In 1993, Camassa and Holm derived the celebrated Camassa-Holm (CH) equation , which is formally integrable, since it admits Lax pair formalism , bi-Hamiltonian structure  as well as infinitely many con- servation laws . 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 . A three-component model with peakon solutions has been studied by Geng and Xue .
Very recently, in Ref. , 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  of Equations (2) and (3). According to the work in , 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.
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.