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.
In 1993, Camassa and Holm derived the celebrated Camassa-Holm (CH) equation [
Very recently, in Ref. [
Particularly, in the case of
and in the case of
Xia and Qiao have presented bi-Hamiltonian structures and single peakon solutions [
We assume that the system Equation (2) admits two peakon solitons of the form
where
Without loss of generality, we assume that
where
Thus we have
where
Thus, we can get
where
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
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
By means of the similar calculation as those in the Section 2.1, taking
where
Supposing that
Thus, we obtain the double peakon solutions of Equation (3)
where
We provide an approach to obtain the double peakon solutions for the four-component CH type Equations (2) and (3) in the case of
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.
Yuanli Li,Qilao Zha, (2016) Double-Peakon Solutions of Two Four-Component Camassa-Holm Type Equations. Journal of Applied Mathematics and Physics,04,1305-1310. doi: 10.4236/jamp.2016.47138