1. Introduction
Symmetry plays a key role in the development of physics. Supersymmetry, a concept introduced in the early seventies of last century, is a symmetry between bosons and fermions [1]. Since its inception, the theory related to supersymmetry has been developed considerably both in physics and mathematics.
Interestingly, a little earlier, the modern theory of integrable systems marked its birth in 1965 with the landmark work of Zabusky and Kruskal on the Korteweg–de Vries (KdV) equation. In particular, the inverse scattering transform, also known as nonlinear Fourier transform, was invented by Kruskal and his collaborators and was applied to solve a Cauchy problem for the KdV equation.
The marriage between supersymmetry and integrable systems leads to the development of super or supersymmetric integrable systems. For the celebrated KdV equation, its first super version and first supersymmetric version were introduced by Kupershmidt [2] and Manin and Radul [3], respectively. Those equations have been studied extensively and a large amount of literature exists (see [4–16] and the references therein). More precisely, the supersymmetric KdV (SKdV) equation of Manin and Radul is actually the N = 1 SKdV equation. In the literature, there is also the N = 2 supersymmetric KdV equation1 ), originally proposed by Laberge and Mathieu [17, 19], actually is a family of equations depending on a parameter a, so in the sequel, it will be referred to as the SKdVa equation.
$\begin{eqnarray}\begin{array}{rcl}{\phi }_{t} & = & -{\phi }_{{xxx}}+3{\left(\phi {{ \mathcal D }}_{1}{{ \mathcal D }}_{2}\phi \right)}_{x}\\ & & +\displaystyle \frac{1}{2}(a-1){\left({{ \mathcal D }}_{1}{{ \mathcal D }}_{2}{\phi }^{2}\right)}_{x}+3a{\phi }^{2}{\phi }_{x},\end{array}\end{eqnarray}$
where φ = φ(x, t, θ1, θ2) is a superboson function depending on temporal variable t, spatial variable x and its fermionic counterparts θi(i = 1, 2). ${{ \mathcal D }}_{1}$ and ${{ \mathcal D }}_{2}$ are the super derivatives defined by ${{ \mathcal D }}_{1}={\partial }_{{\theta }_{1}}+{\theta }_{1}{\partial }_{x},{{ \mathcal D }}_{2}={\partial }_{{\theta }_{2}}+{\theta }_{2}{\partial }_{x}$ and a is a free parameter. Equation (The SKdVa equation (1 ) may be rewritten in component form as follows
$\begin{eqnarray}\,{w}_{t}=-{w}_{{xxx}}+(a+2){\left({wu}\right)}_{x}+(a-1){\left(\zeta \eta \right)}_{x}+3{{aw}}^{2}{w}_{x},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{u}_{t}\,=\,-{u}_{{xxx}}+6{{uu}}_{x}-3{{aw}}_{x}{w}_{{xx}}-(a+2){{ww}}_{{xxx}}\\ \,-3\eta {\eta }_{{xx}}-3\zeta {\zeta }_{{xx}}\\ \,+\,3a{\left({w}^{2}u+2w\zeta \eta \right)}_{x},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\,{\zeta }_{t}\,=\,-{\zeta }_{{xxx}}+3{\left(\zeta u\right)}_{x}+(2a+1){w}_{x}{\eta }_{x}\\ \,+\,(a+2)w{\eta }_{{xx}}+(a-1){w}_{{xx}}\eta \\ \,+\,3a{\left({w}^{2}\zeta \right)}_{x},\end{array}\,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\eta }_{t} & = & -{\eta }_{{xxx}}+3{\left(\eta u\right)}_{x}-(2a+1){w}_{x}{\zeta }_{x}-(a+2)w{\zeta }_{{xx}}-(a-1){w}_{{xx}}\zeta \\ & & +\,3a{\left({w}^{2}\eta \right)}_{x},\end{array}\end{eqnarray}$
where φ = w + θ1ζ + θ2(θ1u + η), w = w(x, t), u = u(x, t) are bosonic and ζ = ζ(x, t), η = η(x, t) are fermionic. Now it is easy to see that on the bosonic sector, namely setting ζ = η = 0, one may obtain a coupled KdV-mKdV system (also known as the Kersten–Krasil'shchik coupled KdV-mKdV system) and the situation is quite different from the N = 1 SKdV case, where one has the KdV equation when the fermionic component vanishes. In this sense, the N = 2 SKdV equation is more interesting.It has been shown that the SKdVa equation (1 ) is integrable only for three values of the parameter a. Laberge and Mathieu in [17] obtained Lax representations and Hamiltonian structures and infinite conservation laws for both SKdV−2 and SKdV4 equations. Popowicz found a Lax representation for the SKdV1 equation [18]. The Hirota bilinear forms were worked and applied to construct solutions for both SKdV4 and SKdV1 systems [20].
Darboux transformations are very effective tools in studying integrable systems, so for a given integrable system it is always interesting to construct its Darboux transformations [21–23]. A brief survey of Darboux transformations of supersymmetric integrable systems is available [24]. As the SKdV4 equation in N = 1 form shares the same hierarchy with the supersymmetric two-boson equation, its Darboux transformations were worked out in [25]. For the SKdV−2 equation, a proper Darboux transformation and the related Bäcklund transformation was constructed in [26]. However, the Darboux transformation is still missing for the SKdV1 equation.
The main purpose of this paper is to fill the gap and present a Darboux transformation and the related Bäcklund transformation for the SKdV1 equation. The paper is organized as follows. In section 2 , we recall the Lax representation of the SKdV1 equation and reformulate it in a way that its Darboux and Bäcklund transformation may be worked out. We also compare the Bäcklund transformation with the bilinear Bäcklund transformation in [20]. In section 3 , we build the nonlinear superposition formula for the Bäcklund transformation. As applications, we calculate some solutions of SKdV1 equation. In the final section, we provide some comments on our results and show that the proper reduction allows us to recover the Bäcklund transformation and some solutions for the Kersten–Krasil'shchik coupled KdV-mKdV system.
2. Darboux and Bäcklund transformations
The SKdV1 equation may be read from (1 ), namely3 ) can be brought to
$\begin{eqnarray}{\phi }_{t}={\left(-{\phi }_{{xx}}+3\phi ({{ \mathcal D }}_{1}{{ \mathcal D }}_{2}\phi )+{\phi }^{3}\right)}_{x}.\end{eqnarray}$
We now rewrite it into N = 1 formalism, thus by assuming $\begin{eqnarray}\phi =v+{\theta }_{2}\alpha ,\end{eqnarray}$
where v = v(t, x, θ) is bosonic (even) while α = α(t, x, θ) is fermionic (odd) with θ ≡ θ1 and ${{ \mathcal D }}_{1}\equiv { \mathcal D }$, ( $\begin{eqnarray}{v}_{t}={\left(-{v}_{{xx}}+3v\alpha ^{\prime} +{v}^{3}\right)}_{x},\,\end{eqnarray}$
$\begin{eqnarray}{\alpha }_{t}={\left(-{\alpha }_{{xx}}-3{vv}{{\prime} }_{x}+3\alpha ^{\prime} \alpha +3{v}^{2}\alpha \right)}_{x},\end{eqnarray}$
where for brevity the superscript ${}^{{\prime} }$ denotes the super derivative ${ \mathcal D }$. It is convenient to work with the potential form of system (5), so we introduce the new variables β, φ via α = − βx, v = iφx(i2 = − 1), and obtain $\begin{eqnarray}{\varphi }_{t}=\mbox{--}{\varphi }_{{xxx}}-{\varphi }_{x}^{3}-3{\varphi }_{x}{\beta }_{x}^{\prime} ,\end{eqnarray}$
$\begin{eqnarray}{\beta }_{t}=-{\beta }_{{xxx}}-3{\varphi }_{x}{\varphi }_{{xx}}^{\prime} -3{\beta }_{x}{\beta }_{x}^{\prime} -3{\varphi }_{x}^{2}{\beta }_{x}.\end{eqnarray}$
The spectral problem of (3 ), constructed by Popowicz [18], is4 ), and ψ = − ig + θ2ξ into (7) with g = g(t, x, θ), ξ = ξ(t, x, θ), we find
$\begin{eqnarray}{\psi }_{x}-{\partial }^{-1}{{ \mathcal D }}_{1}{{ \mathcal D }}_{2}(\phi \psi )=\lambda \psi ,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\psi }_{t} & = & -[{\psi }_{{xxx}}-3({{ \mathcal D }}_{1}{{ \mathcal D }}_{2}(\phi {\psi }_{x}))-3{\phi }^{2}{\psi }_{x}\\ & & -3\phi ({{ \mathcal D }}_{2}\phi )({{ \mathcal D }}_{2}\psi )-3\phi ({{ \mathcal D }}_{1}\phi )({{ \mathcal D }}_{1}\psi )],\end{array}\end{eqnarray}$
from it, we can present the spectral problem for system (6). To this end, substituting ( $\begin{eqnarray*}\begin{array}{rcl}{\xi }_{x} & = & \lambda \xi -({\varphi }_{x}g)^{\prime} ,\quad {g}_{x}^{{\prime} }=\lambda g^{\prime} -({\beta }_{x}g+{\varphi }_{x}\xi ),\\ {\xi }_{t} & = & -{\xi }_{{xxx}}-3{\left({\varphi }_{x}{g}_{x}\right)}_{x}^{{\prime} }+3\left({\varphi }_{x}{\beta }_{x}g^{\prime} -{\varphi }_{x}({\varphi }_{x}\xi )^{\prime} \right)^{\prime} ,\\ {g}_{t} & = & -{g}_{{xxx}}-3({\varphi }_{x}{\xi }_{x}+{\beta }_{x}{g}_{x})^{\prime} +3{\varphi }_{x}{\beta }_{x}\xi -3{\varphi }_{x}({\varphi }_{x}g^{\prime} )^{\prime} .\end{array}\end{eqnarray*}$
Furthermore, we define new variables f, h by $f^{\prime} =\xi $, $h^{\prime} ={\beta }_{x}g+{\varphi }_{x}\xi $, and obtain $\begin{eqnarray*}\begin{array}{rcl}{f}_{x} & = & \lambda f-{\varphi }_{x}g,\quad {g}_{x}=\lambda g-h,\quad h^{\prime} ={\beta }_{x}g+{\varphi }_{x}f^{\prime} ,\\ {f}_{t} & = & -{f}_{{xxx}}-3{\left({\varphi }_{x}{g}_{x}\right)}_{x}+3{\varphi }_{x}{\beta }_{x}g^{\prime} -3{\varphi }_{x}({\varphi }_{x}f^{\prime} )^{\prime} ,\\ {g}_{t} & = & -{g}_{{xxx}}-3({\varphi }_{x}{f}_{x}^{{\prime} }+{\beta }_{x}{g}_{x})^{\prime} +3{\varphi }_{x}{\beta }_{x}f^{\prime} -3{\varphi }_{x}({\varphi }_{x}g^{\prime} )^{\prime} ,\\ {h}_{t} & = & \lambda {g}_{t}-{\left({g}_{t}\right)}_{x},\end{array}\end{eqnarray*}$
which constitute a spectral problem for (6). Letting ${\boldsymbol{\chi }}={(f,g,h,f^{\prime} ,g^{\prime} )}^{{\rm{T}}}$, we get the following matrix form of the spectral problem $\begin{eqnarray}\begin{array}{l}{\boldsymbol{\chi }}^{\prime} ={\boldsymbol{M}}{\boldsymbol{\chi }},\quad {{\boldsymbol{\chi }}}_{t}={\boldsymbol{N}}{\boldsymbol{\chi }},\end{array}\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{l}\quad {\boldsymbol{M}}=\left(\begin{array}{ccccc}0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1\\ 0 & {\beta }_{x} & 0 & {\varphi }_{x} & 0\\ \lambda & -{\varphi }_{x} & 0 & 0 & 0\\ 0 & \lambda & -1 & 0 & 0\end{array}\right),\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{\boldsymbol{N}}=\left(\begin{array}{lcccc}-{\lambda }^{3}-\lambda {\varphi }_{x}^{2} & {n}_{12} & \lambda {\varphi }_{x}+{\varphi }_{{xx}} & -{\varphi }_{x}^{{\prime} }{\varphi }_{x} & {\beta }_{x}{\varphi }_{x}\\ \lambda {\varphi }_{{xx}}-{\lambda }^{2}{\varphi }_{x} & {n}_{22} & {\lambda }^{2}+{\varphi }_{x}^{2}+2{\beta }_{x}^{{\prime} } & -\lambda {\varphi }_{x}^{{\prime} }+{\varphi }_{{xx}}^{\prime} +{\beta }_{x}{\varphi }_{x} & {n}_{25}\\ {n}_{31} & {n}_{32} & \lambda {\varphi }_{x}^{2}+{n}_{25}^{\prime} & {n}_{34} & \lambda {n}_{25}+{n}_{52}\\ -\lambda {\varphi }_{x}^{{\prime} }{\varphi }_{x} & {n}_{42} & \lambda {\varphi }_{x}^{{\prime} }+{\varphi }_{{xx}}^{\prime} +{\beta }_{x}{\varphi }_{x} & -{\lambda }^{3} & {n}_{12}+({\varphi }_{x}{\beta }_{x})^{\prime} \\ -\lambda {\beta }_{x}{\varphi }_{x} & {n}_{52} & \lambda {\beta }_{x}+{\beta }_{{xx}}+{\varphi }_{x}^{\prime} {\varphi }_{x} & {n}_{12}+({\varphi }_{x}{\beta }_{x})^{\prime} & -{\lambda }^{3}\end{array}\right)\end{array}\end{eqnarray*}$
with $\begin{eqnarray*}\begin{array}{rcl}{n}_{12} & = & {\varphi }_{{xxx}}+{\varphi }_{x}^{3}+2{\varphi }_{x}{\beta }_{x}^{{\prime} },\\ {n}_{22} & = & -{\lambda }^{3}-\lambda {\beta }_{x}^{{\prime} }+{\beta }_{{xx}}^{\prime} +{\varphi }_{{xx}}{\varphi }_{x},\\ {n}_{25} & = & \lambda {\beta }_{x}-{\beta }_{{xx}}-{\varphi }_{x}^{{\prime} }{\varphi }_{x},\\ {n}_{31} & = & -\lambda {n}_{12}+{\lambda }^{2}{\varphi }_{{xx}}-{\lambda }^{3}{\varphi }_{x},\\ {n}_{32} & = & {n}_{25,x}^{{\prime} }-{\lambda }^{2}{\beta }_{x}^{{\prime} }+\lambda {\varphi }_{{xx}}{\varphi }_{x}-2{\beta }_{x}^{{\prime} 2}+{\varphi }_{x}^{4}+{\varphi }_{x}^{2}{\beta }_{x}^{{\prime} }\\ & & \quad \quad -{\beta }_{{xx}}{\beta }_{x}+{\varphi }_{{xx}}^{\prime} {\varphi }_{x}^{{\prime} }-2{\varphi }_{x}^{{\prime} }{\beta }_{x}{\varphi }_{x},\\ {n}_{34} & = & -{\lambda }^{2}{\varphi }_{x}^{{\prime} }+\lambda {\beta }_{x}{\varphi }_{x}+\lambda {\varphi }_{{xx}}^{\prime} -{n}_{12}^{\prime} -{\beta }_{x}{\varphi }_{{xx}}-{\varphi }_{x}^{2}{\varphi }_{x}^{{\prime} },\\ {n}_{42} & = & {n}_{12}^{{\prime} }-{\varphi }_{x}^{2}{\varphi }_{x}^{{\prime} }+{\beta }_{x}{\varphi }_{{xx}},\\ {n}_{52} & = & {\beta }_{{xxx}}+{\varphi }_{{xx}}{\varphi }_{x}^{{\prime} }+2{\varphi }_{x}{\varphi }_{{xx}}^{{\prime} }+2{\varphi }_{x}^{2}{\beta }_{x}+2{\beta }_{x}{\beta }_{x}^{{\prime} }.\end{array}\end{eqnarray*}$
The matrices M, N whose entries involve both bosonic and fermionic variables are super matrices. Following [27], we introduce an involution on the algebra of super matrices as follows: given any matrix ${\boldsymbol{A}}={({a}_{{ij}})}_{i,j\in {\mathbb{Z}}}$, we define ${{\boldsymbol{A}}}^{\dagger }={({a}_{{ij}}^{\dagger })}_{i,j\in {\mathbb{Z}}}$ and ${a}_{{ij}}^{\dagger }={(-1)}^{p({a}_{{ij}})}{a}_{{ij}}$ with p(aij) denoting the parity of aij.Our main aim is to construct a Darboux transformation for (8 ). To this end, we search for a gauge matrix T such that8 ) covariant, namely it solves
$\begin{eqnarray}\widetilde{{\boldsymbol{\chi }}}={\boldsymbol{T}}{\boldsymbol{\chi }}\end{eqnarray}$
keeps the spectral problem ( $\begin{eqnarray}\widetilde{{\boldsymbol{\chi }}}^{\prime} =\widetilde{{\boldsymbol{M}}}\widetilde{{\boldsymbol{\chi }}},\end{eqnarray}$
and $\begin{eqnarray*}{\widetilde{{\boldsymbol{\chi }}}}_{t}=\widetilde{{\boldsymbol{N}}}\widetilde{{\boldsymbol{\chi }}},\end{eqnarray*}$
where $\widetilde{{\boldsymbol{M}}}$ and $\widetilde{{\boldsymbol{N}}}$ are the matrices M and M but with φ, β replaced by the new field variables $\widetilde{\varphi },\widetilde{\beta }$. This imposes conditions for the gauge matrix T, that is, $\begin{eqnarray}{\boldsymbol{T}}^{\prime} +{{\boldsymbol{T}}}^{\dagger }{\boldsymbol{M}}-\widetilde{{\boldsymbol{M}}}{\boldsymbol{T}}=0,\quad {{\boldsymbol{T}}}_{t}+{\boldsymbol{TN}}-\widetilde{{\boldsymbol{N}}}{\boldsymbol{T}}=0.\end{eqnarray}$
Thus, finding a proper Darboux transformation means that we have to search a solution of the system (11 ). For simplicity, let us denote
$\begin{eqnarray}\begin{array}{l}z\equiv \widetilde{\varphi }-\varphi ,\quad \rho \equiv \widetilde{\beta }-\beta .\end{array}\end{eqnarray}$
Through tedious calculations, we obtain the following result. 11 ) if only if z and ρ satisfy
Let
$\begin{eqnarray}{\boldsymbol{T}}=\lambda {\boldsymbol{H}}+{\boldsymbol{G}},\end{eqnarray}$
where the matrices ${\boldsymbol{H}}$ and ${\boldsymbol{G}}$ are given as follows: $\begin{eqnarray*}\begin{array}{rcl}{\boldsymbol{H}} & = & \left(\begin{array}{ccccc}-c & s & 0 & 0 & 0\\ s & -c & 0 & 0 & 0\\ -{lb} & b+\rho ^{\prime} -2{\lambda }_{1} & 1 & 0 & 0\\ \omega -\rho & \sigma & 0 & -c & s\\ \sigma -z^{\prime} & \omega & 0 & s & -c\end{array}\right),\\ {\boldsymbol{G}} & = & \left(\begin{array}{ccccc}-{\lambda }_{1} & s{\rm{\Delta }} & -s & \omega -c\rho & -\sigma +{cz}^{\prime} \\ 0 & {\lambda }_{1}-(1+c){\rm{\Delta }} & 1+c & \sigma +s\rho & -\omega -{sz}^{\prime} \\ 0 & {\lambda }_{1}(\rho ^{\prime} -2{\lambda }_{1})+b{\rm{\Delta }} & {\lambda }_{1}-b & -\displaystyle \frac{1}{2}b\left(z^{\prime} +l\rho \right) & \displaystyle \frac{1}{2}b\left(\rho +{lz}^{\prime} \right)\\ 0 & \sigma {\rm{\Delta }} & -\sigma & \displaystyle \frac{1}{2}s\rho z^{\prime} -{\lambda }_{1}c & {\lambda }_{1}s-\displaystyle \frac{1}{2}c\rho z^{\prime} \\ 0 & \omega {\rm{\Delta }} & -\omega & {\lambda }_{1}s-\displaystyle \frac{1}{2}c\rho z^{\prime} & \displaystyle \frac{1}{2}s\rho z^{\prime} -{\lambda }_{1}c\end{array}\right),\end{array}\end{eqnarray*}$
with $\begin{eqnarray*}\begin{array}{rcl}c & = & \cosh z,\quad \,\,s=\sinh z,\quad \,\,l=\tanh \displaystyle \frac{z}{2},\\ \sigma & = & \displaystyle \frac{1}{2}\left((1+c)z^{\prime} -s\rho \right),\qquad \,\,b=-s{\varphi }_{x}+c(2{\lambda }_{1}-\rho ^{\prime} ),\\ \omega & = & \displaystyle \frac{1}{2}\left((1+c)\rho -{sz}^{\prime} \right),\qquad \,\,{\rm{\Delta }}=l{\varphi }_{x}+\displaystyle \frac{2{\lambda }_{1}-\rho ^{\prime} }{1+c}.\end{array}\end{eqnarray*}$
Then, ${\boldsymbol{T}}$ satisfies ( $\begin{eqnarray}{z}_{x}=-2{\varphi }_{x}+(2{\lambda }_{1}-\rho ^{\prime} )\tanh \displaystyle \frac{z}{2},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{x} & = & -(1+\cosh z){\beta }_{x}+{\varphi }_{x}^{{\prime} }\sinh z\\ & & +\displaystyle \frac{1}{2}(2{\lambda }_{1}-\rho ^{\prime} )\left(\rho -z^{\prime} \tanh \displaystyle \frac{z}{2}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{z}_{t} & = & -{z}_{{xxx}}-3{\lambda }_{1}^{2}{z}_{x}+3{\lambda }_{1}{\left(z+2\varphi \right)}_{{xx}}+3{\lambda }_{1}\rho ^{\prime} {z}_{x}-\displaystyle \frac{1}{4}({z}_{x}^{3}+3{z}_{x}{\left(\rho ^{\prime} \right)}^{2})\\ & & -\displaystyle \frac{3}{2}{z}_{x}{\left(\rho +2\beta \right)}_{x}^{{\prime} }-\displaystyle \frac{3}{2}\rho ^{\prime} {\left(z+2\varphi \right)}_{{xx}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{t} & = & -{\rho }_{{xxx}}-3{\lambda }_{1}^{2}{\rho }_{x}+3{\lambda }_{1}{\left(\rho +2\beta \right)}_{{xx}}\\ & & -\,\displaystyle \frac{3}{4}((\rho ^{\prime} {z}_{x}^{2})^{\prime} +{\left(\rho ^{\prime} \right)}^{2}{\rho }_{x})\\ & & +\,\displaystyle \frac{3}{2}\left[{\lambda }_{1}{\left(\rho ^{\prime} \right)}^{2}+{\lambda }_{1}{z}_{x}^{2}-\rho ^{\prime} {\left(\rho +2\beta \right)}_{x}^{{\prime} }-{z}_{x}{\left(z+2\varphi \right)}_{{xx}}\right]^{\prime} .\end{array}\end{eqnarray}$
It is a tedious but direct calculation. Indeed, it may be checked that the first equation of (
The Darboux transformation presented in this proposition is defined in terms of z and ρ, which are governed by the system of differential equations (14a –14d ). It is interesting to note that we may relate z, ρ to a particular solution of (8 ) by the following Proposition.
Let ${{\boldsymbol{\chi }}}_{1}={\left({f}_{1},{g}_{1},{h}_{1},{f}_{1}^{\prime} ,{g}_{1}^{\prime} \right)}^{{\rm{T}}}$ be one solution of (
$\begin{eqnarray}\begin{array}{l}z=\mathrm{ln}\displaystyle \frac{{g}_{1}+{f}_{1}}{{g}_{1}-{f}_{1}},\quad \rho =\left(\mathrm{ln}\left|{f}_{1}^{2}-{g}_{1}^{2}\right|\right)^{\prime} \end{array}\end{eqnarray}$
solve (14).We point out that (14a –14d ) constitutes a Bäcklund transformation of (6). For instance, inserting (12 ) into the first two of (14) leads to
$\begin{eqnarray}\begin{array}{rcl}{\widetilde{\varphi }}_{x} & = & -{\varphi }_{x}+(2{\lambda }_{1}-(\widetilde{\beta }-\beta )^{\prime} )\tanh \left(\displaystyle \frac{\widetilde{\varphi }-\varphi }{2}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\widetilde{\beta }}_{x}\,=\,-{\beta }_{x}\cosh (\widetilde{\varphi }-\varphi )+{\varphi }_{x}^{{\prime} }\sinh (\widetilde{\varphi }-\varphi )\\ \,+\,{\lambda }_{1}(\widetilde{\beta }-\beta )-\displaystyle \frac{1}{2}(\widetilde{\beta }-\beta )^{\prime} (\widetilde{\beta }-\beta )-\displaystyle \frac{1}{2}(\widetilde{\varphi }-\varphi )^{\prime} {\left(\widetilde{\varphi }+\varphi \right)}_{x},\end{array}\end{eqnarray}$
which is the spatial part of Bäcklund transformation.In the remaining part of this section, we shall show the obtained Bäcklund transformation coincides with the one constructed in [20] by Hirota's bilinear method.
According to [20], the following transformation for the dependent variables
$\begin{eqnarray*}\varphi =\mathrm{ln}\displaystyle \frac{{\tau }_{1}}{{\tau }_{2}},\quad \beta =(\mathrm{ln}{\tau }_{1}{\tau }_{2})^{\prime} ,\quad \widetilde{\varphi }=\mathrm{ln}\displaystyle \frac{\widetilde{{\tau }_{1}}}{\widetilde{{\tau }_{2}}},\quad \widetilde{\beta }=(\mathrm{ln}\widetilde{{\tau }_{1}}\widetilde{{\tau }_{2}})^{\prime} \end{eqnarray*}$
brings (6) into the Hirota bilinear form: $\begin{eqnarray}({{\rm{D}}}_{t}+{{\rm{D}}}_{x}^{3}){\tau }_{1}\cdot {\tau }_{2}=0,\quad {\rm{S}}({{\rm{D}}}_{t}+{{\rm{D}}}_{x}^{3}){\tau }_{1}\cdot {\tau }_{2}=0.\end{eqnarray}$
Then, the corresponding bilinear Bäcklund transformation is given by $\begin{eqnarray}({{\rm{D}}}_{x}+{\lambda }_{1}){\tau }_{2}\cdot \widetilde{{\tau }_{1}}-({{\rm{D}}}_{x}+{\lambda }_{1}){\tau }_{1}\cdot \widetilde{{\tau }_{2}}=0,\end{eqnarray}$
$\begin{eqnarray}({\mathrm{SD}}_{x}+{\lambda }_{1}{\rm{S}}){\tau }_{2}\cdot \widetilde{{\tau }_{1}}+({\mathrm{SD}}_{x}+{\lambda }_{1}{\rm{S}}){\tau }_{1}\cdot \widetilde{{\tau }_{2}}=0,\end{eqnarray}$
$\begin{eqnarray}({{\rm{D}}}_{t}+{{\rm{D}}}_{x}^{3}+3{\lambda }_{1}{{\rm{D}}}_{x}^{2}+3{\lambda }_{1}^{2}{{\rm{D}}}_{x}){\tau }_{2}\cdot \widetilde{{\tau }_{2}}=0,\end{eqnarray}$
$\begin{eqnarray}({{\rm{D}}}_{t}+{{\rm{D}}}_{x}^{3}+3{\lambda }_{1}{{\rm{D}}}_{x}^{2}+3{\lambda }_{1}^{2}{{\rm{D}}}_{x}){\tau }_{1}\cdot \widetilde{{\tau }_{1}}=0.\end{eqnarray}$
Here, the bilinear operators are defined by
$\begin{eqnarray*}\begin{array}{l}{{\rm{S}}}^{k}{{\rm{D}}}_{t}^{m}{{\rm{D}}}_{x}^{n}a\cdot b\\ \,={({{ \mathcal D }}_{0}-{ \mathcal D })}^{k}{\left({\partial }_{{t}_{0}}-{\partial }_{t}\right)}^{m}{\left({\partial }_{{x}_{0}}-{\partial }_{x}\right)}^{n}a({x}_{0},{t}_{0},{\theta }_{0})b(x,t,\theta )\left|{}_{\begin{array}{l}{t}_{0}=t\\ {x}_{0}=x\\ {\theta }_{0}=\theta \end{array}}\right.\end{array}\end{eqnarray*}$
with ${{ \mathcal D }}_{0}={\partial }_{{\theta }_{0}}+{\theta }_{0}{\partial }_{{x}_{0}}.$ Now, let $\begin{eqnarray*}\begin{array}{rcl} & & p\,=\mathrm{ln}\displaystyle \frac{{\tau }_{2}}{\widetilde{{\tau }_{1}}},\quad \,q=\mathrm{ln}({\tau }_{2}\widetilde{{\tau }_{1}}),\overline{p}\,=\mathrm{ln}\displaystyle \frac{{\tau }_{1}}{\widetilde{{\tau }_{2}}},\quad \overline{q}\,=\mathrm{ln}(\widetilde{{\tau }_{2}}{\tau }_{1}),\\ & & {p}_{1}=\mathrm{ln}\displaystyle \frac{{\tau }_{1}}{\widetilde{{\tau }_{1}}},\quad {q}_{1}=\mathrm{ln}({\tau }_{1}\widetilde{{\tau }_{1}}),\quad {p}_{2}=\mathrm{ln}\displaystyle \frac{{\tau }_{2}}{\widetilde{{\tau }_{2}}},\quad {q}_{2}=\mathrm{ln}({\tau }_{2}\widetilde{{\tau }_{2}}).\end{array}\end{eqnarray*}$
Then, (18) leads to $\begin{eqnarray}{\overline{p}}_{x}+{\lambda }_{1}-({p}_{x}+{\lambda }_{1}){e}^{q-\overline{q}}=0,\end{eqnarray}$
$\begin{eqnarray}{\overline{q}}_{x}^{{\prime} }+{\overline{p}}_{x}\overline{p}^{\prime} +{\lambda }_{1}\overline{p}^{\prime} +({q}_{x}^{{\prime} }+{p}_{x}p^{\prime} +{\lambda }_{1}p^{\prime} ){e}^{q-\overline{q}}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{p}_{2,t}+{p}_{2,x}^{3}+3{p}_{2,x}{q}_{2,{xx}}+{p}_{2,{xxx}}+3{\lambda }_{1}({p}_{2,x}^{2}+{q}_{2,{xx}})\\ +3{\lambda }_{1}^{2}{p}_{2,x}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}{p}_{1,t}+{p}_{1,x}^{3}+3{p}_{1,x}{q}_{1,{xx}}+{p}_{1,{xxx}}+3{\lambda }_{1}({p}_{1,x}^{2}+{q}_{1,{xx}})+3{\lambda }_{1}^{2}{p}_{1,x}=0.\end{eqnarray}$
Due to
$\begin{eqnarray*}p=\displaystyle \frac{1}{2}({{ \mathcal D }}^{-1}(\beta -\widetilde{\beta })-\widetilde{\varphi }-\varphi ),\quad q=\displaystyle \frac{1}{2}({{ \mathcal D }}^{-1}(\beta +\widetilde{\beta })+\widetilde{\varphi }-\varphi ),\end{eqnarray*}$
$\begin{eqnarray*}\overline{p}=\displaystyle \frac{1}{2}({{ \mathcal D }}^{-1}(\beta -\widetilde{\beta })+\widetilde{\varphi }+\varphi ),\quad \overline{q}=\displaystyle \frac{1}{2}({{ \mathcal D }}^{-1}(\beta +\widetilde{\beta })-\widetilde{\varphi }+\varphi ),\end{eqnarray*}$
$\begin{eqnarray*}{p}_{2}=\displaystyle \frac{1}{2}({{ \mathcal D }}^{-1}(\beta -\widetilde{\beta })+\widetilde{\varphi }-\varphi ),\quad {q}_{2}=\displaystyle \frac{1}{2}({{ \mathcal D }}^{-1}(\beta +\widetilde{\beta })-\widetilde{\varphi }-\varphi ),\end{eqnarray*}$
$\begin{eqnarray*}{p}_{1}=\displaystyle \frac{1}{2}({{ \mathcal D }}^{-1}(\beta -\widetilde{\beta })-\widetilde{\varphi }+\varphi ),\quad {q}_{1}=\displaystyle \frac{1}{2}({{ \mathcal D }}^{-1}(\beta +\widetilde{\beta })+\widetilde{\varphi }+\varphi ),\end{eqnarray*}$
(19) is nothing but (14), this shows that the Bäcklund transformation resulted from the Darboux transformation is the same as the Bäcklund transformation derived from the framework of Hirota's method.3. Nonlinear superposition formula
One of the important ingredients in Bäcklund transformation is the nonlinear superposition formula. Indeed, a Bäcklund transformation for a given nonlinear partial differential equation is meant to supply a way to construct its solutions, but Bäcklund transformation is differential equation itself and solving it may not be easy in general. An well-known way to overcome this difficulty is to derive a nonlinear superposition formula.
The main aim of this section is to construct the corresponding nonlinear superposition formula for the Bäcklund transformation presented in last section.
For clarity, the gauge matrix in (13 ) is denoted by ${\boldsymbol{T}}({\lambda }_{1};\varphi ,\beta ,\widetilde{\varphi },\widetilde{\beta })$. Consider a pair of Darboux transformations 
we have22 ), after some tedious calculations we obtain the following nonlinear superposition formula
$\begin{eqnarray}\widetilde{{\boldsymbol{\chi }}}={{\boldsymbol{T}}}_{1}{\boldsymbol{\chi }},\quad {{\boldsymbol{T}}}_{1}\equiv {\boldsymbol{T}}({\lambda }_{1};\varphi ,\beta ,\widetilde{\varphi },\widetilde{\beta }),\end{eqnarray}$
$\begin{eqnarray}\widehat{{\boldsymbol{\chi }}}={{\boldsymbol{T}}}_{2}{\boldsymbol{\chi }},\quad {{\boldsymbol{T}}}_{2}\equiv {\boldsymbol{T}}({\lambda }_{2};\varphi ,\beta ,\widehat{\varphi },\widehat{\beta }).\end{eqnarray}$
Then, with the help of the Bianchi's permutability theorem, depicted by the diagram below $\begin{eqnarray}\begin{array}{l}{\widehat{{\boldsymbol{T}}}}_{1}{{\boldsymbol{T}}}_{2}={\widetilde{{\boldsymbol{T}}}}_{2}{{\boldsymbol{T}}}_{1},\end{array}\end{eqnarray}$
where ${\widehat{{\boldsymbol{T}}}}_{1}\equiv {\boldsymbol{T}}({\lambda }_{1};\widehat{\varphi },\widehat{\beta }$, $\widehat{\widetilde{\varphi }},\widehat{\widetilde{\beta }})$, ${\widetilde{{\boldsymbol{T}}}}_{2}\equiv {\boldsymbol{T}}({\lambda }_{2}$; $\widetilde{\varphi },\widetilde{\beta }$, $\widetilde{\widehat{\varphi }},\widetilde{\widehat{\beta }})$. From $\widehat{\widetilde{\varphi }}=\widetilde{\widehat{\varphi }},\widehat{\widetilde{\beta }}=\widetilde{\widehat{\beta }}$ and ( $\begin{eqnarray}\begin{array}{rcl}{e}^{\widehat{\widetilde{\varphi }}-\varphi } & = & \displaystyle \frac{{k}_{0}}{{{\ell }}_{0}}+{{\ell }}_{1}[(\widetilde{\beta }-\beta )(\widehat{\beta }-\beta )+(\widehat{\varphi }^{\prime} -\varphi ^{\prime} )(\widetilde{\varphi }^{\prime} -\varphi ^{\prime} )]\\ & & +\,{{\ell }}_{2}[(\widetilde{\varphi }^{\prime} -\varphi ^{\prime} )(\widehat{\beta }-\beta )+(\widehat{\varphi }^{\prime} -\varphi ^{\prime} )(\widetilde{\beta }-\beta )]\\ & & +\,{{\ell }}_{3}(\widetilde{\beta }-\beta )(\widehat{\beta }-\beta )(\widetilde{\varphi }^{\prime} -\varphi ^{\prime} )(\widehat{\varphi }^{\prime} -\varphi ^{\prime} ),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\widehat{\widetilde{\beta }}-\beta & = & {k}_{1}(\widetilde{\beta }-\beta )+{k}_{2}(\widehat{\beta }-\beta )\\ & & +{k}_{3}(\widetilde{\varphi }^{\prime} -\varphi ^{\prime} )+{k}_{4}(\widehat{\varphi }^{\prime} -\varphi ^{\prime} )\\ & & +\,{k}_{5}(\widehat{\varphi }^{\prime} -\varphi ^{\prime} )(\widetilde{\varphi }^{\prime} -\varphi ^{\prime} )(\widetilde{\beta }-\widehat{\beta })+{k}_{6}(\widehat{\varphi }^{\prime} -\widetilde{\varphi }^{\prime} )(\widehat{\beta }-\beta )(\widetilde{\beta }-\beta ),\end{array}\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}{r}_{1} & = & {{\rm{e}}}^{\widetilde{\varphi }-\varphi },\quad \,{r}_{2}={{\rm{e}}}^{\widehat{\varphi }-\varphi },\quad \,R=({r}_{2}+1)({r}_{1}+1),\,\,\epsilon ={\lambda }_{1}+{\lambda }_{2},\\ A & = & R({r}_{1}-{r}_{2}){\varphi }_{x}-{r}_{1}{\left({r}_{2}+1\right)}^{2}(\widetilde{\beta }-\beta )^{\prime} \\ & & +{r}_{2}{\left({r}_{1}+1\right)}^{2}(\widehat{\beta }-\beta )^{\prime} ,\\ {k}_{0} & = & A-{\lambda }_{1}({r}_{2}+1)({r}_{1}^{2}-2{r}_{1}{r}_{2}-{r}_{1})+{\lambda }_{2}({r}_{1}+1)({r}_{2}^{2}-2{r}_{1}{r}_{2}-{r}_{2}),\\ {l}_{0} & = & A+{\lambda }_{1}({r}_{2}+1)({r}_{1}{r}_{2}+2{r}_{1}-{r}_{2})\\ & & -{\lambda }_{2}({r}_{1}+1)({r}_{1}{r}_{2}+2{r}_{2}-{r}_{1}),\\ {k}_{1} & = & -\displaystyle \frac{\epsilon R({r}_{1}{{\ell }}_{0}+{k}_{0}{r}_{2})}{2{{\ell }}_{0}{k}_{0}},\qquad {k}_{2}=\displaystyle \frac{\epsilon R({r}_{2}{{\ell }}_{0}+{k}_{0}{r}_{1})}{2{{\ell }}_{0}{k}_{0}},\\ {k}_{3} & = & -\displaystyle \frac{\epsilon R({r}_{1}{{\ell }}_{0}-{k}_{0}{r}_{2})}{2{{\ell }}_{0}{k}_{0}},\qquad {k}_{4}=\displaystyle \frac{\epsilon R({r}_{2}{{\ell }}_{0}-{k}_{0}{r}_{1})}{2{{\ell }}_{0}{k}_{0}},\\ {k}_{5} & = & \displaystyle \frac{\epsilon {r}_{1}{r}_{2}{R}^{2}({{\ell }}_{0}^{2}+{k}_{0}^{2})}{4{{\ell }}_{0}^{2}{k}_{0}^{2}},\qquad \,\,{k}_{6}=\displaystyle \frac{{\epsilon }^{2}{r}_{1}{r}_{2}{R}^{3}({r}_{1}-{r}_{2})({{\ell }}_{0}+{k}_{0})}{4{{\ell }}_{0}^{2}{k}_{0}^{2}},\\ {{\ell }}_{1} & = & \displaystyle \frac{\epsilon {R}^{2}({r}_{1}^{2}-{r}_{2}^{2})}{4{{\ell }}_{0}^{2}},\qquad \qquad \,\,{{\ell }}_{2}=\displaystyle \frac{\epsilon {R}^{2}{\left({r}_{1}-{r}_{2}\right)}^{2}}{4{{\ell }}_{0}^{2}},\\ {{\ell }}_{3} & = & \displaystyle \frac{\epsilon {r}_{1}{r}_{2}{R}^{3}({r}_{1}-{r}_{2})}{2{{\ell }}_{0}^{3}}.\end{array}\end{eqnarray*}$
While the nonlinear superposition formula (23 )-(24 ) takes a rather complicated form, it is of algebro-differential in the sense that with three known solutions of the potential SKdV equation (6), one can construct the fourth solution via differentiating and algebraic manipulations.
As a simple application of Darboux transformation and nolinear superposition formula, we now calculate some solutions for SKdV1 equation.
We begin with the trivial seed solution φ = 0, β = 0. Then the linear spectral problem becomes15 ) together with (12 ), and obtain
$\begin{eqnarray*}\begin{array}{rcl}{f}_{x} & = & \lambda f,\qquad {g}_{x}=\lambda g-h,\quad h^{\prime} =0,\\ {f}_{t} & = & -{\lambda }^{3}f,\quad {g}_{t}=-{\lambda }^{3}g,\quad \,{h}_{t}=0,\end{array}\end{eqnarray*}$
which may be solved by $\begin{eqnarray*}f={{\rm{e}}}^{\lambda x-{\lambda }^{3}t+\theta \mu +{c}_{1}},\quad g={{\rm{e}}}^{\lambda x-{\lambda }^{3}t+\theta \nu +{c}_{2}}+{c}_{3},\quad h={c}_{3}\lambda ,\end{eqnarray*}$
where ci (i = 1, 2, 3) are even constants, μ, ν are fermionic constants. To recover the one-soliton solution obtained in [20], we take $\begin{eqnarray*}{f}_{1}={\rm{i}}{e}^{{P}_{1}},\quad {g}_{1}=1,\quad {h}_{1}={\lambda }_{1},\quad {P}_{1}={\lambda }_{1}x-{\lambda }_{1}^{3}t+\theta {\mu }_{1}+{c}_{1},\end{eqnarray*}$
and substitute them into ( $\begin{eqnarray*}\widetilde{\varphi }=\mathrm{ln}\left[-\coth \displaystyle \frac{2{P}_{1}+\pi {\rm{i}}}{4}\right],\quad \widetilde{\beta }={P}_{1}^{\prime} (1+\tanh \,{P}_{1}).\end{eqnarray*}$
Then $\begin{eqnarray*}\begin{array}{l}\widetilde{v}={\rm{i}}{\widetilde{\varphi }}_{x}=-{\lambda }_{1}{\rm{sech}} \,{P}_{1},\\ \,\widetilde{\alpha }=-{\widetilde{\beta }}_{x}=-{\lambda }_{1}({\mu }_{1}+\theta {\lambda }_{1}){{\rm{sech}} }^{2}{P}_{1}.\end{array}\end{eqnarray*}$
To proceed, we introduce the solutions23 ) and (24 ), we have
$\begin{eqnarray*}\begin{array}{rcl} & & {{\rm{e}}}^{\widetilde{\varphi }}=-\coth \displaystyle \frac{2{P}_{1}+\pi {\rm{i}}}{4},\quad \widetilde{\beta }={P}_{1}^{\prime} (1+\tanh {P}_{1}),\\ & & {{\rm{e}}}^{\widehat{\varphi }}=-\coth \displaystyle \frac{2{P}_{2}-\pi {\rm{i}}}{4},\quad \widehat{\beta }={P}_{2}^{\prime} (1+\tanh {P}_{2}),\end{array}\end{eqnarray*}$
where ${P}_{j}={\lambda }_{j}x-{\lambda }_{j}^{3}t+\theta {\mu }_{j}+{c}_{j},{P}_{j}^{\prime} ={\mu }_{j}+\theta {\lambda }_{j}$, j = 1, 2. Plugging the above solutions and the trivial seed solution into the ( $\begin{eqnarray*}\begin{array}{rcl}\widehat{\widetilde{\varphi }} & = & \mathrm{ln}\left(\displaystyle \frac{{b}_{0}\,\cosh (\tfrac{{P}_{1}+{P}_{2}}{2})-{b}_{1}{{\rm{e}}}^{\tfrac{{P}_{1}+{P}_{2}}{2}}-{\rm{i}}\,\cosh (\tfrac{{P}_{1}-{P}_{2}}{2})}{{b}_{0}\,\cosh (\tfrac{{P}_{1}+{P}_{2}}{2})-{b}_{1}{{\rm{e}}}^{\tfrac{{P}_{1}+{P}_{2}}{2}}+{\rm{i}}\,\cosh (\tfrac{{P}_{1}-{P}_{2}}{2})}\right),\\ \widehat{\widetilde{\beta }} & = & \displaystyle \frac{({P}_{1}^{\prime} -{P}_{2}^{\prime} )\left[{b}_{0}+{b}_{0}{{\rm{e}}}^{{P}_{1}+{P}_{2}}+\sinh ({P}_{1}-{P}_{2})\right]+({P}_{1}^{\prime} +{P}_{2}^{\prime} )[1+\cosh ({P}_{1}-{P}_{2})]}{1+\cosh ({P}_{1}-{P}_{2})+{b}_{0}^{2}(1+\cosh ({P}_{1}+{P}_{2}))},\end{array}\end{eqnarray*}$
where $\begin{eqnarray*}{b}_{0}=\displaystyle \frac{{\lambda }_{1}-{\lambda }_{2}}{{\lambda }_{1}+{\lambda }_{2}},\quad {b}_{1}=\displaystyle \frac{{P}_{1}^{\prime} {P}_{2}^{\prime} }{{\lambda }_{1}+{\lambda }_{2}}.\end{eqnarray*}$
The imaginary part disappears in the form of $\widehat{\widetilde{v}}={\rm{i}}{\widehat{\widetilde{\varphi }}}_{x}$, of which the final expression is $\begin{eqnarray*}\widehat{\widetilde{v}}=\displaystyle \frac{2{b}_{1}({\lambda }_{2}{{\rm{e}}}^{{P}_{1}}+{\lambda }_{1}{{\rm{e}}}^{{P}_{2}})-2{b}_{0}({\lambda }_{2}\sinh {P}_{1}+{\lambda }_{1}\sinh {P}_{2})}{1+\cosh ({P}_{1}-{P}_{2})+{b}_{0}^{2}(1+\cosh ({P}_{1}+{P}_{2}))-2{b}_{0}{b}_{1}(1+{{\rm{e}}}^{{P}_{1}+{P}_{2}})}.\end{eqnarray*}$
This together with $\widehat{\widetilde{\alpha }}=-{\widehat{\widetilde{\beta }}}_{x}$ yields a two-soliton solution of (5).4. Conclusion and discussion
In this paper, we have constructed a Darboux transformation and Bäcklund transformation for SKdV1 equation. As an application, we have derived a superposition formula and calculated the simplist soliton solutions. As mentioned in the introduction, setting the fermionic components ζ = η = 0, one has a coupled system
$\begin{eqnarray}{w}_{t}={\left(-{w}_{{xx}}+3{wu}+{w}^{3}\right)}_{x},\,\end{eqnarray}$
$\begin{eqnarray}{u}_{t}={\left(-{u}_{{xx}}-3{{ww}}_{{xx}}+3{u}^{2}+3{w}^{2}u\right)}_{x},\end{eqnarray}$
which was considered by Kersten and Krasil'shchik [28] and was referred to as Kersten–Krasil'shchik coupled KdV-mKdV equations [29]. By reduction, we may easily recover some results for the Kersten–Krasil'shchik coupled KdV-mKdV system. For example, by setting φ(x, t, θ) = φ0(x, t), β(x, t, θ) = θβ0(x, t) with the relationship w = iφ0,x, u = − β0,x, $\widetilde{w}={\rm{i}}{\widetilde{\varphi }}_{0,x},\widetilde{u}=-{\widetilde{\beta }}_{0,x}$, we obtain $\begin{eqnarray*}\begin{array}{rcl}{\widetilde{\varphi }}_{0,x} & = & -{\varphi }_{0,x}+(2{\lambda }_{1}-({\widetilde{\beta }}_{0}-{\beta }_{0}))\tanh \left(\displaystyle \frac{{\widetilde{\varphi }}_{0}-{\varphi }_{0}}{2}\right),\\ {\widetilde{\beta }}_{0,x} & = & -{\beta }_{0,x}\cosh ({\widetilde{\varphi }}_{0}-{\varphi }_{0})+{\varphi }_{0,{xx}}\sinh ({\widetilde{\varphi }}_{0}-{\varphi }_{0})\\ & & +\,{\lambda }_{1}({\widetilde{\beta }}_{0}-{\beta }_{0})-\displaystyle \frac{1}{2}{\left({\widetilde{\beta }}_{0}-{\beta }_{0}\right)}^{2}\\ & & -\displaystyle \frac{1}{2}{\left({\widetilde{\varphi }}_{0}+{\varphi }_{0}\right)}_{x}{\left({\widetilde{\varphi }}_{0}-{\varphi }_{0}\right)}_{x}.\end{array}\end{eqnarray*}$
which is nothing but the Bäcklund transformation for Kersten–Krasil'shchik coupled KdV-mKdV system [32]. Furthermore, under μ1 = μ2 = 0, the solutions presented in the last section enable us to recover for the system (25) the one-soliton solution [30, 31] $\begin{eqnarray*}\widetilde{w}=-{\lambda }_{1}{\rm{sech}} \,{Q}_{1},\quad \widetilde{u}=-{\lambda }_{1}^{2}{{\rm{sech}} }^{2}{Q}_{1}\end{eqnarray*}$
and two-soliton solution $\begin{eqnarray*}\begin{array}{rcl}\widehat{\widetilde{w}} & = & \displaystyle \frac{-2{b}_{0}({\lambda }_{2}\sinh {Q}_{1}+{\lambda }_{1}\sinh {Q}_{2})}{1+\cosh ({Q}_{1}-{Q}_{2})+{b}_{0}^{2}(1+\cosh ({Q}_{1}+{Q}_{2}))},\\ \widehat{\widetilde{u}} & = & -{\widehat{\widetilde{\beta }}}_{0,x},\quad {\widehat{\widetilde{\beta }}}_{0}=\displaystyle \frac{({\lambda }_{1}-{\lambda }_{2})\left[{b}_{0}+{b}_{0}{e}^{{Q}_{1}+{Q}_{2}}+\sinh ({Q}_{1}-{Q}_{2})\right]+({\lambda }_{1}+{\lambda }_{2})[1+\cosh ({Q}_{1}-{Q}_{2})]}{1+\cosh ({Q}_{1}-{Q}_{2})+{b}_{0}^{2}(1+\cosh ({Q}_{1}+{Q}_{2}))},\end{array}\end{eqnarray*}$
where ${b}_{0}=\tfrac{{\lambda }_{1}-{\lambda }_{2}}{{\lambda }_{1}+{\lambda }_{2}},{Q}_{j}={\lambda }_{j}x-{\lambda }_{j}^{3}t+{c}_{j}$, j = 1, 2.Finally, apart from its usefulness in constructing solutions, Bäcklund transformation also provides an effective way to build discrete integrable systems [33–35]. Thus, considering further applications of our results is a problem worthy of study. Also, for a given nonlinear partial differential equation, various methods are available for constructing its particular solutions (see [36, 37] and the references therein), it may be interesting to apply those methods to the N = 2 SKdV equation.