Water has been known as a constant reminder that life repeats and the only element that has a visible cycle [1]. The water cycle has been believed to result in the distribution of water on land surface, to purify water, to support plant growth, to facilitate agriculture and to sustain aquatic ecosystem [2]. Water waves have been seen as one of the most common phenomena in nature, the studies of which help the energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc [3–10]. On the water waves, each year, hundreds of papers have been published, a few of which, e.g. are [3–10] recently. More recent nonlinear-physics contributions have been seen as well [11–13].
For the propagation of shallow water waves in a lake or near an ocean beach, people have investigated the Boussinesq–Burgers system [14–19], $ \begin{eqnarray}{u}_{t}=\displaystyle \frac{\beta -1}{2}\,{u}_{{xx}}+2{{uu}}_{x}+\displaystyle \frac{1}{2}{v}_{x},\,\end{eqnarray}$ $ \begin{eqnarray}{v}_{t}=\left(1-\displaystyle \frac{\beta }{2}\right)\beta {u}_{{xxx}}+2{\left({uv}\right)}_{x}+\displaystyle \frac{1-\beta }{2}\,{v}_{{xx}},\,\,\end{eqnarray}$ where x and t are the normalized space and time, the subscripts denote the partial derivatives, the differentiable function u = u(x, t) represents the horizontal velocity, and the differentiable function v = v(x, t) denotes the height of the water surface above the bottom level [16], while β is a real constant representing the dispersive power [15].
For system (1 ), finite-band solutions have been investigated with the help of the Lax representations of some stationary evolution equations [14], rational solutions have been obtained via the bilinear method and Kadomtsev–Petviashvili hierarchy reduction [15], non-auto-Bäcklund transformations, nonlocal symmetries, conservation laws and interaction solutions have been constructed [16], Darboux transformations and multi-soliton solutions have been presented [17, 18], bilinear form and multi-soliton solutions have been studied [19]. By the way, other issues related to system (1 ) have been seen [20–28].
However, to our knowledge, Bell-polynomial investigation on the bilinear forms and N solitons for system (1 ) has not been carried out, where N is a positive integer. Auto-Bäcklund-transformation work on system (1 ), which is different from that in [16], has not been discussed, either. In this paper, we will briefly review the Bell-polynomial concepts. With the binary Bell polynomials and symbolic computation[29] we will work out the bilinear forms for system (1 ), which are different from those in [15, 19], and N solitons for system (1 ), which are different from those in [15–19]. Also with symbolic computation, we will construct two auto-Bäcklund transformations with some soliton features. Conclusions will come out last.
Let us begin the work with the Bell-polynomial preliminary: Bell polynomials have been said to provide a relatively-direct way to get the bilinear forms for certain nonlinear evolution equations, instead of the dependent variable transformations [30–32]. References [30–32] have presented the following:
References [33–35] have linked the ${\mathscr{Y}}$ polynomials to the Hirota D operators as $ \begin{eqnarray}{{\mathscr{Y}}}_{{mx},{nt}}\left[p=\mathrm{ln}\left(\displaystyle \frac{f}{g}\right),q=\mathrm{ln}\left({fg}\right)\right]={\left({fg}\right)}^{-1}{D}_{x}^{m}{D}_{t}^{n}f\cdot g,\end{eqnarray}$ where f(x, t) and g(x, t) are the ${C}^{\infty }$ functions of x and t, while Dx and Dt are the Hirota D operators defined by $ \begin{eqnarray}\begin{array}{l}{D}_{x}^{m}{D}_{t}^{n}f(x,t)\cdot g(x,t)\\ \equiv \,{\left(\displaystyle \frac{\partial }{\partial x}-\displaystyle \frac{\partial }{\partial x^{\prime} }\right)}^{m}\,{\left.{\left(\displaystyle \frac{\partial }{\partial t}-\displaystyle \frac{\partial }{\partial t^{\prime} }\right)}^{n}f(x,t)g(x^{\prime} ,t^{\prime} )\right|}_{x^{\prime} =x,t^{\prime} =t},\end{array}\end{eqnarray}$ with $x^{\prime} $ and $t^{\prime} $ being the formal variables.
| • | The two-dimensional Bell polynomials: |
| • | The binary Bell polynomials: |
Hereby, through the binary Bell polynomials, we need to work out the bilinear forms and N solitons for system (1 ).
To begin with, what is reported in [19] can be considered as a special case of our results in this part.
Similar to the work in [36, 37], the scaling transformation, $ \begin{eqnarray*}\begin{array}{l}x\to \varphi x,\quad t\to {\varphi }^{2}t,\quad \beta \to {\varphi }^{0}\beta ,\\ u\to {\varphi }^{-1}u,\quad v\to {\varphi }^{-2}v,\end{array}\end{eqnarray*}$ $ \begin{eqnarray}u(x,t)={{\rm{\Upsilon }}}_{1}\,{p}_{x}(x,t),\,\end{eqnarray}$ $ \begin{eqnarray}v(x,t)={{\rm{\Upsilon }}}_{2}\,{p}_{{xx}}(x,t)+{{\rm{\Upsilon }}}_{3}\,{q}_{{xx}}(x,t)+\sigma ,\end{eqnarray}$ where ϒ1, ϒ2, ϒ3 and Σ are all the real constants. With the substitution of assumptions (8 ) and the assumptions that $ \begin{eqnarray*}{{\rm{\Upsilon }}}_{3}=2{{\rm{\Upsilon }}}_{1}^{2},\quad {{\rm{\Upsilon }}}_{2}=(1-\beta ){{\rm{\Upsilon }}}_{1}\quad \mathrm{and}\quad {{\rm{\Upsilon }}}_{1}=\pm \displaystyle \frac{1}{2}\end{eqnarray*}$ back into system (1 ), Bell-polynomial procedure and symbolic computation help us to reduce assumptions (8 ) to $ \begin{eqnarray}u(x,t)=\pm \displaystyle \frac{1}{2}{p}_{x}(x,t),\,\end{eqnarray}$ $ \begin{eqnarray}v(x,t)=\pm \displaystyle \frac{1-\beta }{2}{p}_{{xx}}(x,t)+\displaystyle \frac{1}{2}{q}_{{xx}}(x,t)+\sigma ,\end{eqnarray}$ and then to convert system (1 ) into two sets of the coupled systems of the ${\mathscr{Y}}$ polynomials: $ \begin{eqnarray}\pm \displaystyle \frac{1}{2}{{\mathscr{Y}}}_{2x}(p,q)-{{\mathscr{Y}}}_{t}(p)=0,\,\end{eqnarray}$ $ \begin{eqnarray}\pm 2{{\mathscr{Y}}}_{x,t}(p,q)-{{\mathscr{Y}}}_{3x}(p,q)-4\sigma {{\mathscr{Y}}}_{x}(p)=0\ .\end{eqnarray}$
leads to our assumptions
Further, with $ \begin{eqnarray}p(x,t)=\mathrm{ln}\left[\displaystyle \frac{f(x,t)}{g(x,t)}\right],\,\end{eqnarray}$ $ \begin{eqnarray}q(x,t)=\mathrm{ln}\left[f(x,t)g(x,t)\right],\end{eqnarray}$ we transform system (1 ), through systems (10 ), into the following two branches of bilinear forms with the binary Bell polynomials: $ \begin{eqnarray}\left(\pm \displaystyle \frac{1}{2}{D}_{x}^{2}-{D}_{t}\right)f\cdot g=0,\,\end{eqnarray}$ $ \begin{eqnarray}\left(\pm 2{D}_{x}{D}_{t}-{D}_{x}^{3}-4\sigma {D}_{x}\right)f\cdot g=0\ .\end{eqnarray}$
The reason for the existence of two branches of bilinear forms (12 ) is that there appear the ‘±’ signs.
Expanding f(x, t) and g(x, t) in bilinear forms (12 ) with respect to a small parameter ε as $ \begin{eqnarray}f(x,t)=1+\displaystyle \sum _{\varsigma =1}^{N}{\epsilon }^{\varsigma }{f}_{\varsigma }(x,t),\,\end{eqnarray}$ $ \begin{eqnarray}g(x,t)=1+\displaystyle \sum _{\varpi =1}^{N}{\epsilon }^{\varpi }{g}_{\varpi }(x,t),\end{eqnarray}$ and then setting ε = 1, we obtain the N-soliton solutions for system (1 ) as $ \begin{eqnarray}\left.\begin{array}{l}u(x,t)=\\ v(x,t)=\end{array}\right\}\ \mathrm{expressions}\ (9)\ \mathrm{and}\ (11),\mathrm{with}\,\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{rcl}f(x,t) & \,= & \displaystyle \sum _{{\mu }_{i},{\mu }_{j}=0,1}\exp \left[\displaystyle \sum _{i=1}^{N}{\mu }_{i}({\lambda }_{i}x+{\tau }_{i}t+{\omega }_{i})\right.\\ & & +\,\left.\displaystyle \sum _{1\leqslant i\lt j}^{(N)}{\mu }_{i}{\mu }_{j}{\delta }_{{ij}}\right],\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{rcl}g(x,t) & \,= & \displaystyle \sum _{{\mu }_{i},{\mu }_{j}=0,1}\exp \left[\displaystyle \sum _{i=1}^{N}{\mu }_{i}({\lambda }_{i}x+{\tau }_{i}t+{\kappa }_{i})\right.\\ & & +\,\left.\displaystyle \sum _{1\leqslant i\lt j}^{(N)}{\mu }_{i}{\mu }_{j}{\delta }_{{ij}}\right],\end{array}\end{eqnarray}$ $ \begin{eqnarray}{{\rm{e}}}^{{\delta }_{{ij}}}\,={\left(\displaystyle \frac{\sqrt{{a}_{i}{b}_{j}}-\sqrt{{a}_{j}{b}_{i}}}{\sqrt{{a}_{i}{a}_{j}}-\sqrt{{b}_{i}{b}_{j}}}\right)}^{2},\,\end{eqnarray}$ $ \begin{eqnarray}{{\rm{e}}}^{{\omega }_{i}}\,={a}_{i},\,\end{eqnarray}$ $ \begin{eqnarray}{{\rm{e}}}^{{\kappa }_{i}}\,={b}_{i},\,\end{eqnarray}$ $ \begin{eqnarray}{\tau }_{i}\,=\pm \displaystyle \frac{({a}_{i}+{b}_{i}){\lambda }_{i}^{2}}{2({a}_{i}-{b}_{i})},\,\end{eqnarray}$ $ \begin{eqnarray}{\lambda }_{i}\,=({a}_{i}-{b}_{i})\sqrt{\displaystyle \frac{\sigma }{{a}_{i}{b}_{i}}},\,\end{eqnarray}$ ς, ϖ, i and j being the positive integers with ς ≤ N, ϖ ≤ N, i ≤ N and j ≤ N, ωi's and κi's denoting the real constants, ${f}_{\varsigma }(x,t)$ 's and ${g}_{\varpi }(x,t)$ 's being all the real differentiable functions of x and t, ai and bi representing the parameters characterizing the i-th soliton, ai Σ > 0, bi Σ > 0, the sum ${\sum }_{{\mu }_{i},{\mu }_{j}=0,1}$ taken over all the possible combinations of μj = 0,1 while ${\sum }_{1\leqslant i\lt j}^{(N)}$ being the summation over all the possible pairs chosen from the N elements under the condition i < j.
It is noted that there exist two branches of N-soliton solutions (14 ) because of the ‘±’ signs. Both of the branches are dependent on β, the water-wave dispersive power.
Our next goal should be auto-Bäcklund transformations (17 )–(25 ), which are different from those reported in [16], and the relevant soliton features for system (1 ), to be seen below.
For system (1 ), we need to consider the Painlevé expansions in the form of the generalized Laurent series [38, 39] (and references therein), i.e. $ \begin{eqnarray}u(x,t)={\phi }^{-{\rm{\Xi }}}(x,t)\displaystyle \sum _{\xi =0}^{\infty }{u}_{\xi }(x,t){\phi }^{\xi }(x,t),\end{eqnarray}$ $ \begin{eqnarray}v(x,t)={\phi }^{-{ \mathcal X }}(x,t)\displaystyle \sum _{\chi =0}^{\infty }{v}_{\chi }(x,t){\phi }^{\chi }(x,t),\end{eqnarray}$ and to balance the powers of φ at the lowest orders, so as to get $ \begin{eqnarray}{ \mathcal X }=2,\qquad {\rm{\Xi }}=1,\end{eqnarray}$ where ξ and ${ \mathcal X }$ are the natural numbers, uξ's, vχ's and φ are all the analytic functions with ${u}_{0}\ne 0$ , ${v}_{0}\ne 0$ and ${\phi }_{x}\ne 0$ .
With symbolic computation, we will truncate Painlevé expansions (15 ) at the constant level terms [38, 39], as $ \begin{eqnarray}u(x,t)=\displaystyle \frac{{u}_{0}(x,t)}{\phi (x,t)}+{u}_{1}(x,t),\end{eqnarray}$ $ \begin{eqnarray}v(x,t)=\displaystyle \frac{{v}_{0}(x,t)}{{\phi }^{2}(x,t)}+\displaystyle \frac{{v}_{1}(x,t)}{\phi (x,t)}+{v}_{2}(x,t),\,\end{eqnarray}$ and substitute expansions (17 ) into system (1 ). Then, we require that the coefficients of like powers of φ vanish, and see the Painlevé–Bäcklund equations: $ \begin{eqnarray}\,{v}_{0}=\displaystyle \frac{\pm (\beta -1)-1}{2}{\phi }_{x}^{2},\,\end{eqnarray}$ $ \begin{eqnarray}\,\,{u}_{0}=\pm \displaystyle \frac{{\phi }_{x}}{2},\,\end{eqnarray}$ $ \begin{eqnarray}\,{v}_{1}=\pm \displaystyle \frac{2{\phi }_{t}-4{\phi }_{x}{u}_{1}-(\beta -1){\phi }_{{xx}}}{2},\,\end{eqnarray}$ $ \begin{eqnarray}\,\pm {\phi }_{{xx}}+4{\phi }_{x}{u}_{1}-2{\phi }_{t}=0,\end{eqnarray}$ $ \begin{eqnarray}\,\begin{array}{l}\pm 4\beta {\phi }_{x}^{2}{u}_{1,x}\mp 4{\phi }_{x}^{2}{u}_{1,x}\pm 16{\phi }_{x}{\phi }_{t}{u}_{1}\\ \mp \,16{\phi }_{x}^{2}{u}_{1}^{2}\pm 4{\phi }_{x}^{2}{v}_{2}\mp 4{\phi }_{t}^{2}\pm 3{\phi }_{{xx}}^{2}\\ \pm 4{\phi }_{x}{\phi }_{{xxx}}+12{\phi }_{x}^{2}{u}_{1,x}+24{\phi }_{x}{\phi }_{{xx}}{u}_{1}\\ -\,4{\phi }_{{xx}}{\phi }_{t}-8{\phi }_{x}{\phi }_{{xt}}=0,\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{l}\,4\beta {\phi }_{{xx}}{u}_{1,x}+4\beta {\phi }_{x}{u}_{1,{xx}}\\ \,\,\,-\,32{\phi }_{x}{u}_{1,x}{u}_{1}+8{\phi }_{x}{u}_{1,t}+8{\phi }_{t}{u}_{1,x}-4{\phi }_{{xx}}{u}_{1,x}\\ \,\,\,-\,4{\phi }_{x}{u}_{1,{xx}}-16{\phi }_{{xx}}{u}_{1}^{2}+16{\phi }_{{xt}}{u}_{1}+4{\phi }_{x}{v}_{2,x}\\ \,\,+\,4{\phi }_{{xx}}{v}_{2}-4{\phi }_{{tt}}+{\phi }_{{xxxx}}=0,\end{array}\end{eqnarray}$ $ \begin{eqnarray}\,{u}_{1,t}=\displaystyle \frac{\beta -1}{2}{u}_{1,{xx}}+2{u}_{1}{u}_{1,x}+\displaystyle \frac{1}{2}{v}_{2,x},\,\end{eqnarray}$ $ \begin{eqnarray}\,{v}_{2,t}=\left(1-\displaystyle \frac{\beta }{2}\right)\beta {u}_{1,{xxx}}+2{\left({u}_{1}{v}_{2}\right)}_{x}+\displaystyle \frac{1-\beta }{2}{v}_{2,{xx}}.\end{eqnarray}$
Hereby, u1(x, t) and v2(x, t) can be treated as the seed solutions for system (1 ) [38, 39]. The sets of equations (17 )–(25 ) constitute two auto-Bäcklund transformations, since there exist the ‘±’ signs, and the whole sets are mutually consistent, or, explicitly solvable with respect to φ(x, t), u0(x, t), u1(x, t), v0(x, t), v1(x, t) and v2(x, t), to be seen right below.
Each of auto-Bäcklund transformations (17 )–(25 ) works as a system of equations relating a set of the solutions of system (1 ), e.g. one set of solutions (27 ) or (28 ), to another set of the solutions of system (1 ) itself. Therefore, we could, in principle at least, be able to progressively construct more and more complicated solutions of system (1 ).
Each of auto-Bäcklund transformations (17 )–(25 ) is dependent on β, the water-wave dispersive power.
We will try to find the explicitly-solvable soliton examples, assuming that $ \begin{eqnarray}\left\{\begin{array}{l}\phi (x,t)={{\rm{e}}}^{{\zeta }_{1}(t)x+{\zeta }_{2}(t)}+1,\\ {u}_{1}(x,t)={\zeta }_{3}(t)+{\zeta }_{4}(t)x,\\ {v}_{2}(x,t)={\zeta }_{5}(t)+{\zeta }_{6}(t)x+{\zeta }_{7}(t){x}^{2},\end{array}\right.\end{eqnarray}$ where ζl(t)'s (l = 1, ..., 7) are the real differentiable functions, with ${\zeta }_{1}(t)\ne 0$ since ${\phi }_{x}\ne 0$ .
With symbolic computation, we substitute assumptions (26 ) into equations (18 )–(25 ), and work out
$ \begin{eqnarray*}\begin{array}{l}{\zeta }_{7}(t)=\displaystyle \frac{{[{\zeta }_{1}^{\prime} (t)-2{\zeta }_{1}(t){\zeta }_{4}(t)]}^{2}}{{\zeta }_{1}{(t)}^{2}}\end{array}\end{eqnarray*}$ with the “′” sign denoting the derivative, with respect to t. There exist two cases, ζ1(t) ≠ constant or ζ1(t) = constant.
Case (1): ${\zeta }_{1}(t)\,\ne $ constant
$ \begin{eqnarray*}\begin{array}{r}\\ {\zeta }_{1}(t)= {\zeta }_{8}\,{{\rm{e}}}^{2\displaystyle \int {\zeta }_{4}(t){\rm{d}}{t}},\\ {\zeta }_{4}(t) = \displaystyle \frac{1}{{\zeta }_{9}-2t}\ {\rm{with}}\ {\zeta }_{9}-2t\ne 0,\\ {\zeta }_{6}(t) = 0,\\ {\zeta }_{3}(t) = \displaystyle \frac{{\rho }_{1}}{{\zeta }_{9}-2t},\\ {\zeta }_{5}(t) = \displaystyle \frac{{\rho }_{2}}{{\zeta }_{9}-2t},\\ {\zeta }_{2}(t) = \displaystyle \frac{{\zeta }_{8}(4{\rho }_{1}\pm {\zeta }_{8})}{4({\zeta }_{9}-2t)}+{2\rho }_{3},\\ {\rho }_{2} = 1-\beta \pm 1,\end{array}\end{eqnarray*}$
where${\zeta }_{8}\ne 0$ , ζ9, ρ1, ρ2 and ρ3 are all the real constants.
where
Computing with expressions (17 ) as well, we can obtain the following β-dependent soliton solutions of system (1 ): $ \begin{eqnarray}\begin{array}{rcl}u(x,t) & = & \pm \displaystyle \frac{{\zeta }_{8}}{4({\zeta }_{9}-2t)}\,\\ & & \times \mathrm{Tanh}\,\left[\displaystyle \frac{{\zeta }_{8}x}{2({\zeta }_{9}-2t)}+\displaystyle \frac{{\zeta }_{8}(4{\rho }_{1}\pm {\zeta }_{8})}{8({\zeta }_{9}-2t)}+{\rho }_{3}\right]\\ & & +\,\displaystyle \frac{x}{{\zeta }_{9}-2t}+\displaystyle \frac{4{\rho }_{1}\pm {\zeta }_{8}}{4({\zeta }_{9}-2t)},\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{rcl}\quad v(x,t) & = & \displaystyle \frac{{\zeta }_{8}^{2}(\pm \beta \mp 1-1)}{8{\left({\zeta }_{9}-2t\right)}^{2}}\,\\ & & \times {\mathrm{Tanh}}^{2}\,\left[\displaystyle \frac{{\zeta }_{8}x}{2({\zeta }_{9}-2t)}+\displaystyle \frac{{\zeta }_{8}(4{\rho }_{1}\pm {\zeta }_{8})}{8({\zeta }_{9}-2t)}+{\rho }_{3}\right]\\ & & +\,\displaystyle \frac{{\zeta }_{8}^{2}(1\mp \beta \pm 1)+8(1-\beta \pm 1)({\zeta }_{9}-2t)}{8{\left({\zeta }_{9}-2t\right)}^{2}}.\end{array}\end{eqnarray}$ There exist two branches of those solutions because of the ‘±’ signs.
Case (2): ζ1(t) = η1 = non-zero constant
Similarly, we can obtain the following β-dependent solitonic solutions of system (1 ): $ \begin{eqnarray}\begin{array}{rcl}u(x,t) & = & \pm \displaystyle \frac{{\eta }_{1}}{4}\,\mathrm{Tanh}\left[\displaystyle \frac{{\eta }_{1}x}{2}+\displaystyle \frac{{\eta }_{1}\left(4{\eta }_{2}\pm {\eta }_{1}\right)t}{4}+\displaystyle \frac{{\eta }_{3}}{2}\right]\\ & & +\,{\eta }_{2}\pm \displaystyle \frac{{\eta }_{1}}{4},\,\end{array}\end{eqnarray}$ $ \begin{eqnarray}\begin{array}{rcl}v(x,t) & = & \displaystyle \frac{{\eta }_{1}^{2}(\pm \beta \mp 1-1)}{8}\,\\ & & \times \,{\mathrm{Tanh}}^{2}\,\left[\displaystyle \frac{{\eta }_{1}x}{2}+\displaystyle \frac{{\eta }_{1}\left(4{\eta }_{2}\pm {\eta }_{1}\right)t}{4}+\displaystyle \frac{{\eta }_{3}}{2}\right]\\ & & -\,\displaystyle \frac{{\eta }_{1}^{2}(\pm \beta \mp 1-1)}{8},\,\end{array}\end{eqnarray}$ where η2 and η3 are also the real constants. There exist two branches of those solutions because of the ‘±’ signs.
Right now, let us finish up the paper. Water has been known as a constant reminder that life repeats and the only element that has a visible cycle. The water cycle has been believed to result in the distribution of water on land surface, to purify water, to support plant growth, to facilitate agriculture and to sustain aquatic ecosystem. Water waves have been seen as one of the most common phenomena in nature, the studies of which help the energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, on system (1 ), the Boussinesq–Burgers system for the shallow water waves in a lake or near an ocean beach, symbolic computation has been performed. For u(x, t), the water-wave horizontal velocity, and v(x, t), the height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials, i.e. bilinear forms (12 ), and N-soliton solutions (14 ) have been worked out, while two sets of auto-Bäcklund transformations (17 )–(25 ) have been constructed together with solitonic solutions (27 ) and (28 ). It has been noted that bilinear forms (12 ), N-soliton solutions (14 ) and Bäcklund transformations (17 )–(25 ) are different from those in the existing literatures. All of our results have been shown to be dependent on β, the water-wave dispersive power.