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.     

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.     

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.     

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.     

Higher-Dimensional KdV Equations and Their Soliton Solutions

A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of bell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.     

Conservation Laws and Analytic Soliton Solutions for Coupled Integrable Dispersionless Equations with Symbolic Computation

Under investigation in this paper are two coupled integrable dispersionless （CID） equations modeling the dynamics of the current-fed string within an external magnetic field. Through a set of the dependent variable transformations, the bilinear forms for the CID equations are derived. Based on the Hirota method and symbolic computation, the analytic N-soliton solutions are presented. Infinitely many conservation laws for the CID equations are given through the known spectral problem. Propagation characteristics and interaction behaviors of the solitons are analyzed graphically.     

Conservation Laws and Analytic Soliton Solutions for Coupled Integrable Dispersionless Equations with Symbolic Computation

Under investigation in this paper are two coupled integrable dispersionless (CID) equations modeling the dynamics of the current-fed string within an external magnetic field. Through a set of the dependent variable transformations, the bilinear forms for the CID equations are derived. Based on the Hirota method and symbolic computation, the analytic N-soliton solutions are presented. Infinitely many conservation laws for the CID equations are given through the known spectral problem. Propagationcharacteristics and interaction behaviors of the solitons are analyzed graphically.     

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.     

Soliton Solutions to Coupled Integrable Dispersionless Equations

In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless (CID) equations is derived. It will be shown that this bilinear form is easier to perform the standard Hirota process. One-, two-, and three-soliton solutions are presented. Furthermore, the N-soliton solutions are derived.     

(2+1)维Camassa-Holm方程的相似约化与解析解

将Clarkson和Krushal引入的直接约化方法推广并应用到(2+1)维CamassaHolm方程组,获得了该方程的若干相似约化和解析解,其中包括Logistic方程和Bernoulli方程.约化结果得到了Peakon解、Cuspon解和关于时间t的奇异解.该方法也适用于其他有重要物理背景的非线性演化方程 关键词： Camassa-Homl方程 相似约化 直接方法 解析解     

Analytical Investigation of Soliton Solutions to Three Quantum Zakharov-Kuznetsov Equations

In the present paper, the two-dimensional quantum Zakharov-Kuznetsov (QZK) equation, three-dimensional quantum Zakharov-Kuznetsov equation and the three-dimensional modified quantum Zakharov-Kuznetsov equation are analytically investigated for exact solutions using the modified extended tanh-expansion based method. A variety of new and important soliton solutions are obtained including the dark soliton solution, singular soliton solution, combined dark-singular soliton solution and many other trigonometric function solutions. The used method is implemented on the Mathematica software for the computations as well as the graphical illustrations.     

On Peakon and Kink-peakon Solutions to a (2 + 1) Dimensional Generalized Camassa-Holm Equation

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.     

Soliton Solutions to Coupled Integrable Dispersionless Equations

In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless （CID） equations is derived. It will be shown that this bilineax form is easier to perform the standard Hirota process. One-, two-, and three-soliton solutions are presented. Furthermore, the N-soliton solutions axe derived.     

Soliton Solutions for Nonisospectral AKNS Equation by Hirota's Method

Bilinear form of the nonisospectral AKNS equation is given. The N-soliton solutions are obtained through Hirota's method.     

Relationship Between Soliton-like Solutions and Soliton Solutions to a Class of Nonlinear Partial Differential Equations

By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m→1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.     

Soliton Solutions for a Negative Order AKNS Equation

In this short paper, bilinear form of a negative order AKNS equation is given. The N-soliton solutions are obtained through Hiorta's direct method.     

Soliton Solutions for a Negative Order AKNS Equationt

In this short paper, bilinear form of a negative order AKNS equation is given. The N-soliton solutions are obtained through Hiorta＇s direct method.     

Multiple Soliton Solutions of the High Order Broer-Kaup Equations

Using the extended homogeneous balance method, which is very concise and primary, we find the multiple soliton solutions of the high order Broer-Kaup equations. The method can be generalized to dealing with high-dimensional Broer-Kaup equations and other class of nonlinear equations.     

A Two-component Generalization of the Camassa-Holm Equation and its Solutions

An explicit reciprocal transformation between a two-component generalization of the Camassa–Holm equation, called the 2-CH system, and the first negative flow of the AKNS hierarchy is established. This transformation enables one to obtain solutions of the 2-CH system from those of the first negative flow of the AKNS hierarchy. Interesting examples of peakon and multi-kink solutions of the 2-CH system are presented Mathematics Subject Classifications (2000). 35Q53, 37K35     

Soliton Solutions for a Negative Order AKNS Equation Hierarchy

In this paper, bilinear form of a negative order AKNS equation hierarchy is given. The soliton solutions are obtained through Hiorta's direct method.