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-Holm 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.     

Analytic Approximations for Soliton Solutions of Short-Wave Models for Camassa-Holm and Degasperis-Procesi Equations

In this paper, the short-wave model equations are investigated, which are associated with the Camassa- Holm （CH） and Degasperis Procesi （DP） shallow-water wave equations. Firstly, 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. Secondly, the equation is solved by homotopy analysis method. Lastly, by the transformatioas back to the original independent variables, the solution of the original partial differential equation is obtained. The two types of solutions of the short-wave models are obtained in parametric form, one is one-cusp soliton for the CH equation while the other one is one-loop soliton for the DP equation. The approximate analytic solutions expressed by a series of exponential functions agree well with the exact solutions. It demonstrates the validity and great potential of homotopy analysis method for complicated nonlinear solitary wave problems.     

Novel approach for modified forms of Camassa-Holm and Degasperis-Procesi equations using fractional operator

In the present study, we consider the q-homotopy analysis transform method to find the solution for modified Camassa-Holm and Degasperis-Procesi equations using the Caputo fractional operator. Both the considered equations are nonlinear and exemplify shallow water behaviour. We present the solution procedure for the fractional operator and the projected solution procedure gives a rapidly convergent series solution. The solution behaviour is demonstrated as compared with the exact solution and the response is plotted in 2D plots for a diverse fractional-order achieved by the Caputo derivative to show the importance of incorporating the generalised concept. The accuracy of the considered method is illustrated with available results in the numerical simulation. The convergence providence of the achieved solution is established in $\hslash $-curves for a distinct arbitrary order. Moreover, some simulations and the important nature of the considered model, with the help of obtained results, shows the efficiency of the considered fractional operator and algorithm, while examining the nonlinear equations describing real-world problems.     

An Integrable Matrix Camassa-Holm Equation

We present an integrable sl(2)-matrix Camassa-Holm(CH) equation.The integrability means that the equation possesses zero-curvature representation and infinitely many conservation laws.This equation includes two undetermined functions,which satisfy a system of constraint conditions and may be reduced to a lot of known multicomponent peakon equations.We find a method to construct constraint condition and thus obtain many novel matrix CH equations.For the trivial reduction matrix CH equation we construct its N-peakon solutions.     

Exact Solutions to Degasperis-Procesi Equation

In this paper, dependent and independent variable transformations are introduced to solve the Degasperis-Procesi equation. It is shown that different kinds of solutions can be obtained to the Degasperis-Procesi equation.     

Peakon Solutions to Supersymmetric Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper,the supersymmetric Camassa-Holm equation and Degasperis-Procesi equation are derived from a general superfield equations by choosing different parameters.Their peakon-type solutions are shown in weak sense.At the same time,the dynamic behaviors are analyzed particularly when the two peakons collide elastically,and some results are compared with each other between the two equations.     

Compacton-Like Solutions in a Camassa-Holm Type Equation

This paper shows that a Camassa-Holm type equation can be reduced to Hamiltonian system by transformation of variables. The hamiltonian system is studied by making use of the dynamical systems theory and the qualitative behavior of degenerate singular points is presented. In particular, new type of compacton-like solutions is obtained by setting the partial differential equation under boundary condition →±∞ψ(ξ)=A.     

Adomian Decomposition Method for Nonlinear Differential-Difference Equation

Adomian decomposition method is applied to find the analytical and numerical solutions for the discretized mKdV equation. A numerical scheme is proposed to solve the long-time behavior of the discretized mKdV equation. The procedure presented here can be used to solve other differential-difference equations.     

Peaked Traveling Wave Solutions to a Generalized Novikov Equation with Cubic and Quadratic Nonlinearities

The Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation are the three typical integrable evolution equations admitting peaked solitons. In this paper, a generalized Novikov equation with cubic and quadratic nonlinearities is studied, which is regarded as a generalization of these three well-known studied equations. It is shown that this equation admits single peaked traveling wave solutions, periodic peaked traveling wave solutions, and multi-peaked traveling wave solutions.     

Numerical Solutions of Fractional Boussinesq Equation

Based upon the Adomian decomposition method, a scheme is developed to obtain numerical solutions of a fractional Boussinesq equation with initial condition, which is introduced by replacing some order time and space derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the traditional Adomian decomposition method for differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional differential equations. The solutions of our model equation are calculated in the form of convergent series with easily computable components.     

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 structure similar to the Camassa Holm equation. It has peakon solitons and conserves H^1-norm conservation law.     

The limiting behavior of smooth periodic waves for the Degasperis-Procesi equation

We study the limiting behavior of smooth periodic waves for the Degasperis-Procesi equation as the parameters trend to some special values. Using an improved qualitative method, the existence of smooth periodic waves is considered. For some limiting values of parameters of smooth periodic waves, one obtains peaked solitary waves.     

Reciprocal link for a three-component Camassa-Holm type equation

A reciprocal transformation is introduced for a three-component Camassa-Holm type equation and it is showed that the transformed system is a reduction of the first negative flow in a generalized MKdV hierarchy.     

Multi-Peakon Solutions for Two New Coupled Camassa-Holm Equations

We study the multi-peakon solutions for two new coupled Camassa-Holm equations, which include two-component and three-component Camassa-Holm equations. These multi-peakon solutions are shown in weak sense. In particular, the double peakon solutions of both equations are investigated in detail. At the same time, the dynamic behaviors of three types double peakon solutions are analyzed by some figures.     

Numerical Complexiton Solutions of Complex KdV Equation

In this paper, we directly extend the applications of the Adomian decomposition method to investigate the complex KdV equation. By choosing different forms of wave functions as the initial values, three new types of realistic numerical solutions： numerical positon, negaton solution, and particularly the numerical analytical complexiton solution are obtained, which can rapidly converge to the exact ones obtained by Lou et al. Numerical simulation figures are used to illustrate the efficiency and accuracy of the proposed method.     

Numerical Complexiton Solutions of Complex KdV Equation

In this paper, we directly extend the applications of the Adomian decomposition method to investigate the complex KdV equation. By choosing different forms of wave functions as the initial values, three new types of realistic numerical solutions: numerical positon, negaton solution, and particularly the numerical analytical complexiton solution are obtained, which can rapidly converge to the exact ones obtained by Lou et al. Numerical simulation figures are used to illustrate the efficiency and accuracy of the proposed method.     

On Camassa-Holm Equation with Self-Consistent Sources and Its Solutions

Regarded as the integrable generalization of Camassa-Holm （CH） equation, the CH equation with self-consistent sources （CHESCS） is derived. The Lax representation of the CHESCS is presented. The conservation laws for CHESCS are constructed. The peakon solution, N-soliton, N-cuspon, N-positon, and N-negaton solutions of CHESCS are obtained by using Darboux transformation and the method of variation of constants.     

New exact solutions for two nonlinear equations

In this Letter, we investigate two nonlinear equations given by ut−uxxt+3u²ux=2uxuxx+uuxxx and ut−uxxt+4u²ux=3uxuxx+uuxxx. Through some special phase orbits we obtain four new exact solutions for each equation above. Some previous results are extended.     

On a spectral analysis of scattering data for the Camassa-Holm equation

Physical details of the Camassa–Holm (CH) equation that are difficult to obtain in space-time simulation can be explored by solving the Lax pair equations within the direct and inverse scattering analysis context. In this spectral analysis of the completely integrable CH equation we focus solely on the direct scattering analysis of the initial condition defined in the physical space coordinate through the time-independent Lax equation. Both of the continuous and discrete spectrum cases for the initial condition under current investigation are analytically derived. The scattering data derived from the direct scattering transform for non-reflectionless case are also discussed in detail in spectral domain from the physical viewpoint.     

CRE Method for Solving m Kd V Equation and New Interactions Between Solitons and Cnoidal Periodic Waves

In nonlinear physics, the modified Korteweg de-Vries(m Kd V) as one of the important equation of nonlinear partial differential equations, its various solutions have been found by many methods. In this paper, the CRE method is presented for constructing new exact solutions. In addition to the new solutions of the m Kd V equation, the consistent Riccati expansion(CRE) method can unearth other equations.