Gaihua Wang, Nianhua Li, Q P Liu^,31Department of Mathematics, China University of Mining and Technology, Beijing, 100083, China
²School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China

First author contact: ³Author to whom any correspondence should be addressed.
Received:2019-12-9Revised:2020-02-4Accepted:2020-02-11Online:2020-03-24


Abstract
We propose and develop another approach to constructing multi-soliton solutions of an integrable two-component Camassa–Holm (CH2) system. With the help of a reciprocal transformation and a gauge transformation, we relate the CH2 system to a negative flow of the Broer–Kaup or two-boson hierarchy. The solutions of this negative flow are given in terms of Wronskians via Darboux transformation. Then the multi-soliton solutions of the CH2 system are recovered in parametric form by inverting the reciprocal transformation and the gauge transformation.
Keywords： Darboux transformation;reciprocal transformation;two-component Camassa–Holm system;multi-soliton solutions


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Gaihua Wang, Nianhua Li, Q P Liu. Multi-soliton solutions of a two-component Camassa–Holm system: Darboux transformation approach. Communications in Theoretical Physics, 2020, 72(4): 045003- doi:10.1088/1572-9494/ab7706

1. Introduction

The Camassa–Holm (CH) equation(1)$\begin{eqnarray}{m}_{t}+{{um}}_{x}+2{{mu}}_{x}=0,\ m=u-{u}_{{xx}}+{k}^{2},\end{eqnarray}$derived by Camassa and Holm [1] in 1993 as a shallow water wave model, has been studied extensively (see [2–13] and references therein). While the CH equation appears in the paper by Fuchssteiner and Fokas [14], it becomes popular only after the work of Camassa and Holm. In the case of k=0, the CH equation (1) has the remarkable property of possessing the peaked solutions or peakons [1].

A two-component generalization of CH (CH2) equation was proposed by Olver and Rosenau within the framework of tri-Hamiltonian duality [15]. It reads as(2a)$\begin{eqnarray}{m}_{t}+{{um}}_{x}+2{{mu}}_{x}+\rho {\rho }_{x}=0,\end{eqnarray}$(2b)$\begin{eqnarray}{\rho }_{t}+{\left(u\rho \right)}_{x}=0,\end{eqnarray}$where $m=u-{u}_{{xx}}+{k}^{2}$. Obviously, the CH2 system (2) reduces to CH equation (1) when ρ=0. By construction, the CH2 system is a bi-Hamiltonian system, thus it is integrable. Immediately after, Schiff obtained a Lax representation for it [16]. It is mentioned that another two-component extension of the CH equation, which is similar to (2) but with the replacement of ρρ_x by $-\rho {\rho }_{x}$, was proposed [17–19] in a different context. Two-component CH equations with both plus sign and minus sign in front of ρρ_x have been studied considerably and a large number of results has been accumulated (see [20–29] and the references therein). One may consult [27, 28] for more details on the history of the CH2 systems. We just mention here that the CH equation has been extended into multicomponent by various methods and in particular a classification study was conducted for two-component generalizations most recently [30]. While a physical interpretation of the two-component CH equation in [19] is not available till now, the CH2 system (2) is indeed interesting in the physical context [22, 23].

As mentioned above, the CH2 system is a completely integrable system and its linear spectral problem or Lax representation reads as follows [16](3a)$\begin{eqnarray}{{\rm{\Psi }}}_{{xx}}=\left(-{\lambda }^{2}{\rho }^{2}+m\lambda +\displaystyle \frac{1}{4}\right){\rm{\Psi }},\end{eqnarray}$(3b)$\begin{eqnarray}{{\rm{\Psi }}}_{t}=\left(\displaystyle \frac{1}{2\lambda }-u\right){{\rm{\Psi }}}_{x}+\displaystyle \frac{{u}_{x}}{2}{\rm{\Psi }},\end{eqnarray}$where λ is the spectral parameter (see also [22, 23]). For the CH2 system, different methods have been developed to construct its solutions and study their properties [22, 23, 25–29]. In particular, for the constructions to multi-soliton solutions, we mention following works. The first one is the work of Holm and Ivanov who developed an inverse scattering method and the multi-soliton solutions are constructed analytically [27]. The second work is done by Matsuno very recently. He obtained the multi-soliton solutions to the CH2 system systematically within the framework of Hirota bilinear method and the analysis of these solutions was further conducted [28].

It has been known that Darboux transformations play vital roles in the study of integrable equations and are very effective in constructing multi-soliton solutions [31–33]. For the CH equation, Li and Zhang made important contributions and proposed a way to calculate its multi-solution solutions [8]. Recently Xia et al refined the method of Li and Zhang and made the approach more direct [13]. The ideas were applied successfully to two more Camassa–Holm type equations, namely the Novikov equation and Degasperis–Procesi equation and their multi-solition solutions are obtained [34, 35].

The purpose of this paper is to develop a Darboux transformation approach to the CH2 system (2) so that its multi-soliton solutions will be recovered. As always, for the Camassa–Holm type equations, reciprocal transformation will play a key role. In addition, we also introduce further gauge transformation so that the CH2 system is related to a negative flow of the Broer–Kaup (or the dispersive water waves, or the two-boson) hierarchy. In this way, we are able to give a compact and neat representation for the multi-soliton solutions of the CH2 system in terms of Wronskians. For the two-component CH equation with minus sign in the second equation, we notice that both Wu [20] and Lin et al [21], based on Darboux transformations, derived some solutions including lower order solitons.

This paper is organized as follows. In section 2, we recall the reciprocal transformation and then introduce a gauge transformation. These transformations establish a link between the CH2 system and a particular negative flow of the Broer–Kaup hierarchy. In section 3, we consider a Darboux transformation for the negative flow of the Broer–Kaup hierarchy. The iteration of this Darboux transformation leads to a representation of solutions in Wronskians. These results are adopted to find the multi-soliton solutions in parametric form. In section 4, we give two explicit examples and explain their relationship with the known results.

2. Reciprocal transformation

It is well known that the transformations of reciprocal type play key roles in the study of the Camassa–Holm type equations and this also is the case for the CH2 system. Next, we will take a reciprocal transformation in conjugation with a gauge transformation to establish a link between the CH2 system and a negative flow of the Broer–Kaup hierarchy.

From the conservation law (2b), we consider the following reciprocal transformation $(x,t)\to (y,\tau )$ defined by(4)$\begin{eqnarray}{\rm{d}}y=\rho {\rm{d}}x-\rho u{\rm{d}}t,\end{eqnarray}$(5)$\begin{eqnarray}{\rm{d}}\tau ={\rm{d}}t.\end{eqnarray}$Thus, we have(6)$\begin{eqnarray}\displaystyle \frac{\partial }{\partial x}=\rho \displaystyle \frac{\partial }{\partial y},\ \ \displaystyle \frac{\partial }{\partial t}=\displaystyle \frac{\partial }{\partial \tau }-\rho u\displaystyle \frac{\partial }{\partial y}.\end{eqnarray}$According to (6), the variables ρ and u may be expressed by x=x(y, τ):(7a)$\begin{eqnarray}{x}_{y}=\displaystyle \frac{1}{\rho },\end{eqnarray}$(7b)$\begin{eqnarray}{x}_{\tau }=u.\end{eqnarray}$Through the reciprocal transformation, the system (2) is brought into(8a)$\begin{eqnarray}{\left(\displaystyle \frac{m}{{\rho }^{2}}\right)}_{\tau }+{\rho }_{y}=0,\end{eqnarray}$(8b)$\begin{eqnarray}{\rho }_{\tau }+{\rho }^{2}{u}_{y}=0,\end{eqnarray}$where(9)$\begin{eqnarray}m=u+\rho {\left(\mathrm{ln}\rho \right)}_{y\tau }+{k}^{2}.\end{eqnarray}$In addition, the associated Lax representation (3) is transformed to(10a)$\begin{eqnarray}{{\rm{\Psi }}}_{{yy}}=\left(-{\lambda }^{2}+\displaystyle \frac{m}{{\rho }^{2}}\lambda +\displaystyle \frac{1}{4{\rho }^{2}}\right){\rm{\Psi }}-\displaystyle \frac{{\rho }_{y}}{\rho }{{\rm{\Psi }}}_{y},\end{eqnarray}$(10b)$\begin{eqnarray}{{\rm{\Psi }}}_{\tau }=\displaystyle \frac{\rho }{2\lambda }{{\rm{\Psi }}}_{y}+\displaystyle \frac{\rho {u}_{y}}{2}{\rm{\Psi }}.\end{eqnarray}$It is noted that the gauge transformation(11)$\begin{eqnarray}{\rm{\Psi }}=\displaystyle \frac{1}{\sqrt{\rho }}{{\rm{e}}}^{{\rm{i}}\int \left(\displaystyle \frac{m}{2{\rho }^{2}}-\lambda \right){\rm{d}}y}{\rm{\Phi }}\end{eqnarray}$brings the spectral problem (10a, 10b) to the following energy dependent spectral problem(12)$\begin{eqnarray}{{\rm{\Phi }}}_{{yy}}=(2{\rm{i}}\lambda +U){{\rm{\Phi }}}_{y}+({U}_{y}-V){\rm{\Phi }},\end{eqnarray}$(13)$\begin{eqnarray}{{\rm{\Phi }}}_{\tau }=\displaystyle \frac{\rho }{2\lambda }{{\rm{\Phi }}}_{y}-\left(\displaystyle \frac{{\rm{i}}\rho }{2}+\displaystyle \frac{U\rho }{4\lambda }+\displaystyle \frac{{\rho }_{y}}{4\lambda }-\displaystyle \frac{1}{2}\int {U}_{\tau }{\rm{d}}y\right){\rm{\Phi }},\end{eqnarray}$where(14)$\begin{eqnarray}U=-\displaystyle \frac{m{\rm{i}}}{{\rho }^{2}},\ \ V=-\displaystyle \frac{{\rm{i}}}{2}{\left(\displaystyle \frac{m}{{\rho }^{2}}\right)}_{y}-\displaystyle \frac{{m}^{2}}{4{\rho }^{4}}-\displaystyle \frac{1}{4{\rho }^{2}}+\displaystyle \frac{{\rho }_{y}^{2}}{4{\rho }^{2}}-\displaystyle \frac{{\rho }_{{yy}}}{2\rho }.\end{eqnarray}$

Thus, equations (12) and (13) constitute a Lax representation of the CH2 equation in the new coordinates. After a simple calculation, we find the corresponding system as follows(15a)$\begin{eqnarray}{U}_{\tau }={\rm{i}}{\rho }_{y},\quad \quad \quad {V}_{\tau }=\displaystyle \frac{{\rm{i}}}{2}({\rho }_{{yy}}-U{\rho }_{y}-\rho {U}_{y}),\end{eqnarray}$(15b)$\begin{eqnarray}{\rho }_{{yyy}}=\rho {{UU}}_{y}+{\rho }_{y}{U}^{2}+2{\rho }_{y}{U}_{y}+\rho {U}_{{yy}}-2\rho {V}_{y}-4{\rho }_{y}V.\end{eqnarray}$It is remarked that taking the first equation of (15) into consideration implies that the temporal part of the spectral problem may be simplified as(16)$\begin{eqnarray}{{\rm{\Phi }}}_{\tau }=\displaystyle \frac{1}{4\lambda }\left(2\rho {{\rm{\Phi }}}_{y}-\rho U{\rm{\Phi }}-{\rho }_{y}{\rm{\Phi }}\right),\end{eqnarray}$and we will take (12) and (16) as the spectral problems for (15) in the sequel.

The system (15) is a bit complicated and to get a better understanding of it, we introduce $\sigma =\tfrac{1}{2}({\rho }_{y}-\rho U)$, then it may be rewritten as(17a)$\begin{eqnarray}{\left(\begin{array}{c}U\\ V\end{array}\right)}_{\tau }={\rm{i}}\left(\begin{array}{cc}0 & {\partial }_{y}\\ {\partial }_{y} & 0\end{array}\right)\left(\begin{array}{c}\sigma \\ \rho \end{array}\right),\end{eqnarray}$(17b)$\begin{eqnarray}\left(\begin{array}{cc}2{\partial }_{y} & {\partial }_{y}U-{\partial }_{y}^{2}\\ U{\partial }_{y}+{\partial }_{y}^{2} & V{\partial }_{y}+{\partial }_{y}V\end{array}\right)\left(\begin{array}{c}\sigma \\ \rho \end{array}\right)=0,\end{eqnarray}$since the two matrix differential operators appeared in (17a) and (17b) are the first and second Hamiltonian operators of the Broer–Kaup or two-boson system respectively [36], we succeed in relating the CH2 system to a negative flow of Broer–Kaup hierarchy. The system will be referred as the negative Broer–Kaup (NBK) equation.

3. Darboux transformation and multi-soliton solutions

Having made a connection between the CH2 system (2) and the NBK equation (17), we now need to consider Darboux transformation and its iteration for the latter. In general, for a Lax operator $L=\sum _{i}{u}_{i}{\partial }_{x}^{i}$, it is known that three different classes of integrable nonlinear hierarchies may be constructed as follows$\begin{eqnarray*}\displaystyle \frac{\partial L}{\partial {t}_{q}}=\left[{\left({L}^{q}\right)}_{\geqslant k},L\right],\end{eqnarray*}$where k=0, 1, 2. For the explanation of notations and a thorough discussion of these hierarchies, we refer to the papers of Oevel and his collaborators [37, 38], which also contain the corresponding Darboux transformations. As far as we are concerned, the Lax representation for the Broer–Kaup hierarchy corresponds to the case k=1, which sometimes is dubbed as the nonstandard Lax representation. We now recall the relevant Darboux transformation. Let f=f₁(x, t) be a particular solution of (12) and (16) at λ=λ₁, then the transforms$\begin{eqnarray*}\begin{array}{rcl}{\rm{\Phi }}[1] & = & {\rm{\Phi }}-\displaystyle \frac{{f}_{1}}{{f}_{1y}}{{\rm{\Phi }}}_{y},\qquad \qquad U[1]=U-{\left(\mathrm{ln}\displaystyle \frac{{f}_{1y}}{{f}_{1}}\right)}_{y},\\ V[1] & = & V-{U}_{y}+{\left(\mathrm{ln}{f}_{1}\right)}_{{yy}},\qquad \qquad \rho [1]=\rho +{\rm{i}}{\left(\mathrm{ln}\displaystyle \frac{{f}_{1y}}{{f}_{1}}\right)}_{\tau },\end{array}\end{eqnarray*}$keep the spectral problems (12) and (16) covariant.

It is interesting that the iteration of above Darboux transformation or N-fold Darboux transformation may be represented in terms of Wronksians [39]. Thus, let ${f}_{1},{f}_{2}$, ..., ${f}_{N}$ be solutions of (12), (16) at $\lambda ={\lambda }_{1},{\lambda }_{2}$, ..., ${\lambda }_{N}$ respectively. Then with these solutions, the N-fold Darboux transformation of the NKB equation reads as(18)$\begin{eqnarray}{\rm{\Phi }}[N]=\displaystyle \frac{{\left(-1\right)}^{N}W({f}_{1},{f}_{2},\,\ldots ,\,{f}_{N},{\rm{\Phi }})}{W({f}_{1y},{f}_{2y},\,\ldots ,\,{f}_{{Ny}})},\end{eqnarray}$(19)$\begin{eqnarray}U[N]=U-{\left(\mathrm{ln}\displaystyle \frac{W({f}_{1y},{f}_{2y},\ldots ,{f}_{{Ny}})}{W({f}_{1},{f}_{2},\ldots ,{f}_{N})}\right)}_{y},\end{eqnarray}$(20)$\begin{eqnarray}V[N]=V-{{NU}}_{y}+{\left(\mathrm{ln}W({f}_{N},{f}_{N-1},\ldots ,{f}_{1},)\right)}_{{yy}},\end{eqnarray}$(21)$\begin{eqnarray}\rho [N]=\rho +{\rm{i}}{\left(\mathrm{ln}\displaystyle \frac{W({f}_{1y},{f}_{2y},\ldots ,{f}_{{Ny}})}{W({f}_{1},{f}_{2},\ldots ,{f}_{N})}\right)}_{\tau },\end{eqnarray}$where $W({f}_{1},{f}_{2},\,\ldots ,\,{f}_{N})$ denotes the standard Wronskian.

To obtain multi-soliton solutions for the NBK equation, we take $U=-\tfrac{{\rm{i}}{k}^{2}}{{\rho }_{0}^{2}},$ $V=-\tfrac{{k}^{4}+{\rho }_{0}^{2}}{4{\rho }_{0}^{4}}$ as a seed solution, which corresponds to a seed solution $\rho ={\rho }_{0},u=0$ of the CH2 equation. Then, we calculate the needed f_j's for the N-fold Darboux transformation. To this end, we have to solve the following system$\begin{eqnarray*}\begin{array}{l}{f}_{j,{yy}}-{\rm{i}}\left(2{\lambda }_{j}-\displaystyle \frac{{k}^{2}}{{\rho }_{0}^{2}}\right){f}_{j,y}-\displaystyle \frac{{k}^{4}+{\rho }_{0}^{2}}{4{\rho }_{0}^{4}}{f}_{j}=0,\\ {f}_{j,\tau }-\displaystyle \frac{1}{4{\lambda }_{j}}\left(2{\rho }_{0}{f}_{j,y}+\displaystyle \frac{{\rm{i}}{k}^{2}}{{\rho }_{0}}{f}_{j}\right)=0.\end{array}\end{eqnarray*}$Suppose that α_j, β_j are two solutions of the quadratic equation(22)$\begin{eqnarray}{\mu }^{2}-\left(2{\lambda }_{j}-\displaystyle \frac{{k}^{2}}{{\rho }_{0}^{2}}\right){\rm{i}}\mu -\displaystyle \frac{{k}^{4}+{\rho }_{0}^{2}}{4{\rho }_{0}^{4}}=0,\end{eqnarray}$then functions ${f}_{1},{f}_{2},...,{f}_{N}$ may be found as(23)$\begin{eqnarray}{f}_{j}={c}_{2j-1}{{\rm{e}}}^{{\xi }_{j}}+{c}_{2j}{{\rm{e}}}^{{\eta }_{j}},\end{eqnarray}$(24)$\begin{eqnarray}\begin{array}{rcl}{\xi }_{j} & = & {\alpha }_{j}y+\displaystyle \frac{2{\rho }_{0}^{2}{\alpha }_{j}+{\rm{i}}{k}^{2}}{4{\lambda }_{j}{\rho }_{0}}\tau +{\xi }_{j0},\\ {\eta }_{j} & = & {\beta }_{j}y+\displaystyle \frac{2{\rho }_{0}^{2}{\beta }_{j}+{\rm{i}}{k}^{2}}{4{\lambda }_{j}{\rho }_{0}}\tau +{\eta }_{j0},\end{array}\end{eqnarray}$and ${c}_{2j-1},{c}_{2j},{\xi }_{j0},{\eta }_{j0}$ are arbitrary constants.

Let ${p}_{j}={\alpha }_{j}-{\beta }_{j}$, (23) may be rewritten as(25)$\begin{eqnarray}{f}_{j}={{\rm{e}}}^{\tfrac{{\xi }_{j}+{\eta }_{j}}{2}}\left({c}_{2j-1}{{\rm{e}}}^{{\tilde{\xi }}_{j}}+{c}_{2j}{{\rm{e}}}^{-{\tilde{\xi }}_{j}}\right),\end{eqnarray}$with ${\tilde{\xi }}_{j}=\tfrac{{p}_{j}}{2}(y+\tfrac{{\rho }_{0}}{2{\lambda }_{j}}\tau +{\tilde{\xi }}_{{j}_{0}})$ and ${\tilde{\xi }}_{{j}_{0}}$ arbitrary constants. To obtain the real solutions for the CH2 equation, we should assume that p_j and ${\lambda }_{j}\ne 0$ are real numbers. Thus we have ${k}^{4}+{\rho }_{0}^{2}-{p}_{j}^{2}{\rho }_{0}^{4}\gt 0$ and hence ${\lambda }_{j}=\tfrac{{k}^{2}\mp \sqrt{{k}^{4}+{\rho }_{0}^{2}-{p}_{j}^{2}{\rho }_{0}^{4}}}{2{\rho }_{0}^{2}}$. Furthermore, without loss of generality, we assume ${p}_{j}\gt 0,{\rho }_{0}\gt 0$.

Remark. In general, $-\tfrac{{\rho }_{0}}{2{\lambda }_{j}}$ is the velocity of jth soliton in the coordinate $(y,\tau )$, which is denoted by c_j in [28]. From the expression of λ_j, it follows that $-\tfrac{{\rho }_{0}}{2{\lambda }_{j}}$ solves a quadratic equation which is nothing but the equation $(2.23e)$ of [28] up to some identifications of parameters.

Above we presented the solutions of the NBK equation in terms of the independent variables y, τ by means of Darboux transformation. To construct solution u(x, t), ρ(x, t) of the CH2 equation, we now follow the idea of Li and Zhang [8] (see [13] also) and find the coordinate transformation between x, t and y, τ. To this end, we first consider two fundamental solutions of equation (12) with $U[N],V[N]$ and $\rho [N]$ at $\lambda =0$ which may be generated as(26)$\begin{eqnarray}\left\{\begin{array}{l}{{\rm{\Phi }}}_{1}=\tfrac{{\left(-1\right)}^{N}W({f}_{1},{f}_{2},\ldots ,{f}_{N},{{\rm{e}}}^{\tfrac{{\rho }_{0}-{\rm{i}}{k}^{2}}{2{\rho }_{0}^{2}}y})}{W({f}_{1y},{f}_{2y},\ldots ,{f}_{{Ny}})}\triangleq \tfrac{{\left(-1\right)}^{N}{\bar{W}}_{N}}{{W}_{N}},\\ \\ {{\rm{\Phi }}}_{2}=\tfrac{{\left(-1\right)}^{N}W\left({f}_{1},{f}_{2},\ldots ,{f}_{N},{{\rm{e}}}^{-\tfrac{{\rho }_{0}+{\rm{i}}{k}^{2}}{2{\rho }_{0}^{2}}y}\right)}{W({f}_{1y},{f}_{2y},\ldots ,{f}_{{Ny}})}\triangleq \tfrac{{\left(-1\right)}^{N}{\hat{W}}_{N}}{{W}_{N}}.\end{array}\right.\end{eqnarray}$Their asymptotic properties are:(27)$\begin{eqnarray}{{\rm{\Phi }}}_{1}\sim \prod _{j=1}^{N}\left(1-\displaystyle \frac{{\rho }_{0}-{\rm{i}}{k}^{2}}{2{\rho }_{0}^{2}{\alpha }_{j}}\right){{\rm{e}}}^{\tfrac{{\rho }_{0}-{\rm{i}}{k}^{2}}{2{\rho }_{0}^{2}}y},\ (y\to +\infty ),\end{eqnarray}$(28)$\begin{eqnarray}{{\rm{\Phi }}}_{1}\sim \prod _{j=1}^{N}\left(1-\displaystyle \frac{{\rho }_{0}-{\rm{i}}{k}^{2}}{2{\rho }_{0}^{2}{\beta }_{j}}\right){{\rm{e}}}^{\tfrac{{\rho }_{0}-{\rm{i}}{k}^{2}}{2{\rho }_{0}^{2}}y},\ (y\to -\infty ),\end{eqnarray}$(29)$\begin{eqnarray}{{\rm{\Phi }}}_{2}\sim \prod _{j=1}^{N}\left(\displaystyle \frac{{\rho }_{0}+{\rm{i}}{k}^{2}}{2{\rho }_{0}^{2}{\alpha }_{j}}+1\right){{\rm{e}}}^{-\tfrac{{\rho }_{0}+{\rm{i}}{k}^{2}}{2{\rho }_{0}^{2}}y},\ (y\to +\infty ),\end{eqnarray}$(30)$\begin{eqnarray}{{\rm{\Phi }}}_{2}\sim \prod _{j=1}^{N}\left(\displaystyle \frac{{\rho }_{0}+{\rm{i}}{k}^{2}}{2{\rho }_{0}^{2}{\beta }_{j}}+1\right){{\rm{e}}}^{-\tfrac{{\rho }_{0}+{\rm{i}}{k}^{2}}{2{\rho }_{0}^{2}}y},\ (y\to -\infty ).\end{eqnarray}$

Then, it is noted that when λ=0, equations (3a) and (12) yield(31)$\begin{eqnarray}{{\rm{\Psi }}}_{{xx}}=\displaystyle \frac{1}{4}{\rm{\Psi }},\end{eqnarray}$(32)$\begin{eqnarray}{{\rm{\Phi }}}_{{yy}}=U[N]{{\rm{\Phi }}}_{y}+\left(U{\left[N\right]}_{y}-V[N]\right){\rm{\Phi }}.\end{eqnarray}$The general solution of the equation (31) reads as ${\rm{\Psi }}={d}_{1}{{\rm{e}}}^{\tfrac{x}{2}}+{d}_{2}{{\rm{e}}}^{-\tfrac{x}{2}}$. For equation (32), its general solution is a linear combination of the two fundamental solutions Φ₁ and Φ₂ given by (26). Taking the gauge transformation (11) and the asymptotic properties (27)–(30) into consideration, we may infer that$\begin{eqnarray*}\begin{array}{rcl}{{\rm{e}}}^{\tfrac{x}{2}} & = & \displaystyle \frac{{d}_{3}}{\sqrt{\rho [N]}}{{\rm{e}}}^{-\tfrac{1}{2}\displaystyle \int U[N]{\rm{d}}y}{{\rm{\Phi }}}_{1},\\ {{\rm{e}}}^{-\tfrac{x}{2}} & = & \displaystyle \frac{{d}_{4}}{\sqrt{\rho [N]}}{{\rm{e}}}^{-\tfrac{1}{2}\displaystyle \int U[N]{\rm{d}}y}{{\rm{\Phi }}}_{2},\end{array}\end{eqnarray*}$where d₃, d₄ are constants. Thus, we have ${{\rm{e}}}^{x}=\tfrac{{{\rm{e}}}^{\tfrac{x}{2}}}{{{\rm{e}}}^{-\tfrac{x}{2}}}=\tfrac{{d}_{3}{\bar{W}}_{N}}{{d}_{4}{\hat{W}}_{N}}$. It follows immediately that$\begin{eqnarray*}x=d+\mathrm{ln}\displaystyle \frac{{\bar{W}}_{N}}{{\hat{W}}_{N}},\quad \quad t=\tau ,\quad \quad \quad d={\rm{const.}}\end{eqnarray*}$Summarizing we have the following

Proposition 1. The N-soliton solutions of the CH2 system can be written in parametric form:(33)$\begin{eqnarray}x=d+\mathrm{ln}\displaystyle \frac{{\bar{W}}_{N}}{{\hat{W}}_{N}},\quad \quad t=\tau \end{eqnarray}$(34)$\begin{eqnarray}u={x}_{\tau },\ \ \ \rho =\displaystyle \frac{1}{{x}_{y}},\end{eqnarray}$where ${\bar{W}}_{N}$ and ${\hat{W}}_{N}$ are defined by (26).

4. Examples

In this section, we shall give the parametric representations for one-soliton and two-soliton solutions of the CH2 system explicitly. These solutions are same as the early results obtained by using a direct method combined with a reciprocal transformation [28]. For convenience, we set ${z}_{1}=\tfrac{{\rho }_{0}-{{ik}}^{2}}{2{\rho }_{0}^{2}},{z}_{2}=-\tfrac{{\rho }_{0}+{{ik}}^{2}}{2{\rho }_{0}^{2}}$.

Example 1In this case, $N=1$ and(35)$\begin{eqnarray}{f}_{1}={{\rm{e}}}^{\tfrac{{\xi }_{1}+{\eta }_{1}}{2}}\left({c}_{1}{{\rm{e}}}^{{\tilde{\xi }}_{1}}+{c}_{2}{{\rm{e}}}^{-{\tilde{\xi }}_{1}}\right),\end{eqnarray}$where$\begin{eqnarray*}\begin{array}{rcl}{\tilde{\xi }}_{1} & = & \displaystyle \frac{{p}_{1}}{2}\left(y+\displaystyle \frac{{\rho }_{0}}{2{\lambda }_{1}}\tau +{\tilde{\xi }}_{10}\right),\\ {\lambda }_{1} & = & \displaystyle \frac{{k}^{2}\mp \sqrt{{k}^{4}+{\rho }_{0}^{2}-{p}_{1}^{2}{\rho }_{0}^{4}}}{2{\rho }_{0}^{2}}.\end{array}\end{eqnarray*}$

Notice that because of ${\lambda }_{1}\ne 0$, if ${\lambda }_{1}=\tfrac{{k}^{2}-\sqrt{{k}^{4}+{\rho }_{0}^{2}-{p}_{1}^{2}{\rho }_{0}^{4}}}{2{\rho }_{0}^{2}}$, we should take ${p}_{1}{\rho }_{0}\ne 1$.

Now taking the constants c₁ and c₂ as$\begin{eqnarray*}\begin{array}{rcl}\displaystyle \frac{{c}_{1}}{{c}_{2}} & = & \displaystyle \frac{{z}_{1}-{\beta }_{1}}{{z}_{1}-{\alpha }_{1}}\sqrt{\displaystyle \frac{({z}_{1}-{\alpha }_{1})({z}_{2}-{\beta }_{1})}{({z}_{2}-{\alpha }_{1})({z}_{1}-{\beta }_{1})}}\\ & = & \displaystyle \frac{({\rho }_{0}-{\mathrm{ik}}^{2}-2{\rho }_{0}^{2}{\beta }_{1})}{({\rho }_{0}-{\mathrm{ik}}^{2}-2{\rho }_{0}^{2}{\alpha }_{1})}\sqrt{\displaystyle \frac{1-{p}_{1}{\rho }_{0}+2{k}^{2}{\lambda }_{1}}{1+{p}_{1}{\rho }_{0}+2{k}^{2}{\lambda }_{1}}},\end{array}\end{eqnarray*}$then from equation (33) we find that(36)$\begin{eqnarray}x=\displaystyle \frac{y}{{\rho }_{0}}+\mathrm{ln}\displaystyle \frac{1+\sqrt{\tfrac{1-{p}_{1}{\rho }_{0}+2{k}^{2}{\lambda }_{1}}{1+{p}_{1}{\rho }_{0}+2{k}^{2}{\lambda }_{1}}}{{\rm{e}}}^{2{\tilde{\xi }}_{1}}}{1+\sqrt{\tfrac{1+{p}_{1}{\rho }_{0}+2{k}^{2}{\lambda }_{1}}{1-{p}_{1}{\rho }_{0}+2{k}^{2}{\lambda }_{1}}}{{\rm{e}}}^{2{\tilde{\xi }}_{1}}}+d,\end{eqnarray}$and from (34) we have(37)$\begin{eqnarray}u={x}_{\tau }=\displaystyle \frac{-{p}_{1}^{2}{\rho }_{0}^{2}}{2{\lambda }_{1}{\gamma }_{1}}{\left(\cosh 2{\tilde{\xi }}_{1}+\displaystyle \frac{1+2{k}^{2}{\lambda }_{1}}{{\gamma }_{1}}\right)}^{-1},\end{eqnarray}$(38)$\begin{eqnarray}\rho =\displaystyle \frac{1}{{x}_{y}}={\rho }_{0}+\displaystyle \frac{{p}_{1}^{2}{\rho }_{0}^{3}}{{\gamma }_{1}}{\left(\cosh 2{\tilde{\xi }}_{1}+\displaystyle \frac{(1+2{k}^{2}{\lambda }_{1}-{p}_{1}^{2}{\rho }_{0}^{2})}{{\gamma }_{1}}\right)}^{-1},\end{eqnarray}$where ${\gamma }_{1}=\sqrt{(1-{p}_{1}{\rho }_{0}+2{k}^{2}{\lambda }_{1})(1+{p}_{1}{\rho }_{0}+2{k}^{2}{\lambda }_{1})}$.

Two remarks are in order. First, for ${p}_{1}{\rho }_{0}\ne 1$, $1-{p}_{1}{\rho }_{0}+2{k}^{2}{\lambda }_{1}=\tfrac{1}{2}{\left({p}_{1}{\rho }_{0}-1\right)}^{2}\,+\,2{\rho }_{0}^{2}{\lambda }_{1}^{2}\gt 0$, thus $\sqrt{(1-{p}_{1}{\rho }_{0}+2{k}^{2}{\lambda }_{1})/(1+{p}_{1}{\rho }_{0}+2{k}^{2}{\lambda }_{1})}\gt 0$. Second, one-soliton solution presented here is identical to the one constructed by Matsuno with N=1 (see equation (2.18) and theorem 2.2 of [28]). It is mentioned that the parameters ${p}_{j},k$ and $\tfrac{{\rho }_{0}}{2{\lambda }_{j}}$ here correspond to the parameters ${k}_{j},\kappa $ and $-{c}_{j}$ in [28] respectively.

Example 2For $N=2$, we take(39)$\begin{eqnarray}{f}_{j}={{\rm{e}}}^{\tfrac{{\xi }_{j}+{\eta }_{j}}{2}}\left({c}_{2j-1}{{\rm{e}}}^{{\tilde{\xi }}_{j}}+{c}_{2j}{{\rm{e}}}^{-{\tilde{\xi }}_{j}}\right),(j=1,2)\end{eqnarray}$where$\begin{eqnarray*}{\tilde{\xi }}_{j}=\displaystyle \frac{{p}_{j}}{2}\left(y+\displaystyle \frac{{\rho }_{0}}{2{\lambda }_{j}}\tau +{\tilde{\xi }}_{{j}_{0}}\right),\quad {\lambda }_{j}=\displaystyle \frac{{k}^{2}\mp \sqrt{{k}^{4}+{\rho }_{0}^{2}-{p}_{j}^{2}{\rho }_{0}^{4}}}{2{\rho }_{0}^{2}}.\end{eqnarray*}$Similar to N=1 case, when ${\lambda }_{j}=\tfrac{{k}^{2}-\sqrt{{k}^{4}+{\rho }_{0}^{2}-{p}_{j}^{2}{\rho }_{0}^{4}}}{2{\rho }_{0}^{2}}$, we need ${p}_{j}{\rho }_{0}\ne 1.$ Then choose(40)$\begin{eqnarray}\displaystyle \frac{{c}_{1}}{{c}_{2}}=\displaystyle \frac{({\beta }_{2}-{\beta }_{1})({z}_{1}-{\beta }_{1})}{({\beta }_{2}-{\alpha }_{1})({z}_{1}-{\alpha }_{1})}\sqrt{\displaystyle \frac{({z}_{1}-{\alpha }_{1})({z}_{2}-{\beta }_{1})}{({z}_{2}-{\alpha }_{1})({z}_{1}-{\beta }_{1})}},\end{eqnarray}$(41)$\begin{eqnarray}\displaystyle \frac{{c}_{3}}{{c}_{4}}=\displaystyle \frac{({\beta }_{2}-{\beta }_{1})({z}_{1}-{\beta }_{2})}{({\alpha }_{2}-{\beta }_{1})({z}_{1}-{\alpha }_{2})}\sqrt{\displaystyle \frac{({z}_{1}-{\alpha }_{2})({z}_{2}-{\beta }_{2})}{({z}_{2}-{\alpha }_{2})({z}_{1}-{\beta }_{2})}}.\end{eqnarray}$In this case, (33) and (34) yield(42)$\begin{eqnarray}x=\displaystyle \frac{y}{{\rho }_{0}}+d+\mathrm{ln}\displaystyle \frac{{h}_{1}}{{h}_{2}},\end{eqnarray}$(43)$\begin{eqnarray}u={x}_{\tau }={\left(\mathrm{ln}\displaystyle \frac{{h}_{1}}{{h}_{2}}\right)}_{\tau }=\displaystyle \frac{{h}_{1\tau }}{{h}_{1}}-\displaystyle \frac{{h}_{2\tau }}{{h}_{2}},\end{eqnarray}$(44)$\begin{eqnarray}\rho =\displaystyle \frac{1}{{x}_{y}}=\displaystyle \frac{{h}_{1}{h}_{2}{\rho }_{0}}{{h}_{1}{h}_{2}+{\rho }_{0}({h}_{1y}{h}_{2}-{h}_{1}{h}_{2y})},\end{eqnarray}$with ${h}_{1}=1+{{\rm{e}}}^{2{\tilde{\xi }}_{1}-{{\rm{\Omega }}}_{1}}+{{\rm{e}}}^{2{\tilde{\xi }}_{2}-{{\rm{\Omega }}}_{2}}+\delta {{\rm{e}}}^{2{\tilde{\xi }}_{1}+2{\tilde{\xi }}_{2}-{{\rm{\Omega }}}_{1}-{{\rm{\Omega }}}_{2}}$, ${h}_{2}\,=1+{{\rm{e}}}^{2{\tilde{\xi }}_{1}+{{\rm{\Omega }}}_{1}}+{{\rm{e}}}^{2{\tilde{\xi }}_{2}+{{\rm{\Omega }}}_{2}}+\delta {{\rm{e}}}^{2{\tilde{\xi }}_{1}+2{\tilde{\xi }}_{2}+{{\rm{\Omega }}}_{1}+{{\rm{\Omega }}}_{2}}$, ${{\rm{e}}}^{-{{\rm{\Omega }}}_{j}}=\sqrt{\tfrac{1-{p}_{j}{\rho }_{0}+2{k}^{2}{\lambda }_{j}}{1+{p}_{j}{\rho }_{0}+2{k}^{2}{\lambda }_{j}}}$ ($j=1,2$), $\delta =\tfrac{({\alpha }_{2}-{\alpha }_{1})({\beta }_{2}-{\beta }_{1})}{({\alpha }_{2}-{\beta }_{1})({\beta }_{2}-{\alpha }_{1})}$ =$\,\tfrac{4{\left({\lambda }_{2}-{\lambda }_{1}\right)}^{2}+{\left({p}_{1}-{p}_{2}\right)}^{2}}{4{\left({\lambda }_{2}-{\lambda }_{1}\right)}^{2}+{\left({p}_{1}+{p}_{2}\right)}^{2}}$, which is the same as the solution of N=2 in [28] (equations (2.18), (3.22a), (3.22b), (3.24b), (3.24c)).

For the general $N$, the multi-soliton solution of the CH2 equation may be obtained similarly.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (Grant Nos. 11871471, 11931017, 11505064 and 11805071), Natural Science Foundation of Fujian Province, China (Grant No. 2016J05008), the Yue Qi Outstanding Scholar Project, China University of Mining and Technology, Beijing (Grant No. 00-800015Z1177) and the Fundamental Research Funds for the Central Universities.

Reference View Option By original order
By published year
By cited within times
By Impact factor

[1]Camassa R Holm D D 1993 An integrable shallow water equation with peaked soliton
Phys. Rev. Lett. 71 1661 1664
DOI:10.1103/PhysRevLett.71.1661 [Cited within: 2]

[2]Constantin A 2001 On the scattering problem for the Camassa-Holm equation
Proc. R. Soc. A 457 953 970
DOI:10.1098/rspa.2000.0701 [Cited within: 1]

[3]Beals R Sattinger D H Szmigielski J 1998 Acoustic scattering and the extended Korteweg–de Vries hierarchy
Adv. Math. 140 190 206
DOI:10.1006/aima.1998.1768

[4]Schiff J 1998 The Camassa-Holm equation: a loop group approach
Physica D 121 24 43
DOI:10.1016/S0167-2789(98)00099-2

[5]Constantin A Strauss W A 2000 Stability of peakons
Commun. Pure Appl. Math. 53 603 610
DOI:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L

[6]Constantin A Strauss W A 2002 Stability of the Camassa–Holm solitons
J. Nonlinear Sci. 12 415 422
DOI:10.1007/s00332-002-0517-x

[7]Johnson R S 2003 On solutions of the Camassa-Holm equation
Proc. R. Soc. A 459 1687 1708
DOI:10.1098/rspa.2002.1078

[8]Li Y S Zhang J E 2004 The multiple-soliton solution of the Camassa-Holm equation
Proc. R. Soc. A 460 2617 2627
DOI:10.1098/rspa.2004.1331 [Cited within: 2]

[9]Parker A 2004 On the Camassa-Holm equation and a direct method of the solution: I. Bilinear form and solitary waves
Proc. R. Soc. A 460 2929 2957
DOI:10.1098/rspa.2004.1301

[10]Parker A 2005 On the Camassa-Holm equation and a direct method of the solution: II. Soliton solutions
Proc. R. Soc. A 461 3611 3632
DOI:10.1098/rspa.2005.1536

[11]Parker A 2005 On the Camassa-Holm equation and a direct method of the solution: III. N-soliton solutions
Proc. R. Soc. A 461 3893 3911
DOI:10.1098/rspa.2005.1537

[12]Matsuno Y 2005 Parametric representation for the multisoliton solution of the Camassa-Holm equation
J. Phys. Soc. Japan 74 1983 1987
DOI:10.1143/JPSJ.74.1983

[13]Xia B Q Zhou R G Qiao Z J 2016 Darboux transformation and multi-soliton solutions of the Camassa-Holm equation and modified Camassa–Holm equation
J. Math. Phys. 57 103502
DOI:10.1063/1.4964256 [Cited within: 3]

[14]Fuchssteiner B Fokas A S 1981 Symplectic structures, their Bäcklund transformation and hereditary symmetries
Physica D 4 47 66
DOI:10.1016/0167-2789(81)90004-X [Cited within: 1]

Fuchssteiner B 1981 The Lie algebra structure of nonlinear evolution equations admitting infinite dimensional Abelian symmetry groups
Progr. Theor. Phys. 65 861 876
DOI:10.1016/0167-2789(81)90004-X [Cited within: 1]

[15]Olver P J Rosenau P 1996 Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support
Phys. Rev. E 53 1900 1906
DOI:10.1103/PhysRevE.53.1900 [Cited within: 1]

[16]Schiff J 1996 Zero curvature formulations of dual hierarchies
J. Math. Phys. 37 1928 1938
DOI:10.1063/1.531486 [Cited within: 2]

[17]Liu S Q Zhang Y J 2005 Deformation of semisimple bihamiltonian structures of hydrodynamic type
J. Geom. Phys. 54 427 453
DOI:10.1016/j.geomphys.2004.11.003 [Cited within: 1]

[18]Falqui G 2006 On a Camassa–Holm type equation with two dependent variables
J. Phys. A: Math. Gen. 39 327 342
DOI:10.1088/0305-4470/39/2/004

[19]Chen M Liu S Q Zhang Y J 2006 A two-component generalization of the Camassa-Holm equation and its solutions
Lett. Math. Phys. 75 1 15
DOI:10.1007/s11005-005-0041-7 [Cited within: 2]

[20]Wu C Z 2006 On solutions of the two-component Camassa–Holm system
J. Math. Phys. 47 083513
DOI:10.1063/1.2234729 [Cited within: 2]

[21]Lin J Ren B Li H M Li Y S 2008 Soliton solutions for two nonlinear partial differential equations using a Darboux transformation of the Lax pairs
Phys. Rev. E 77 036605
DOI:10.1103/PhysRevE.77.036605 [Cited within: 1]

[22]Constantin A Ivanov R I 2008 On an integrable two-component Camassa-Holm shallow water system
Phys. Lett. A 372 7129 7132
DOI:10.1016/j.physleta.2008.10.050 [Cited within: 3]

[23]Ivanov R I 2009 Two-component integrable systems modelling shallow water waves: the constant vorticity case
Wave Motion 46 389 396
DOI:10.1016/j.wavemoti.2009.06.012 [Cited within: 3]

[24]Holm D D Tronci C 2009 Geodesic Vlasov equation and their integrable moment closures
J. Geom. Mech. 1 181 208
DOI:10.3934/jgm.2009.1.181

[25]Li J B Li Y S 2008 Bifurcations of travelling wave solutions for a two-component Camassa-Holm equation
Acta. Math. Sin. 24 1319 1330
DOI:10.1007/s10114-008-6207-3 [Cited within: 1]

[26]Dutykh D Kruse D I 2016 Travelling wave solutions for some two-component shallow water models
J. Diff. Eqs. 261 1099 1114
DOI:10.1016/j.jde.2016.03.035

[27]Holm D D Ivanov R I 2011 Two-component CH system: inverse scattering, peakons and geometry
Inverse Prob. 27 045013
DOI:10.1088/0266-5611/27/4/045013 [Cited within: 2]

[28]Matsuno Y 2017 Multisoliton solutions of the two-component Camassa-Holm system and their reductions
J. Phys. A: Math. Theor. 50 345202
DOI:10.1088/1751-8121/aa7e8d [Cited within: 8]

[29]Gui G Liu Y 2010 On the global existence and wave-breaking criteria for the two-component Camassa-Holm system
J. Funct. Anal. 258 4251 4278
DOI:10.1016/j.jfa.2010.02.008 [Cited within: 2]

[30]Hone A N W Novikov V Wang J P 2017 Two-component generalizations of the Camassa-Holm equation
Nonlinearity 30 622 658
DOI:10.1088/1361-6544/aa5490 [Cited within: 1]

[31]Matveev V B Salle M A 1991 Darboux Transformations and SolitonsBerlinSpringer
[Cited within: 1]

[32]Gu C H Hu H S Zhou Z X 2005 Darboux Transformations in Integrable Systems: Theory and their Applications to GeometryBerlinSpringer


[33]Rogers C Schief W 2002 Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton TheoryCambridgeCambridge University Press
[Cited within: 1]

[34]Wu L H Li C Li N H 2019 Soliton solutions to the Novikov equation and a negative flow of the Novikov hierarchy
Appl. Math. Lett. 87 134 140
DOI:10.1016/j.aml.2018.07.036 [Cited within: 1]

[35]Li N H Wang G H Kuang Y H 2020 Darboux transformation for the Degasperis–Procesi equation and its short-wave limit
Theor. Math. Phys.(in press)
[Cited within: 1]

[36]Kupershmidit B A 1985 Mathematics of dispersive water waves
Commun. Math. Phys. 99 51 73
DOI:10.1007/BF01466593 [Cited within: 1]

[37]Oevel W Rogers C 1993 Gauge transformations and reciprocal links in 2 + 1 dimensions
Rev. Math. Phys. 5 299 330
DOI:10.1142/S0129055X93000073 [Cited within: 1]

[38]Konopenchenko B Oevel W 1993 An r-matrix approach to nonstandard classes of integrable equations
Publ. RIMS 29 581 666
DOI:10.2977/prims/1195166743 [Cited within: 1]

[39]Das A Saleem U 2011 Darboux transformation and multi-soliton solutions of two-boson hierarchy
Mod. Phys. Lett. A 26 625 636
DOI:10.1142/S0217732311035249 [Cited within: 1]