR FOR CAMASSA-HOLM TYPE EQUATIONS WITH DISSIPATIVE TERM

In this paper, the global existence of classical solution and global attractor for Camassa-Holm type equations with dissipative term are established by using fixed point theorem and a priori estimates.

GLOBAL ASYMPTOTICS TOWARD THE REREFACTION WAVES FOR A PARABOLIC-ELLIPTIC SYSTEM RELATED TO THE CAMASSA-HOLM SHALLOW WATER EQUATION

This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut＋（u^2/2）x＋px=εuxx, t〉0,x∈R, -αPxx＋P=f（u）＋α/2ux^2-1/2u^2, t〉0,x∈R, （E） with the initial data u（0,x）=u0（x）→u±, as x→±∞ （I） Here, u_ 〈 u＋ are two constants and f（u） is a sufficiently smooth function satisfying f＂ （u） 〉 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u_ 〈 u＋, the above Riemann problem admits a unique global entropy solution u^R（x/t） u^R（x/t）={u_,（f′）^-1（x/t）,u＋, x≤f′（u_）t, f′（u_）t≤x≤f′（u＋）t, x≥f′（u＋）t. Let U（t, x） be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u0（x） - U（0,x） ∈ H^1（R） and u_ 〈 u＋, the above Cauchy problem （E） and （I） admits a unique global classical solution u（t, x） which tends to the rarefaction wave u^R（x/t） as → ＋∞ in the maximum norm. The proof is given by an elementary energy method.

NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE

In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt ＋ umx ＋ 2ux m = 0, m =（1- δx^2）^2u,u（x, 0） = u0（x） ∈ H^s（R）, x ∈ R, t 〉 0,and show that the solution map is not uniformly continuous in Sobolev spaces H^s（R） for s 〉 7/2. Compared with the periodic problem, the non-periodic problem is more difficult,e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.

Exact Traveling Wave Solutions for Generalized Camassa-Holm Equation by Polynomial Expansion Methods

We formulate efficient polynomial expansion methods and obtain the exact traveling wave solutions for the generalized Camassa-Holm Equation. By the methods, we obtain three types traveling wave solutions for the generalized Camassa-Holm Equation: hyperbolic function traveling wave solutions, trigonometric function traveling wave solutions, and rational function traveling wave solutions. At the same time, we have shown graphical behavior of the traveling wave solutions.

An Approach for Solving Short-Wave Models for Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper, to construct exact solution of nonlinear partial differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived. We investigate the short wave model for the Camassa-Holm equation and the Degasperis-Procesi equation respectively. One-cusp soliton solution of the Camassa-Flolm equation is obtained. One-loop soliton solution of the Degasperis- Procesi equation is also obtained, the approximation of which in a closed form can be obtained firstly by the Adomian decomposition method. The obtained results in a parametric form coincide perfectly with those given in the present reference. This illustrates the efficiency and reliability of our approach.

THE RELATION OF TWO-DIMENSIONAL VISCOUS CAMASSA-HOLM EQUATIONS AND THE NAVIER-STOKES EQUATIONS

In this article, the authors show the existence of global solution of two-dimensional viscous Camassa-Holm (Navier-Stokes-alpha) (NS-α) equations. The authors also prove that the solution of the NS-α equations converges to the solution of the 2D NS equations in the inviscid limit and give the convergence rate of the difference of the solution.

Blow-up of the Solutions of the Initial Boundary Value Problem of Camassa-Holm Equation

Any classical non-null solution to the initial boundary value problem of Camassa-Holm equation on finite interval with homogeneous boundary condition must blow up in finite time. An initial boundary value problem of CamassaHolm equation on half axis is also investigated in this paper. When the initial potential is nonnegative,then the classical solution exists globally; if the derivative of initial data on zero point is nonpositire, then the life span of nonzero solution nmst be finite.

Orbital Stability of Peakons for a Generalized Camassa-Holm Equation

We investigate the orbital stability of the peakons for a generalized Camassa-Holm equation (gCH). Using variable transformation, a planar dynamical system is obtained from the gCH equation. It is shown that the planar system has two heteroclinic cycles which correspond two peakon solutions. We then prove that the peakons for the gCH equation are orbitally stable by using the method of Constantin and Strauss.

Blow-Up for a Periodic Two-Component Camassa-Holm Equation with Generalized Weakly Dissipation

In this paper we study a periodic two-component Camassa-Holm equation with generalized weakly dissipation. The local well-posedness of Cauchy problem is investigated by utilizing Kato’s theorem. The blow-up criteria and the blow-up rate are established by applying monotonicity. Finally, the global existence results for solutions to the Cauchy problem of equation are proved by structuring functions.

A New Analytical Study of Modified Camassa-Holm and Degasperis-Procesi Equations

In this letter, variational homotopy perturbation method (VHPM) has been studied to obtain solitary wave solutions of modified Camassa-Holm and Degasperis-Procesi equations. The results show that the VHPM is suitable for solving nonlinear differential equations with fully nonlinear dispersion term. The travelling wave solution for above equation compared with VIM, HPM, and exact solution. Also, it was shown that the present method is effective, suitable, and reliable for these types of equations.

A Multi-Symplectic Compact Method for the Two-Component Camassa-Holm Equation with Singular Solutions

The two-component Camassa–Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa–Holm equation. Based on its multi-symplectic formulation, the proposed method is derived by the sixth-order compact finite difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization. Numerical experiments finely describe the velocity and density variables in the two-component integrable system and distinctly display the evolvement of the singular solutions. Moreover, the proposed method shows good conservative properties during long-time numerical simulation.

Asymptotic Stability of Singular Solution for Camassa-Holm Equation

The aim of this paper is to study singular dynamics of solutions of Camassa-Holm equation. Based on the semigroup theory of linear operators and Banach contraction mapping principle, we prove the asymptotic stability of the explicit singular solution of Camassa-Holm equation.

On a New Three-Component Camassa Holm Equation with Peakons

A new three-component Camassa-Holm equation is introduced. This system is endowed with a structuresimilar to the Camassa-Holm equation. It has peakon solitons and conserves H^1-norm conservation law.

Analytical solutions of simplified modified Camassa-Holm equation with conformable and M-truncated derivatives:A comparative study

This paper extracts some analytical solutions of simplified modified Camassa-Holm(SMCH)equations with various derivative operators,namely conformable and M-truncated derivatives that have been recently introduced.The SMCH equation is used to model the unidirectional propagation of shallowwater waves.The extended rational sine−cosine and sinh−cosh techniques have been successfully implemented to the considered equations and some kinds of the solitons such as kink and singular have been derived.We have checked that all obtained solutions satisfy the main equations by using a computer algebraic system.Furthermore,some 2D and 3D graphical illustrations of the obtained solutions have been presented.The effect of the parameters in the solutions on the wave propagation has been examined and all figures have been interpreted.The derived solutions may contribute to comprehending wave propagation in shallow water.So,the solutions might help further studies in the development of autonomous ships/underwater vehicles and coastal zone management,which are critical topics in the ocean and coastal engineering.

Equal-Time and Equal-Space Poisson Brackets of the N-Component Coupled NLS Equation

Two Poisson brackets for the N-component coupled nonlinear Schrdinger(NLS) equation are derived by using the variantional principle. The first one is called the equal-time Poisson bracket which does not depend on time but only on the space variable. Actually it is just the usual one describing the time evolution of system in the traditional theory of integrable Hamiltonian systems. The second one is equal-space and new. It is shown that the spatial part of Lax pair with respect to the equal-time Poisson bracket and temporal part of Lax pair with respect to the equal-space Poisson bracket share the same r-matrix formulation. These properties are similar to that of the NLS equation.

Explicit Peaked Wave Solutions to the Generalized Camassa-Holm Equation

By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.

The Blow-Up Phenomena for the Camassa-Holm Equation with a Zero Order Dissipation

In this paper, we study the Cauchy problem of the Camassa-Holm equation with a zero order dissipation. We establish local well-posedness and investigate the blow-up phenomena for the equation.