Double-Peakon Solutions of Two Four-Component Camassa-Holm Type Equations


This paper is contributed to study two new integrable four-component systems reduced from a multi-component generation of Camassa-Holm equation. Some double peakon solutions of both systems are described in an explicit formula by the method of variation of constant for ordinary differential equations. These double peakon solutions are established in weak sense. The dynamic behaviors of the obtained double peakon solutions are illustrated by some figures.

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Li, Y. and Zha, Q. (2016) Double-Peakon Solutions of Two Four-Component Camassa-Holm Type Equations. Journal of Applied Mathematics and Physics, 4, 1305-1310. doi: 10.4236/jamp.2016.47138.

1. Introduction

In 1993, Camassa and Holm derived the celebrated Camassa-Holm (CH) equation [1], which is formally integrable, since it admits Lax pair formalism [1], bi-Hamiltonian structure [2] as well as infinitely many con- servation laws [2]. One of the remarkable properties of the equation is that it possesses peakon wave solutions. Subsequently, a large amount of literature was devoted to find new integrable models with peakon solutions, such as Degasperis-Procesi (DP) equation, the Fokas-Olver-Rosenau-Qiao (FORQ) equation, the Novikov’s cubic nonlinear equation and some other completely integrable peakon systems. It is a natural idea to continue studying multi-component generalizations of peakon equations. One of the most popular two-component integrable systems, which admits Lax Pair and infinitely conservation laws, has multi-peakon solitons [3]. A three-component model with peakon solutions has been studied by Geng and Xue [4].

Very recently, in Ref. [5], another multi-component system of Camassa-Holm equation, which admited Lax pair and infinitely many conservation laws, denoted by CH(N,H) with components and an arbitrary smooth function H of and their derivatives, was derived and studied


Particularly, in the case of, Equation (1) becomes the following system


and in the case of, it is reduced into


Xia and Qiao have presented bi-Hamiltonian structures and single peakon solutions [5] of Equations (2) and (3). According to the work in [5], we will investigate the double peakon solutions of Equations (2) and (3) in this paper. Section 2 is devoted to look for double peakon solutions of Equations (2) and (3). Further, we discuss the dynamic behaviors of the obtained peakon solutions by some figures. Some conclusions and open problems are addressed in Section 3.

2. Two-Peakon Solutions

2.1. Two-Peakon Solutions to Equation (2)

We assume that the system Equation (2) admits two peakon solitons of the form


where, , , and , are functions of t to be determined. Moreover, we can obtain their derivatives in the weak sense as follows


Without loss of generality, we assume that. Substituting Equations (4) and (5) into Equation (2) and in the distribution sense, we can obtain the following double peakon differential equations


where, , , and. and taking derivative with respect to t ,we can obtain


Thus we have


where and are arbitrary integration constants. In the following, we assume that.

taking derivative with respect to t and combining with Equation (6), we see


Thus, we can get


where is arbitrary constant.

Combining the Equation (6) with Equations (8) and (10), Equation (6) is reduced to


where m, n are integration constants. According to Equation (11), we can arrive at


where, , and denote integration constants. Moreover, with the help of Equation (11), we easily obtain that


Solving the differential equations of Equation (13), we get


Using the method of variation of constant, we can have


Based on Equations (8) and (10), those constants have relations as


Thus, we establish the double peakon solutions of the Equation (2)


where and satisfy Equation (16).

Figure 1 show the profiles of the double peakon solutions Equation (17). The amplitudes of the peakons grow or decay exponentially with time t. All peak positions don’t change along with the time t and the collision between the two-peakon waves will never happen.

2.2. Double Peakon Solutions to Equation (3)

By means of the similar calculation as those in the Section 2.1, taking, we arrive at the differential equations as follows

Figure 1. 3D graphs of the double peakon solutions for, , defined by Equation (17) with.


where and are constants. From which, it is easy to see that we may have the following relations


Supposing that and letting, we can readily see Thus, the t-dependent functions satisfy the following ordinary differential equations


Thus, we obtain the double peakon solutions of Equation (3)


where, , and.

3. Conclusion

We provide an approach to obtain the double peakon solutions for the four-component CH type Equations (2) and (3) in the case of and respectively. However, its exact double peakon solutions without or are expected to attract more endeavor to study.


This work was supported by the National Natural Science Foundation of China under Grant No. 11261037, the Natural Science Foundation of Inner Mongolia Autonomous Region under Grant No. 2014MS0111, the Caoyuan Yingcai Program of Inner Mongolia Autonomous Region under Grant No. CYYC2011050, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant No. NJYT14A04.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Camassa, R. and Holm, D.D. (1993) An Integrable Shallow Water Equation with Peaked Solitons. Physical Review Letters, 71, 1661-1664.
[2] Fuchssteiner, B. and Fokas, A.S. (1981) Symplectic Structures, Their Backlund Transformations and Hereditary Symmetries. Physcia D, 4, 47-66.
[3] Xia, B.Q. and Qiao, Z.J. (2015) A Synthetical Two-Component Model with Peakon Solutions. Studies in Applied Mathematics, 135, 248-276.
[4] Geng, X.G. and Xue, B. (2011) A three-Component Generation of Camassa-Holm Equation with N-Peakon Solutions. Advances in Mathematics, 226, 827-839.
[5] Xia, B.Q. and Qiao, Z.J. (2015) Multi-Component Generalization of Camassa-Holm Equation. arXiv:1310.0268v2

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