MULTI-PEAKON SOLUTIONS TO A FOUR-COMPONENT CAMASSA-HOLM TYPE SYSTEM ∗

A four-component Camassa-Holm type system with cubic nonlinearity is investigated. It allows an arbitrary function Γ(x, t) to be involved in to include some existing integrable peakon equations as special reductions. We obtain N -peakon solutions of the four-component Camassa-Holm type system with two special cases of Γ(x, t). In particular, we give oneand two-peakon solutions in an explicit formula and are graphically plotted. Further, some interesting peaked solutions are found: some peakon waves possessing positive and negative amplitudes while others decaying and growing amplitudes.


Introduction
In 1993, Camassa and Holm (CH) derived a completely integrable dispersive shallow water equation [2], which has been studied quite extensively in the past two decades. A significant property of this equation is that the CH equation admits peaked soliton (peakon) and multi-peakon wave solutions. As an integrable equation more diverse studies on the CH equation have been remarkably developed in the literatures [1,10,17]. Recently, more integrable equations with peakon properties attract much attention, including the Degasperis-Procesi (DP) equation [3,11], the Fokas-Olver-Rosenau-Qiao(FORQ) equation [4,5,9,[12][13][14], the Novikov equation [7,15], and other CH type equations [6,8,16,18]. The CH and the DP equations are completely integrable peakon systems with quadratical nonlinearity, and the FORQ and the Novikov equations are typical integrable peakon systems with cubic nonlinearity.
Recently, Li, Liu and Popowicz studied the following 3 × 3 spectral problem [8] where m i = m i (x, t), n i = n i (x, t), i = 1, 2. Actually, this spectral problem is a special case of the multi-component problem studied in [17]. The spectral problem (1.1) is interesting because it could cover some 3 × 3 spectral problems for CH type equations as special cases, such as the three-component CH system proposed by Geng and Xue [6], one and two-component Novikov equations [7,15], and one and two-component Qiao equations [12,16]. Based on the spectral problem (1.1), the authors [8] gave the following fourcomponent CH type system and Γ = Γ(x, t) is an arbitrary function. The system (1.2) is integrable in the sense of Lax pair associated with the spectral problem (1.1) and the following auxiliary spectral problem We notice that the system (1.2) contains an arbitrary function Γ, which amazingly leads (1.2) to some different CH type equations through certain choices of Γ. For instance, some cubic systems could be reduced (see [8]). In particular, if Γ = 0, we have where f i , g i , m i and n i , i = 1, 2 are given in (1.3). The system (1.5) can be rewritten in the form of the following bi-Hamiltonian structure The aim of this paper is to construct multi-peakon solutions for the four-component CH type system (1.2) with a special Γ. In the case of Γ = 0, we solve the system (1.5) and obtain its multi-peakon solutions, which are not in the traveling wave type. In the case of Γ = ρ (ρ is a non-zero constant), we find the four-component CH type system (1.2) possesses the traveling wave type multi-peakon solutions.

Multi-peakon solutions
In the following, we will derive multi-peakon solutions to the four-component CH Case 1. Let us suppose that one-peakon solution of the four-component CH type system (1.2) with Γ = ρ is of the following form where p 1 , r 1 , s 1 , τ 1 and q 1 are functions of t to be determined. With the help of distribution theory, we are able to write out u 1x , u 2x , v 1x , v 2x , m 1 , m 2 , n 1 and n 2 as follows Substituting (2.1) and (2.2) into (1.2) with Γ = ρ, we arrive at the following one-peakon dynamical system where ∆ 11 = p 1 s 1 + r 1 τ 1 , θ 11 = p 1 r 1 + s 1 τ 1 . ∆ 11 and θ 11 taking derivative with respect to t, and using Eqs. (2.3), we have the following relations where A 1 and B 1 are arbitrary integration constants. Therefore, Eqs. (2.3) becomes We arrive at the following general solution of Eq. (2.6) , ω 1 are arbitrary integration constants. From (2.1) and (2.7), we obtain one-peakon solution of (1.2) with Γ = ρ: where ξ 1 = x−ρt−ω 1 and λ (1) i (i = 1, 2, 3, 4) are given in (2.8). See Figs. 1-2 for the profile of the one-peakon dynamics for the potentials u i and v i (i = 1, 2) in (2.9). In Fig. 1, (a),(d) and (b),(c) show that the one-peakon with amplitudes exponentially decaying and growing with time t, respectively. And an interesting phenomenon is shown in Fig. 2: the amplitude of u 1 ( or u 2 ) is changed from positive to negative (or negative to positive ) while v 1 (or v 2 ) has positive amplitude which is changed from decaying to growing along the t axis.  Case 2. A two-peakon solution is given in the form of where p i , r i , s i , τ i and q i (i = 1, 2) are functions of t to be determined. In a similar process as case 1, we can find the two-peakon dynamical system, which consists of ten equations. Let us start from the first two equations: q 1t = ρ and q 2t = ρ, which yield where ω 1 and ω 2 are constants. Without loss of generality, we suppose ω 2 > ω 1 . With the help of (2.11), the two-peakon dynamical system can be rewritten as 2 )r i ,  where E (1) and Ω 12 = e ω1−ω2 . Apparently, (2.12) implies the following relations ∆ iit = (p i s i + r i τ i ) t = 0, (i = 1, 2).
Substituting (2.16) into (2.10), we obtain the two-peakon solution of (1.2) where ξ j = x − ρt − ω j (j = 1, 2) and λ  Fig. 3 and Fig. 4 show the right-traveling and left-traveling waves, respectively. The amplitudes of the peakons to equation (2.18) grow/decay exponentially with time t. All two-peakon waves have the same velocity ρ. Namely, the collision between the two-peakon waves will never happen.
Case N. Following the procedure in cases 1 and 2, the N -peakon solutions of the four-component Camassa-Holm type system (1.2) are just linear superpositions where p j , r j , s j and τ j (j = 1, 2, . . . , N ) are N amplitudes of the potentials u 1 , u 2 , v 1 and v 2 , respectively, and q j (j = 1, 2, . . . , N ) are N -peak positions. Functions   p j , r j , s j , τ j and q j (j = 1, 2, . . . , N ) evolve according to the following system: (2.20) where jl = sgn(q j −q l ), jk = sgn(q j −q k ), Λ = e −|qj −q l |−|qj −q k | , ∆ jj = p j s j +r j τ j , (1 ≤ j; k; l ≤ N ). In the above formula, q jt = ρ (ρ = 0) implies that N -peakon waves move at the same velocity ρ in the traveling wave type whereas ρ = 0 implies that all peak positions do not change along with the time t.

Conclusions
In this paper, we study a generalized four-component CH system (1.2) with an arbitrary function Γ(x, t). This model provides a large class of peakon dynamical systems and covers several well-known integrable peakon equations associated with 3 × 3 spectral problems. We obtain two kinds of multi-peakon solutions to the system (1.2) with Γ = ρ: 1) for ρ = 0, the multi-peakon solutions are not in the traveling wave type, and 2) if ρ = 0, the multi-peakon solutions are in the traveling wave type. Furthermore, the peakon solutions (2.9) and (2.18) can be reduced to the solutions of the model (1.5) if ρ = 0.