Smooth Multisoliton Solutions of a 2-Component Peakon System with Cubic Nonlinearity

. We present a reciprocal transformation which links the Geng–Xue equation to a particular reduction of the first negative flow of the Boussinesq hierarchy. We discuss two reductions of the reciprocal transformation for the Degasperis–Procesi and Novikov equations, respectively. With the aid of the Darboux transformation and the reciprocal transformation, we obtain a compact parametric representation for the smooth soliton solutions such as multi-kink solutions of the Geng–Xue equation


Introduction
The Degasperis-Procesi (DP) equation m t + 3u x m + um x = 0, m = u − u xx was derived by applying the method of asymptotic integrability to a many-parameter family of third order dispersive PDEs [9]. It may be viewed as an approximate model describing shallow water wave propagation in the small amplitude and long wavelength regime [7,10,16,17]. The DP equation is a completely integrable equation. It admits a Lax pair, bi-Hamiltonian structure, and is reciprocally related to a negative flow in the Kaup-Kupershmidt hierarchy [8]. Moreover, the DP equation has been studied by the inverse scattering method [1,6], and was shown to possess various periodic-wave solutions and travelling-wave solutions [19,37]. Its smooth multi-soliton solutions were constructed by means of the τ function approach, Riemann-Hilbert method, dressing method and Darboux transformation (DT) [1,5,24,30]. In particular, the DP equation is an equation of Camassa-Holm (CH) type, which has an unusual feature of admitting non-analytic solutions called peakons [26,27]. There exists an interesting connection between the DP peakon lattice and the finite C-Toda lattice [3]. By using the method of perturbative symmetry to classifying integrable equations of generalised CH form, a new equation with cubic nonlinearity m t + u 2 m x + 3uu x m = 0, m = u − u xx was discovered by Vladimir Novikov [33]. Soon afterwards, Hone and Wang [15] confirmed its integrability by presenting a Lax representation, infinitely many conserved quantities as well as a bi-Hamiltonian structure. They also related this equation to a negative flow of the Sawada-Kotera hierarchy via a reciprocal transformation. Smooth multi-soliton solutions of the Novikov equation have been presented via several approaches such as the Hirota bilinear method, Riemann-Hilbert method and DT [2,31,39]. Furthermore, multipeakons of the Novikov equation may be computed by the inverse spectral method [14]. Dynamical system of the multipeakons of the Novikov equation is a Hamiltonian system, which is connected to the finite Toda lattice of BKP type [4,15]. Subsequently, Geng and Xue [12] proposed a two-component generalization of the Novikov equation and the DP equation Indeed, for u = 1 and u = v, the Geng-Xue equation (1.1) reduces to the DP equation and the Novikov equation, respectively. This equation is a completely integrable system with a Lax pair and bi-Hamiltonian structure [12,22]. The homogeneous and local properties of the Hamiltonian functionals were discussed [20]. Also, the Geng-Xue equation is related to a negative flow in a modified Boussinesq hierarchy by a reciprocal transformation [23] and the behaviour of the bi-Hamiltonian structures under the transformation was studied [21]. Moreover, the Geng-Xue equation was shown to admit multi-peakon solutions [28,29,35] and its Cauchy problem was considered [13,36]. However, to the best of our knowledge, smooth solutions such as multisoliton solutions of the Geng-Xue equation have not been constructed. The purpose of this paper is to propose a method for building soliton solutions of the Geng-Xue equation. To this end, we find it is convenient to relate the Geng-Xue equation to a particular reduction of the first negative flow in the Boussinesq hierarchy via a reciprocal transformation. Different from the works of [21,23], this reciprocal transformation may be reduced to that of the DP equation and the Novikov equation. Furthermore, by combining the reciprocal transformation with DT of the negative flow in the Boussinesq hierarchy, we are able to obtain a parametric representation for the multi-kink solutions of the Geng-Xue equation.
The paper is arranged as follows. In Section 2, we introduce a reciprocal transformation and relate the Geng-Xue equation to a particular reduction of the first negative flow of the Boussinesq hierarchy. The reductions of the reciprocal transformation to the DP equation and the Novikov equation will also be discussed. In Section 3, with the aid of the reciprocal transformation and DT, we construct the smooth multisoliton or multi-kink solutions of the Geng-Xue equation. Interestingly the solutions will be represented in terms of Wronksians, and the simplest nontrivial cases will be given explicitly.

A reciprocal transformation and negative flow of the Boussinesq hierarchy
In this section, we present a proper reciprocal transformation and establish a link between the Geng-Xue equation and a special negative flow of the Boussinesq hierarchy. Also, we consider the possible reductions of the reciprocal transformation and show that the reciprocal transformations for both the DP equation and Novikov equation are recovered.

A reciprocal transformation of the Geng-Xue equation
The Geng-Xue equation (1.1) has a Lax representation [12], namely it is the compatibility condition of Use of the above linear spectral problem (2.1) and a standard algorithm lead to infinitely many conservation laws for the Geng-Xue equation. One of them is which allows us to introduce new independent variables y and τ via the following reciprocal transformation Setting w = ( m n ) 1 3 and eliminating φ 1 , φ 2 , we find that the linear system (2.1) may be rewritten in terms of φ ≡ φ 3 as To bring (2.3) into a familiar form, we introduce a gauge transformation and have It is noted that the first equation of (2.5) is the linear spectral problem of the Boussinesq hierarchy. Now, the compatibility condition of the two equations of (2.5) yields the associated Geng-Xue equation, which reads where s 1 = p yy + 3p y q + 3pq 2 + ξp + 1, It is mentioned that the Geng-Xue equation under the transformation (2.2) implies the associated Geng-Xue equation (2.7). As system (2.7) possesses a Lax pair, one may construct infinitely many conserved quantities for it. Furthermore, it passes the standard Painlevé test of Weiss, Tabor and Canevale (WTC) [38].
As mentioned, the spatial part of the spectral problem (2.5) is the one for the Boussinesq hierarchy. The first equation in this hierarchy is the Boussinesq equation [34,41] Thus the associated Geng-Xue equation (2.7) should have connection with a particular flow of this hierarchy. To see it, let us consider a more general spectral problem Its compatibility condition yields is the well-known recursion operator of the Boussinesq hierarchy, and hence the system (2.8) is just the first negative flow in the Boussinesq hierarchy. Now, setting a = p, b = pq, we have the following relation Thus, the associated Geng-Xue equation (2.7) indeed is a particular reduction of the negative Boussinesq equation (2.8).

Reductions of the reciprocal transformation
Recall that the Geng-Xue equation may be reduced to the DP equation and the Novikov equation by setting u = 1 and u = v respectively. So it is interesting to consider the reciprocal transformation (2.2) under these reductions. Case 1: u = 1. In this case, the reciprocal transformation (2.2) becomes where h = n 1 3 . A direct calculation shows that the associated equation (2.7) and the spectral problem (2.5) reduce to 2h yyy + hξ y + 2ξh y = 0, and ϕ yyy + ξϕ y + 1 2 respectively. These are just the associated DP equation and its spectral problem [8,24].
where h = m 2 3 . In this case, the spectral problem (2.5) becomes Meanwhile, (2.7) yields Setting ϕ = ψ y , from (2.9) we have It is easy to see that by integrating the second equation of (2.10) and substituting it into (2.11), we reach the results appeared in [15,39].

Multisoliton solutions of the Geng-Xue equation
In the previous section, we have related the Geng-Xue equation (1.1) to the associated equation (2.7), which was shown to be a reduction of the first negative flow in the Boussinesq hierarchy. In what follows, we will take a similar approach as done for the CH, modified CH and DP equations [24,25,40] and explain that this connection, together with the DT for the associated Geng-Xue equation, allows us to propose an algorithm to build multisoliton solutions of the Geng-Xue equation.
As a first step, we have the following Proposition 3.1. The spectral problem (2.5) is covariant with respect to the following N -DT where f 1 , . . . , f N are solutions of the spectral problem (2.5) at λ = λ 1 , . . . , λ N , respectively. W denotes the Wronskian, Here and in the sequel all determinant notations are adopted according to [11].
Proof . For the Lax operator L = ∂ 3 y + ξ∂ y + η, it is well known that the operator of the 1-DT is given by [18,32]). It is straightforward to check that the transformed variables solve namely the proposition is valid for N = 1 case. For the general case, we may assume that the operator of the N -DT takes the form T = ∂ N y + a 1 ∂ N −1 y + · · · + a N , where coefficients a 1 , . . . , a N are functions of f 1 , . . . , f N and their derivatives with respect to y. Hence, for this iterated DT, we havê Plugging the expressions of T , L into the first equation of (3.2) and considering the coefficients of ∂ N +1 and ∂ N , we obtain The coefficients of the operator T may be determined from the conditions T f i = 0, 1 ≤ i ≤ N . In other words, we have Using Cramer's rule, we obtain from the system (3.4) that

Substituting them into (3.3), we have
In addition, the iterated formulae for p, q may be acquired by substituting ξ  To calculate the solutions f j of the system (2.5) at λ j , j = 1, . . . , N , it is convenient to assume α j , β j , −α j − β j being three distinct roots of the cubic equation ω 3 − k −2 ω = λ 2 j . Now, without loss of generality, from (2.5) we have Herein c 1j , c 2j , δ j are arbitrary constants. To build real solutions of the Geng-Xue equation, let us assume α j − β j = p j > 0 and rewrite (3.5) as It is noted that 1, e y/k and e −y/k constitute a fundamental set of solutions of the first equation of (2.5) at λ = 0 with the seed ξ = −k −2 , p = k 2 , η = q = 0. With the help of the f j 's given by (3.5), it follows immediately from Proposition 3.1 that for the N -th iterated spectral problem at λ = 0 we may calculate its solutions, and in particular we have where ν i , i = 1, 2, 3, are arbitrary constants and (3.8) Next, we work out the coordinate transformation between the independent variables x, t and y, τ . To this end, we consider the spectral problem in (2.1) at λ = 0 which yields for φ = φ 3 . As e x and e −x form a fundamental set of solutions of (3.9), in view of the gauge transformation (2.4), they may be represented as where the coefficients c k , d k , k = 1, 2, 3, are independent of y. Taking account of the asymptotic behaviours (3.8), we find which further imply where c = c(τ ). Differentiating above equation with respect to τ and using the reciprocal transformation (2.2), we have then by taking the limit y → ∞, we find c τ = u 0 v 0 = k 3 , which leads to c = k 3 τ + d with d as an integration constant. Thus we obtain Finally, we need to work out the transformation formulae for the field variables u, v, which can be done by means of (2.6) and the reciprocal transformation (2.2). In particular, we may deduce w from q[N ] = wy w . It is interesting to observe that for p[N ] and q[N ], we have where The proof of (3.12) and (3.13) is presented in the appendix. Summarizing above discussions, we have x y W N W (g 1 , g 2 , . . . , g N ) where x, t are defined by (3.10) and (3.11), ℓ is an integration constant, and ϕ 2 [N ], ϕ 3 [N ] are given by (3.7), respectively.
It may be checked that for the solution presented above both potentials u and v have different asymptotic behaviors as y → ∞ and y → −∞. This is a indication that the solution is of kinktype.
In the rest part of this section, we consider the simplest cases and present two examples.

Example 3.3 (1-kink solution)
. For N = 1, let us take where δ 1 may be chosen as ±1. Direct computations show that the DT (3.1) yields Then, application of Proposition 3.2 allows us to have the following solution of the Geng-Xue equation (3.14) Here and in the sequel, allk i , i ∈ Z, are assumed to be arbitrary constants. We take α 1 , β 1 , λ 2 1 as real constants so that we have the real-valued solutions. Furthermore, assuming that δ 1 = 1, 0 < kp 1 < 1, our solutions will be non singular. Under these assumptions and substituting the expressions of f 1 , g 1 into the solution (3.14), we obtain a parameter representation of 1-kink solution of the Geng-Xue equation wherek 3 = ln −k and θ 1 = 1 2 p 1 y + 3k 2 p 1 We now consider the first a few conserved quantities for the 1-kink solution. The Geng-Xue equation is known to possess infinitely many conserved quantities [12,20,22,23], and among them the first three are Since 1-kink solution of the Geng-Xue equation is not explicit and written in parametric form, we consider I i under (2.2), namely,  Thus, for the 1-kink solution we have For I 3 , using definition of p together with we arrive at Example 3.4 (2-kink solution). For N = 2, we may take where δ 1 , δ 2 are allowed to be ±1. From (3.6), it follows that Notice that g iy = f i , f iτ = k 2 g i , i = 1, 2. Then, after some direct calculations, we find  where Following Proposition 3.2 and after tedious calculations, we may get a parameter representation of exact solution Hereafter, to obtain real solution without singularity, let us assume that α i , β i , λ 2 i are real and δ 1 = δ 2 = 1. Then we may establish the parameter representation of the 2-kink solution of the Geng-Xue equation. A profile of 2-kink solution is plotted in Figures 3 and 4 for the parameters p 1 = 2, p 2 = 3 2 , r 1 = α 1 + β 1 = √ 7, r 2 = α 2 + β 2 = − 91 12 ,k 5 = θ 10 = θ 20 = 0, k = 2 5 ,k 6 = 1.
A Proof of (3.12) and (3.13) For N = 1, 2, the validity of (3.12), (3.13) has been shown in Examples 3.3 and 3.4. In general case, we first introduce the following two matrices M ,M and N -dimensional column vectors g, f , a, b, d as  is used. Next we consider (3.13). Indeed, we have