1. Introduction
The study of integrable partial differential equations (PDEs) is one of the most prominent fields in nonlinear science. Several of the past decades have witnessed the rapid progress of research on integrable PDEs, especially on the (1+1)-dimensional and (2+1)-dimensional cases. In contrast, higher-dimensional integrable PDEs are relatively rare. Thus, the establishment of more high-dimensional integrable PDEs, especially by the extension of lower-dimensional integrable systems, is an important topic in the fields of mathematics and physics.
It is well known that there are many universal and convenient tools which could be used to simplify a complex equation into a simple one. For example, symmetry reduction can reduce the order of an ordinary differential equation (ODE) or the dimension of a PDE under consideration. Specifically, it can simplify a high-dimensional/order equation to a low-dimensional/order equation. In contrast, it is often very hard to study the laws of a complex equation from the results of its associated simple equation. However, in some special cases, it is also possible to transform the results of a simple equation into those of some of its related complex equations. For example, by using the Miura-type transformation from the Korteweg–de Vries (KdV) equation to the modified KdV equation and Schwarz KdV equation, (1+1)-dimensional and (2+1)-dimensional integrable sine-Gordon equations can be derived from the Riccati equation [1]. In addition, starting from the (2+1)-dimensional Kadomtsev–Petviashvili, nonlinear Schrödinger (NLS) and Schwarz KdV equations, some (3+1)-dimensional integrable models were explicitly given via Painlevé analysis [2].
To obtain more integrable high-dimensional equations and study their properties efficiently and conveniently, very recently Lou et al developed a deformation algorithm, which was first proposed as a deformation conjecture [3] and was soon proved by Casati and Zhang [4]. This deformation algorithm, which demonstrated that a lower-dimensional equation can be extended to higher-dimensional equations on condition that the lower-dimensional equation possesses enough conservation laws, providing more possibilities for the study of higher-dimensional equations. It was firstly, in [3], applied to the usual (1+1)-dimensional KdV equation and the (1+1)-dimensional Ablowitz–Kaup–Newell–Segur system and, furthermore, it is found that for the resultant higher-dimensional equations, there are completely different structures and properties from the traditional integrable equations. In [5], the (1+1)-dimensional Kaup–Newell system was extended to a (4+1)-dimensional system whose Lax integrability and symmetry integrability were also proved. Furthermore, the deformation algorithm also refers to many other integrable equations, such as the modified KdV equation [6], Burgers equation [7], Camassa–Holm equation [8] and the NLS-type Boussinesq equation [9].
In this paper, we consider the integrable deformation of the (1+1)-dimensional classical Boussinesq–Burgers (CBB) system [10, 11]1 ) reduces to the Broer–Kaup system, which is related to the classical Boussinesq system with a special transformation [12]. If β = 1 and each of the dependent and independent variables is replaced by its negative counterpart, system (1 ) becomes the Boussinesq–Burgers system [13]. Geng and Wu constructed finite-band solutions of the CBB system (1 ) based on Lax pairs of stationary evolution equations, in [11]. In [14], Rui established two basic Darboux transformations of a spectral problem to system (1 ), and exact solutions of the system were obtained as well. For system (1 ), its rational and special function solutions were derived using Lie symmetries in [15], and there the conservation laws were also established.
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{1}{2}\beta (2-\beta ){v}_{{xxx}}+\displaystyle \frac{1}{2}(1-\beta ){u}_{{xx}}+2{\left({uv}\right)}_{x},\\ {v}_{t}=\displaystyle \frac{1}{2}(\beta -1){v}_{{xx}}+2{{vv}}_{x}+\displaystyle \frac{1}{2}{u}_{x},\end{array}\right.\end{eqnarray}$
where u = u(x, t) is the height deviating from the equilibrium position of water, v = v(x, t) represents the field of horizontal velocity and β is a real constant. For β = 0 (or β = 2, t → − t, x → − x), system (To keep this exposition self-contained, we summarize below the main scheme of Lou's deformation algorithm. An interested reader can find further details in the monographs [3–9]. For a general (1+1)-dimensional m-component integrable evolution system
$\begin{eqnarray*}{{\boldsymbol{u}}}_{t}={\boldsymbol{F}}({\boldsymbol{u}},{{\boldsymbol{u}}}_{x},\cdots ,{{\boldsymbol{u}}}_{{xn}}),\end{eqnarray*}$
endowed with the conservation laws $\begin{eqnarray*}\begin{array}{l}{[{\rho }_{i}({\boldsymbol{u}})]}_{t}={[{J}_{i}({\boldsymbol{u}},{{\boldsymbol{u}}}_{x},\cdots ,{{\boldsymbol{u}}}_{{xN}})]}_{x},\\ i\,=\,1,2,\cdots ,D-1,\end{array}\end{eqnarray*}$
where ${{\boldsymbol{u}}}_{{xn}}={\partial }_{x}^{n}{\boldsymbol{u}},\,{\boldsymbol{u}}={\left({u}_{1},{u}_{2},\cdots ,{u}_{m}\right)}^{{\rm{T}}}$ and ui = ui(x, t), and the deformed (D+1)-dimensional system $\begin{eqnarray*}\hat{T}{\boldsymbol{u}}={\boldsymbol{F}}({\boldsymbol{u}},\hat{L}{\boldsymbol{u}},\cdots ,{\hat{L}}^{n}{\boldsymbol{u}})\end{eqnarray*}$
is integrable, where the time and space deformation operators are, respectively, $\begin{eqnarray*}\begin{array}{l}\hat{T}\equiv {\partial }_{t}+\displaystyle \sum _{i=1}^{D-1}{\bar{J}}_{i}{\partial }_{{x}_{i}},\\ \hat{L}\equiv {\partial }_{x}+\displaystyle \sum _{i=1}^{D-1}{\rho }_{i}{\partial }_{{x}_{i}},\end{array}\end{eqnarray*}$
with the deformed flows ${\bar{J}}_{i}={J}_{i}{| }_{{{\boldsymbol{u}}}_{{xj}}\to {\hat{L}}^{j}{\boldsymbol{u}}}$.The outline of this paper is as follows. In section 2 , the (4+1)-dimensional CBB system is obtained by applying the deformation algorithm to the (1+1)-dimensional system (1 ). In addition, the Lax integrability and symmetry integrability of the higher-dimensional system are given. In section 3 , we study many lower-dimensional reduced integrable systems of the resultant (4+1)-dimensional CBB system. And for illustration, an exact solution of one reduced system is constructed with the help of Lie symmetry analysis and a power series method in section 4 . The last section contains a brief discussion of the results obtained and our concluding remarks.
2. Integrable deformation of the CBB system (1 )
2.1. The higher-dimensional CBB system
For the (1+1)-dimensional CBB system (1 ), there are three conservation laws of the form1 ), we get the (4+1)-dimensional CBB system
$\begin{eqnarray*}{\rho }_{{it}}={J}_{{ix}},\,\,\,\,i=1,2,3,\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{l}{\rho }_{1}=u,\,\,\,{J}_{1}=\displaystyle \frac{\beta (2-\beta )}{2}{v}_{{xx}}+\displaystyle \frac{1-\beta }{2}{u}_{x}+2{uv},\\ {\rho }_{2}=v,\,\,\,{J}_{2}=\displaystyle \frac{\beta -1}{2}{v}_{x}+{v}^{2}+\displaystyle \frac{1}{2}u,\\ {\rho }_{3}={uv},\,\,\,{J}_{3}=\displaystyle \frac{\beta (2-\beta )}{2}{{vv}}_{{xx}}+\displaystyle \frac{\beta (\beta -2)}{4}{v}_{x}^{2}\\ +\displaystyle \frac{1-\beta }{2}({{vu}}_{x}-{{uv}}_{x})+2{{uv}}^{2}+\displaystyle \frac{1}{4}{u}^{2}.\end{array}\end{eqnarray*}$
Taking these conservation laws into consideration and applying Lou's deformation algorithm to system ( $\begin{eqnarray}\left\{\begin{array}{l}\hat{T}u=\hat{L}\left(\displaystyle \frac{\beta (2-\beta )}{2}{\hat{L}}^{2}v+\displaystyle \frac{1-\beta }{2}\hat{L}u+2{uv}\right),\\ {}_{}\hat{T}v=\hat{L}\left(\displaystyle \frac{1-\beta }{2}\hat{L}v+{v}^{2}+\displaystyle \frac{1}{2}u\right),\end{array}\right.\end{eqnarray}$
with the deformation operators $\begin{eqnarray*}\begin{array}{l}\hat{L}\equiv {\partial }_{x}+u{\partial }_{y}+v{\partial }_{z}+{uv}{\partial }_{\xi },\\ \hat{T}\equiv {\partial }_{t}+{\bar{J}}_{1}{\partial }_{y}+{\bar{J}}_{2}{\partial }_{z}+{\bar{J}}_{3}{\partial }_{\xi },\end{array}\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{l}{\bar{J}}_{1}=\displaystyle \frac{\beta (2-\beta )}{2}{\hat{L}}^{2}v+\displaystyle \frac{1-\beta }{2}\hat{L}u+2{uv},\\ {\bar{J}}_{2}=\displaystyle \frac{\beta -1}{2}\hat{L}v+{v}^{2}+\displaystyle \frac{1}{2}u,\\ {\bar{J}}_{3}=\displaystyle \frac{\beta (2-\beta )}{2}v{\hat{L}}^{2}v+\displaystyle \frac{\beta (\beta -2)}{4}{\left(\hat{L}v\right)}^{2}\\ +\displaystyle \frac{1-\beta }{2}(v\hat{L}u-u\hat{L}v)+2{{uv}}^{2}+\displaystyle \frac{1}{4}{u}^{2}.\end{array}\end{eqnarray*}$
2.2. The Lax integrability of the (4+1)-dimensional CBB system (2 )
Now, based on the Lax pair of the previous low-dimensional CBB system (1 ), we consider the Lax integrability of the higher-dimensional system (2 ). The Lax equation of system (1 ) is3 ) yields the Lax pair of system (2 )2 ).
$\begin{eqnarray}\begin{array}{l}{\psi }_{x}=M\psi =\left(\begin{array}{cc}\lambda -v & u+\beta {v}_{x}\\ -1 & -\lambda +v\end{array}\right)\psi ,\\ {\psi }_{t}=N\psi =\left(\begin{array}{cc}{\lambda }^{2}+\displaystyle \frac{1}{2}{v}_{x}-{v}^{2} & \lambda (u+\beta {v}_{x})+\displaystyle \frac{1}{2}{\left(u+\beta {v}_{x}\right)}_{x}+(u+\beta {v}_{x})v\\ -\lambda -v & -{\lambda }^{2}-\displaystyle \frac{1}{2}{v}_{x}+{v}^{2}\end{array}\right)\psi .\end{array}\end{eqnarray}$
Here, $\psi ={\left({\psi }_{1},{\psi }_{2}\right)}^{{\rm{T}}}$, and λ is a spectral parameter. The compatibility condition ψxt = ψtx yields a zero curvature equation $\begin{eqnarray*}{M}_{t}-{N}_{x}+[M,N]=0.\end{eqnarray*}$
Applying the deformation relations $\begin{eqnarray*}{\partial }_{x}\to \hat{L},\,\,\,\,{\partial }_{t}\to \hat{T}\end{eqnarray*}$
to the Lax equation ( $\begin{eqnarray}\begin{array}{l}\hat{M}\psi =\left(\begin{array}{cc}\lambda -v-\hat{L} & u+\beta \hat{L}v\\ -1 & -\lambda +v-\hat{L}\end{array}\right)\psi =0,\\ \hat{N}\psi =\left(\begin{array}{cc}{\lambda }^{2}+\displaystyle \frac{1}{2}\hat{L}v-{v}^{2}-\hat{T} & \lambda (u+\beta \hat{L}v)+\displaystyle \frac{1}{2}\hat{L}(u+\beta \hat{L}v)+{uv}+\beta v\hat{L}v\\ -\lambda -v & -{\lambda }^{2}-\displaystyle \frac{1}{2}\hat{L}v+{v}^{2}-\hat{T}\end{array}\right)\psi =0.\end{array}\end{eqnarray}$
The compatibility condition $[\hat{M},\hat{N}]=\hat{M}\hat{N}-\hat{N}\hat{M}=0$ just gives the (4+1)-dimensional CBB system (2.3. Higher-order symmetries of the extended CBB system (2 )
To construct the higher-order symmetries of the obtained (4+1)-dimensional CBB system (2 ), we now apply the deformation algorithm to the higher-order flow systems of system (1 ). Note that these symmetries cannot be obtained from the direct deformation of the original system.
The simplest higher-order flow system for system (1 ) is given by5 ) exists, and the corresponding conservation law is
$\begin{eqnarray}{u}_{\tau }={K}_{1x},\qquad {v}_{\tau }={K}_{2x},\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{l}{K}_{1}=\displaystyle \frac{1}{4}{u}_{{xx}}+\displaystyle \frac{3}{4}\beta (2-\beta )(2{{vv}}_{{xx}}+{v}_{x}^{2})\\ \,+\displaystyle \frac{3}{2}(1-\beta ){{vu}}_{x}+\displaystyle \frac{3}{4}{u}^{2}+3{{uv}}^{2},\\ {K}_{2}=\displaystyle \frac{1}{4}{v}_{{xx}}+\displaystyle \frac{3}{2}(\beta -1){{vv}}_{x}+\displaystyle \frac{3}{2}{uv}+{v}^{3}.\end{array}\end{eqnarray*}$
In addition to u and v, the derivative-independent conserved density uv of the higher-order flow system ( $\begin{eqnarray*}{\left({uv}\right)}_{\tau }={K}_{3x},\end{eqnarray*}$
with $\begin{eqnarray*}\begin{array}{l}{K}_{3}=\displaystyle \frac{3}{2}\beta (2-\beta ){v}^{2}{v}_{{xx}}\\ +\displaystyle \frac{1}{4}({{vu}}_{{xx}}+{{uv}}_{{xx}}-{u}_{x}{v}_{x})\\ +\displaystyle \frac{3}{2}(\beta -1)({{uvv}}_{x}-{v}^{2}{u}_{x})+3{{uv}}^{3}+\displaystyle \frac{3}{2}{u}^{2}v.\end{array}\end{eqnarray*}$
Applying the deformation algorithm to the higher-order flow system (5 ), we arrive at the following (4+1)-dimensional higher-order flow system
$\begin{eqnarray}\begin{array}{l}\hat{\tau }u=\hat{L}{\bar{K}}_{1},\,\,\,\,\hat{\tau }v=\hat{L}{\bar{K}}_{2},\end{array}\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{l}\hat{\tau }={\partial }_{\tau }+{\bar{K}}_{1}{\partial }_{y}+{\bar{K}}_{2}{\partial }_{z}+{\bar{K}}_{3}{\partial }_{\xi },\\ {\bar{K}}_{1}=\displaystyle \frac{1}{4}{\hat{L}}^{2}u+\displaystyle \frac{3}{4}\beta (2-\beta )(2v{\hat{L}}^{2}v+{\left(\hat{L}v\right)}^{2})\\ +\displaystyle \frac{3}{2}(1-\beta )v\hat{L}u+\displaystyle \frac{3}{4}{u}^{2}+3{{uv}}^{2},\\ {\bar{K}}_{2}=\displaystyle \frac{1}{4}{\hat{L}}^{2}v+\displaystyle \frac{3}{2}(\beta -1)v\hat{L}v+\displaystyle \frac{3}{2}{uv}+{v}^{3},\end{array}\end{eqnarray*}$
with $\begin{eqnarray*}\begin{array}{l}{\bar{K}}_{3}=\displaystyle \frac{3}{2}\beta (2-\beta ){v}^{2}{\hat{L}}^{2}v\\ +\displaystyle \frac{1}{4}(v{\hat{L}}^{2}u+u{\hat{L}}^{2}v-\hat{L}u\hat{L}v)\\ +\displaystyle \frac{3}{2}(\beta -1)({uv}\hat{L}v-{v}^{2}\hat{L}u)\\ +3{{uv}}^{3}+\displaystyle \frac{3}{2}{u}^{2}v.\end{array}\end{eqnarray*}$
Based on the higher-order flow system (6 ), in system (2 ) we obtain the higher-order symmetry of the form
$\begin{eqnarray*}\sigma =\left(\begin{array}{c}\hat{L}{\bar{K}}_{1}-{\bar{K}}_{1}{u}_{y}-{\bar{K}}_{2}{u}_{z}-{\bar{K}}_{3}{u}_{\xi }\\ \hat{L}{\bar{K}}_{2}-{\bar{K}}_{1}{v}_{y}-{\bar{K}}_{2}{v}_{z}-{\bar{K}}_{3}{v}_{\xi }\end{array}\right).\end{eqnarray*}$
3. Reduced systems of the (4+1)-dimensional CBB system (2 )
Generally speaking, it is often very difficult to discuss higher-dimensional systems. To find some new characteristics of system (2 ) more intuitively, we reduce system (2 ) along the coordinate axes and discuss the properties of the reduced systems. Obviously, when u and v are only dependent on time t and space x, system (2 ) degenerates into the original (1+1)-dimensional system (1 ). Therefore, the solutions of system (1 ) are special solutions of system (2 ). This trivial case will not be discussed.
3.1. The (1+1)-dimensional reduced systems of system (2 )
Here, we intend to discuss some (1+1)-dimensional reduced systems of (4+1)-dimensional integrable system (2 ).
3.1.1. Reduction with respect to {y, t}
Taking into consideration that u and v depend only on {y, t}, system (2 ) degenerates into the (1+1)-dimensional system, as below4 ) of system (2 ), we obtain the Lax pair of system (7 )7 ).
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{\beta (2-\beta )}{2}({u}^{3}{v}_{{yyy}}+{u}^{2}{v}_{y}{u}_{{yy}}+2{u}^{2}{u}_{y}{v}_{{yy}})+\displaystyle \frac{1-\beta }{2}{u}^{2}{u}_{{yy}}+2{u}^{2}{v}_{y},\\ {v}_{t}=\displaystyle \frac{\beta (\beta -2)}{2}({u}^{2}{v}_{y}{v}_{{yy}}+{{uu}}_{y}{v}_{y}^{2})+\displaystyle \frac{\beta -1}{2}({u}^{2}{v}_{{yy}}+2{{uu}}_{y}{v}_{y})+\displaystyle \frac{1}{2}{{uu}}_{y}.\end{array}\right.\end{eqnarray}$
Recalling the Lax pair ( $\begin{eqnarray*}\begin{array}{l}\hat{{M}^{y}}\psi =\left(\begin{array}{cc}\lambda -v-u{\partial }_{y} & u+\beta {{uv}}_{y}\\ -1 & -\lambda +v-u{\partial }_{y}\end{array}\right)\psi =0,\\ \hat{{N}^{y}}\psi =\left(\begin{array}{cc}{\lambda }^{2}+\displaystyle \frac{1}{2}{{uv}}_{y}-{v}^{2}-{\bar{J}}_{1}{\partial }_{y}-{\partial }_{t} & {\hat{N}}_{12}^{y}\\ -\lambda -v & -{\lambda }^{2}-\displaystyle \frac{1}{2}{{uv}}_{y}+{v}^{2}-{\bar{J}}_{1}{\partial }_{y}-{\partial }_{t}\end{array}\right)\psi =0,\end{array}\end{eqnarray*}$
where ${\hat{N}}_{12}^{y}=\lambda (u+\beta {{uv}}_{y})+\tfrac{1}{2}{{uu}}_{y}$ + $\tfrac{1}{2}\beta u({u}_{y}{v}_{y}+{{uv}}_{{yy}})\,+{uv}+\beta {{uvv}}_{y},$ and the compatibility condition $[{\hat{M}}^{y},{\hat{N}}^{y}]=0$ is just the (1+1)-dimensional system (The conservation laws of system (7 ) satisfying the deformation theorem are7 ) to obtain a new higher-dimensional system. According to conservation laws (8 ) we introduce the following deformation operators7 ) is of the form9 ) and (2 ) are exactly the same. Hence, system (9 ) is integrable.
$\begin{eqnarray}\begin{array}{l}{\left({u}^{-1}\right)}_{t}=\left(\displaystyle \frac{\beta (\beta -2)}{2}({{uv}}_{{yy}}+{u}_{y}{v}_{y})\right.\\ {\left.+\displaystyle \frac{\beta -1}{2}{u}_{y}-2v\right)}_{y},\\ {v}_{t}={\left(\displaystyle \frac{\beta (\beta -2)}{4}{u}^{2}{v}_{y}^{2}+\displaystyle \frac{\beta -1}{2}{u}^{2}{v}_{y}+\displaystyle \frac{1}{4}{u}^{2}\right)}_{y},\\ {\left({u}^{-1}v\right)}_{t}=\left(\displaystyle \frac{\beta (\beta -2)}{2}({{uvv}}_{{yy}}+{{vu}}_{y}{v}_{y})\right.\\ {\left.+\displaystyle \frac{\beta -1}{2}({{uv}}_{y}+{{vu}}_{y})+\displaystyle \frac{1}{2}u-{v}^{2}\right)}_{y}.\end{array}\end{eqnarray}$
Now, we apply the deformation algorithm to system ( $\begin{eqnarray*}\begin{array}{l}{\hat{L}}_{1}={\partial }_{y}+\displaystyle \frac{1}{u}{\partial }_{x}+\displaystyle \frac{v}{u}{\partial }_{z}+v{\partial }_{\xi },\\ {\hat{T}}_{1}={\partial }_{t}+\left(\displaystyle \frac{\beta (\beta -2)}{2}(u{\hat{L}}_{1}^{2}v+{\hat{L}}_{1}u{\hat{L}}_{1}v)\right.\\ \left.+\displaystyle \frac{\beta -1}{2}{\hat{L}}_{1}u-2v\right){\partial }_{x}\\ +\left(\displaystyle \frac{\beta (\beta -2)}{2}({uv}{\hat{L}}_{1}^{2}v+v{\hat{L}}_{1}u{\hat{L}}_{1}v)\right.\\ \left.+\displaystyle \frac{\beta -1}{2}(u{\hat{L}}_{1}v+v{\hat{L}}_{1}u)+\displaystyle \frac{1}{2}u-{v}^{2}\right){\partial }_{z}\\ +\left(\displaystyle \frac{\beta (\beta -2)}{4}{u}^{2}{\left({\hat{L}}_{1}v\right)}^{2}\right.\\ \left.+\displaystyle \frac{\beta -1}{2}{u}^{2}{\hat{L}}_{1}v+\displaystyle \frac{1}{4}{u}^{2}\right){\partial }_{\xi }.\end{array}\end{eqnarray*}$
Based upon the deformation theorem, the (4+1)-dimensional integrable deformation of the (1+1)-dimensional CBB system ( $\begin{eqnarray}\begin{array}{l}{\hat{T}}_{1}u=-{u}^{2}{\hat{L}}_{1}\left(\displaystyle \frac{\beta (\beta -2)}{2}(u{\hat{L}}_{1}^{2}v+{\hat{L}}_{1}u{\hat{L}}_{1}v)\right.\\ \,\left.+\displaystyle \frac{\beta -1}{2}{\hat{L}}_{1}u-2v\right),\\ {\hat{T}}_{1}v={\hat{L}}_{1}\left(\displaystyle \frac{\beta (\beta -2)}{4}{u}^{2}{\left({\hat{L}}_{1}v\right)}^{2}\right.\\ \,\left.+\displaystyle \frac{\beta -1}{2}{u}^{2}{\hat{L}}_{1}v+\displaystyle \frac{1}{4}{u}^{2}\right).\end{array}\end{eqnarray}$
Direct calculation shows that systems (3.1.2. Reduction with respect to {z, t}
If u and v merely contain one spatial variable {z}, then system (2 ) becomes a (1+1)-dimensional CBB system
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{\beta (2-\beta )}{2}({v}^{3}{v}_{{zzz}}+4{v}^{2}{v}_{z}{v}_{{zz}}+{{vv}}_{z}^{3})+\displaystyle \frac{1-\beta }{2}({v}^{2}{u}_{{zz}}+2{{vu}}_{z}{v}_{z})+{v}^{2}{u}_{z}-\displaystyle \frac{1}{2}{{uu}}_{z}+2{{uvv}}_{z},\\ {v}_{t}=\displaystyle \frac{\beta -1}{2}{v}^{2}{v}_{{zz}}+\displaystyle \frac{1}{2}{{vu}}_{z}+{v}^{2}{v}_{z}-\displaystyle \frac{1}{2}{{uv}}_{z},\end{array}\right.\end{eqnarray}$
which admits the Lax pair $\begin{eqnarray*}\begin{array}{l}\hat{{M}^{z}}\psi =\left(\begin{array}{cc}\lambda -v-v{\partial }_{z} & u+\beta {{vv}}_{z}\\ -1 & -\lambda +v-v{\partial }_{z}\end{array}\right)\psi =0,\\ \hat{{N}^{z}}\psi =\left(\begin{array}{cc}{\lambda }^{2}+\displaystyle \frac{1}{2}{{vv}}_{z}-{v}^{2}-{\bar{J}}_{2}{\partial }_{z}-{\partial }_{t} & {\hat{N}}_{12}^{z}\\ -\lambda -v & -{\lambda }^{2}-\displaystyle \frac{1}{2}{{vv}}_{z}+{v}^{2}-{\bar{J}}_{2}{\partial }_{z}-{\partial }_{t}\end{array}\right)\psi =0,\end{array}\end{eqnarray*}$
where ${\hat{N}}_{12}^{z}=\lambda (u+\beta {{vv}}_{z})+\tfrac{1}{2}{{vu}}_{z}+\tfrac{1}{2}\beta v({v}_{z}^{2}+{{vv}}_{{zz}})+{uv}+\beta {v}^{2}{v}_{z}.$The conservation laws10 ) yield the deformation operators10 )2 ) again.
$\begin{eqnarray*}\begin{array}{l}{u}_{t}=\left(\displaystyle \frac{\beta (2-\beta )}{4}(2{v}^{3}{v}_{{zz}}+{v}^{2}{v}_{z}^{2})\right.\\ {\left.+\displaystyle \frac{1-\beta }{2}{v}^{2}{u}_{z}+{{uv}}^{2}-\displaystyle \frac{1}{4}{u}^{2}\right)}_{z},\\ {\left({v}^{-1}\right)}_{t}={\left(\displaystyle \frac{1-\beta }{2}{v}_{z}-\displaystyle \frac{u}{2v}-v\right)}_{z},\\ {\left({{uv}}^{-1}\right)}_{t}=\left(\displaystyle \frac{\beta (2-\beta )}{2}({v}^{2}{v}_{{zz}}+{{vv}}_{z}^{2})\right.\\ {\left.+\displaystyle \frac{1-\beta }{2}({{uv}}_{z}+{{vu}}_{z})-\displaystyle \frac{{u}^{2}}{2v}+{uv}\right)}_{z},\end{array}\end{eqnarray*}$
of system ( $\begin{eqnarray*}\begin{array}{l}{\hat{L}}_{2}={\partial }_{z}+\displaystyle \frac{1}{v}{\partial }_{x}+\displaystyle \frac{u}{v}{\partial }_{y}+u{\partial }_{\xi },\\ {\hat{T}}_{2}={\partial }_{t}+\left(\displaystyle \frac{\beta (2-\beta )}{2}({v}^{2}{\hat{L}}_{2}^{2}v+v{\left({\hat{L}}_{2}v\right)}^{2})\right.\\ \left.+\displaystyle \frac{1-\beta }{2}(u{\hat{L}}_{2}v+v{\hat{L}}_{2}u)-\displaystyle \frac{{u}^{2}}{2v}+{uv}\right){\partial }_{y}\\ +\left(\displaystyle \frac{\beta (2-\beta )}{4}(2{v}^{3}{\hat{L}}_{2}^{2}v+{v}^{2}{\left({\hat{L}}_{2}v\right)}^{2})\right.\\ \left.+\displaystyle \frac{1-\beta }{2}{v}^{2}{\hat{L}}_{2}u+{{uv}}^{2}-\displaystyle \frac{1}{4}{u}^{2}\right){\partial }_{\xi }\\ +\left(\displaystyle \frac{1-\beta }{2}{\hat{L}}_{2}v-\displaystyle \frac{u}{2v}-v\right){\partial }_{x}.\end{array}\end{eqnarray*}$
With these operators in hand, we have the (4+1)-dimensional integrable deformation of system ( $\begin{eqnarray}\begin{array}{l}{\hat{T}}_{2}u=\displaystyle \frac{\beta (2-\beta )}{2}({v}^{3}{\hat{L}}_{2}^{3}v+4{v}^{2}{\hat{L}}_{2}v{\hat{L}}_{z}^{2}v+v{\left({\hat{L}}_{2}v\right)}^{3})\\ \,+\displaystyle \frac{1-\beta }{2}({v}^{2}{\hat{L}}_{2}^{2}u+2v{\hat{L}}_{2}u{\hat{L}}_{2}v)\\ \,+{v}^{2}{\hat{L}}_{2}u-\displaystyle \frac{u}{2}{\hat{L}}_{2}u+2{uv}{\hat{L}}_{2}v,\\ {\hat{T}}_{2}v=\displaystyle \frac{\beta -1}{2}{v}^{2}{\hat{L}}_{2}^{2}v\\ \,+\displaystyle \frac{v}{2}{\hat{L}}_{2}u-\displaystyle \frac{u}{2}{\hat{L}}_{2}v+{v}^{2}{\hat{L}}_{2}v,\end{array}\end{eqnarray}$
which is system (3.1.3. Reduction with respect to {ξ, t}
When u and v are {y, z}-independent, system (2 ) turns into the following (1+1)-dimensional CBB system12 ).
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{\beta (2-\beta )}{4}(2{u}^{3}{v}^{3}{v}_{\xi \xi \xi }+2{u}^{2}{v}^{3}{v}_{\xi }{u}_{\xi \xi }+4{u}^{2}{v}^{3}{u}_{\xi }{v}_{\xi \xi }+8{u}^{3}{v}^{2}{v}_{\xi }{v}_{\xi \xi }+7{u}^{2}{v}^{2}{v}_{\xi }^{2}{u}_{\xi }+2{u}^{3}{{vv}}_{\xi }^{3})\\ +\displaystyle \frac{1-\beta }{2}({u}^{2}{v}^{2}{u}_{\xi \xi }+2{u}^{2}{{vu}}_{\xi }{v}_{\xi })-\displaystyle \frac{1}{4}{u}^{2}{u}_{\xi }+2{u}^{2}{{vv}}_{\xi },\\ {v}_{t}=\displaystyle \frac{\beta (\beta -2)}{4}(2{u}^{2}{v}^{3}{v}_{\xi }{v}_{\xi \xi }+2{{uv}}^{3}{v}_{\xi }^{2}{u}_{\xi }+{u}^{2}{v}^{2}{v}_{\xi }^{3})+\displaystyle \frac{\beta -1}{2}({u}^{2}{v}^{2}{v}_{\xi \xi }+2{{uv}}^{2}{u}_{\xi }{v}_{\xi })\\ +\displaystyle \frac{1}{2}{{uvu}}_{\xi }-\displaystyle \frac{1}{4}{u}^{2}{v}_{\xi },\end{array}\right.\end{eqnarray}$
which possesses the Lax pair $\begin{eqnarray*}\begin{array}{l}\hat{{M}^{\xi }}\psi =\left(\begin{array}{cc}\lambda -v-{uv}{\partial }_{\xi } & u+\beta {{uvv}}_{\xi }\\ -1 & -\lambda +v-{uv}{\partial }_{\xi }\end{array}\right)\psi =0,\\ \hat{{N}^{\xi }}\psi =\left(\begin{array}{cc}{\lambda }^{2}+\displaystyle \frac{1}{2}{{uvv}}_{\xi }-{v}^{2}-{\bar{J}}_{3}{\partial }_{\xi }-{\partial }_{t} & {\hat{N}}_{12}^{\xi }\\ -\lambda -v & -{\lambda }^{2}-\displaystyle \frac{1}{2}{{uvv}}_{\xi }+{v}^{2}-{\bar{J}}_{3}{\partial }_{\xi }-{\partial }_{t}\end{array}\right)\psi =0,\end{array}\end{eqnarray*}$
where ${\hat{N}}_{12}^{\xi }=\lambda (u+\beta {{uv}}_{\xi })+\tfrac{1}{2}{{uu}}_{\xi }$ + $\tfrac{1}{2}\beta u({u}_{\xi }{v}_{\xi }$ + uvξξ) + uv + βuvvξ, and the compatibility condition $[{\hat{M}}^{\xi },{\hat{N}}^{\xi }]=0$ just gives system (It can be found that system (12 ) has the conservation laws12 ) yields the following (4+1)-dimensional system2 ) and (13 ) are identical.
$\begin{eqnarray*}\begin{array}{l}{\left({u}^{-1}\right)}_{t}=\left(\displaystyle \frac{\beta (\beta -2)}{4}(2{{uv}}^{3}{v}_{\xi \xi }+2{v}^{3}{u}_{\xi }{v}_{\xi }+{{uv}}^{2}{v}_{\xi }^{2})\right.\\ \,{\left.+\displaystyle \frac{\beta -1}{2}{v}^{2}{u}_{\xi }+\displaystyle \frac{1}{4}u-{v}^{2}\right)}_{\xi },\\ {\left({v}^{-1}\right)}_{t}={\left(\displaystyle \frac{\beta (2-\beta )}{4}{u}^{2}{{vv}}_{\xi }^{2}+\displaystyle \frac{1-\beta }{2}{u}^{2}{v}_{\xi }-\displaystyle \frac{{u}^{2}}{4v}\right)}_{\xi },\\ {\left({u}^{-1}{v}^{-1}\right)}_{t}=\left(\displaystyle \frac{\beta (\beta -2)}{4}(2{{uv}}^{2}{v}_{\xi \xi }+2{v}^{2}{u}_{\xi }{v}_{\xi }+{{uvv}}_{\xi }^{2})\right.\\ {\left.+\displaystyle \frac{\beta -1}{2}({{vu}}_{\xi }-{{uv}}_{\xi })-2v-\displaystyle \frac{u}{4v}\right)}_{\xi }.\end{array}\end{eqnarray*}$
Thus, applying the deformation algorithm to the (1+1)-dimensional integrable system ( $\begin{eqnarray}\begin{array}{l}{\hat{T}}_{3}u=\displaystyle \frac{\beta (2-\beta )}{4}(2{u}^{3}{v}^{3}{\hat{L}}_{3}^{3}v\\ +2{u}^{2}{v}^{3}{\hat{L}}_{3}v{\hat{L}}_{3}^{2}u+4{u}^{2}{v}^{3}{\hat{L}}_{3}u{\hat{L}}_{3}^{2}v\\ +8{u}^{3}{v}^{2}{\hat{L}}_{3}v{\hat{L}}_{3}^{2}v+7{u}^{2}{v}^{2}{\left({\hat{L}}_{3}v\right)}^{2}{\hat{L}}_{3}u\\ +2{u}^{3}v{\left({\hat{L}}_{3}v\right)}^{3})\\ +\displaystyle \frac{1-\beta }{2}({u}^{2}{v}^{2}{\hat{L}}_{3}^{2}u+2{u}^{2}v{\hat{L}}_{3}u{\hat{L}}_{3}v)\\ -\displaystyle \frac{1}{4}{u}^{2}{\hat{L}}_{3}u+2{u}^{2}v{\hat{L}}_{3}v,\\ {\hat{T}}_{3}v=\displaystyle \frac{\beta (\beta -2)}{4}(2{u}^{2}{v}^{3}{\hat{L}}_{3}v{\hat{L}}_{3}^{2}v\\ +2{{uv}}^{3}{\left({\hat{L}}_{3}v\right)}^{2}{\hat{L}}_{3}u+{u}^{2}{v}^{2}{\left({\hat{L}}_{3}v\right)}^{3})\\ +\displaystyle \frac{\beta -1}{2}({u}^{2}{v}^{2}{\hat{L}}_{3}^{2}v+2{{uv}}^{2}{\hat{L}}_{3}u{\hat{L}}_{3}v)\\ +\displaystyle \frac{1}{2}{uv}{\hat{L}}_{3}u-\displaystyle \frac{1}{4}{u}^{2}{\hat{L}}_{3}v,\end{array}\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{l}{\hat{L}}_{3}={\partial }_{\xi }+\displaystyle \frac{1}{{uv}}{\partial }_{x}+\displaystyle \frac{1}{u}{\partial }_{z}+\displaystyle \frac{1}{v}{\partial }_{y},\\ {\hat{T}}_{3}={\partial }_{t}+\left(\displaystyle \frac{\beta (\beta -2)}{4}(2{{uv}}^{2}{\hat{L}}_{3}^{2}v\right.\\ +2{v}^{2}{\hat{L}}_{3}u{\hat{L}}_{3}v+{uv}{\left({\hat{L}}_{3}v\right)}^{2})\\ \left.+\displaystyle \frac{\beta -1}{2}(v{\hat{L}}_{3}u-u{\hat{L}}_{3}v)-2v-\displaystyle \frac{u}{4v}\right){\partial }_{x}\\ +\left(\displaystyle \frac{\beta (\beta -2)}{4}(2{{uv}}^{3}{\hat{L}}_{3}^{2}v\right.\\ \left.+2{v}^{3}{\hat{L}}_{3}u{\hat{L}}_{3}v+{{uv}}^{2}{\left({\hat{L}}_{3}v\right)}^{2})+\displaystyle \frac{\beta -1}{2}{v}^{2}{\hat{L}}_{3}u+\displaystyle \frac{1}{4}u-{v}^{2}\right){\partial }_{z}\\ +\left(\displaystyle \frac{\beta (2-\beta )}{4}{u}^{2}v{\left({\hat{L}}_{3}v\right)}^{2}+\displaystyle \frac{1-\beta }{2}{u}^{2}{\hat{L}}_{3}v-\displaystyle \frac{{u}^{2}}{4v}\right){\partial }_{y}.\end{array}\end{eqnarray*}$
Note that systems (3.2. The (2+1)-dimensional reduced systems of system (2 )
We now consider the (2+1)-dimensional reduced systems of the (4+1)-dimensional CBB system. Here, we assume β = 0 for brevity.
Case (i) If u and v are {z, ξ}-independent, system (2 ) becomes the (2+1)-dimensional CBB system as follows
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{1}{2}{u}_{{xx}}+{{uu}}_{{xy}}+\displaystyle \frac{1}{2}{u}^{2}{u}_{{yy}}+2{u}^{2}{v}_{y}+2{{uv}}_{x}+2{u}_{x}v,\\ {v}_{t}=-\displaystyle \frac{1}{2}{v}_{{xx}}-{{uv}}_{{xy}}-\displaystyle \frac{1}{2}{u}^{2}{v}_{{yy}}+\displaystyle \frac{1}{2}{u}_{x}-{u}_{x}{v}_{y}-{{uu}}_{y}{v}_{y}+\displaystyle \frac{1}{2}{{uu}}_{y}+2{{vv}}_{x}.\end{array}\right.\end{eqnarray}$
Case (ii) Assuming uy = uξ = vy = vξ = 0, we arrive at the reduced system
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{1}{2}{u}_{{xx}}+{{vu}}_{{xz}}+\displaystyle \frac{1}{2}{v}^{2}{u}_{{zz}}+2{u}_{x}v+{v}_{x}{u}_{z}+{{vu}}_{z}{v}_{z}+{v}^{2}{u}_{z}-\displaystyle \frac{1}{2}{{uu}}_{z}+2{{uv}}_{x}+2{{uvv}}_{z},\\ {v}_{t}=-\displaystyle \frac{1}{2}{v}_{{xx}}-{{vv}}_{{xz}}-\displaystyle \frac{1}{2}{v}^{2}{v}_{{zz}}+\displaystyle \frac{1}{2}{u}_{x}+\displaystyle \frac{1}{2}{{vu}}_{z}+2{{vv}}_{x}+{v}^{2}{v}_{z}-\displaystyle \frac{1}{2}{{uv}}_{z}.\end{array}\right.\end{eqnarray}$
Case (iii) If u and v are {x, ξ} dependent, system (2 ) is equivalent to
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{1}{2}{u}_{{xx}}+{{uvu}}_{x\xi }+\displaystyle \frac{1}{2}{u}^{2}{v}^{2}{u}_{\xi \xi }+2{{vu}}_{x}+2{{uv}}_{x}+{{uv}}_{x}{u}_{\xi }+{u}^{2}{{vv}}_{\xi }{u}_{\xi }-\displaystyle \frac{1}{4}{u}^{2}{u}_{\xi }+2{u}^{2}{{vv}}_{\xi },\\ {v}_{t}=-\displaystyle \frac{1}{2}{v}_{{xx}}-{{uvv}}_{x\xi }-\displaystyle \frac{1}{2}{u}^{2}{v}^{2}{v}_{\xi \xi }+\displaystyle \frac{1}{2}{u}_{x}-{{vv}}_{\xi }{u}_{x}-{{uv}}^{2}{u}_{\xi }{v}_{\xi }+2{{vv}}_{x}+\displaystyle \frac{1}{2}{{uvu}}_{\xi }-\displaystyle \frac{1}{4}{u}^{2}{v}_{\xi }.\end{array}\right.\end{eqnarray}$
Case (iv) Considering u and v, which are only dependent on {y, z}, leads to
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{1}{2}{u}^{2}{u}_{{yy}}+{{uvu}}_{{yz}}+\displaystyle \frac{1}{2}{v}^{2}{u}_{{zz}}+{{uv}}_{y}{u}_{z}+{{vv}}_{z}{u}_{z}+{v}^{2}{u}_{z}-\displaystyle \frac{1}{2}{{uu}}_{z}+2{u}^{2}{v}_{y}+2{{uvv}}_{z},\\ {v}_{t}=-\displaystyle \frac{1}{2}{u}^{2}{v}_{{yy}}-{{uvv}}_{{yz}}-\displaystyle \frac{1}{2}{v}^{2}{v}_{{zz}}-{{uu}}_{y}{v}_{y}+\displaystyle \frac{1}{2}{{uu}}_{y}-{{vv}}_{y}{u}_{z}+\displaystyle \frac{1}{2}{{vu}}_{z}+{v}^{2}{v}_{z}-\displaystyle \frac{1}{2}{{uv}}_{z}.\end{array}\right.\end{eqnarray}$
Case (v) If u and v are {x, y}-independent, system (2 ) turns out to be
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{1}{2}{u}^{2}{v}^{2}{u}_{\xi \xi }+{{uv}}^{2}{u}_{\xi z}+\displaystyle \frac{1}{2}{v}^{2}{u}_{{zz}}+{u}^{2}{{vu}}_{\xi }{v}_{\xi }+{{uvv}}_{z}{u}_{\xi }-\displaystyle \frac{1}{4}{u}^{2}{u}_{\xi }+{{uvv}}_{\xi }{u}_{z}+{{vv}}_{z}{u}_{z}+{v}^{2}{u}_{z}\\ -\displaystyle \frac{1}{2}{{uu}}_{z}+2{u}^{2}{{vv}}_{\xi }+2{{uvv}}_{z},\\ {v}_{t}=-\displaystyle \frac{1}{2}{u}^{2}{v}^{2}{v}_{\xi \xi }-{{uv}}^{2}{v}_{\xi z}-\displaystyle \frac{1}{2}{v}^{2}{v}_{{zz}}-{{uv}}^{2}{v}_{\xi }{u}_{\xi }+\displaystyle \frac{1}{2}{{uvu}}_{\xi }-{v}^{2}{v}_{\xi }{u}_{z}+\displaystyle \frac{1}{2}{{vu}}_{z}\\ -\displaystyle \frac{1}{4}{u}^{2}{v}_{\xi }+{v}^{2}{v}_{z}-\displaystyle \frac{1}{2}{{uv}}_{z}.\end{array}\right.\end{eqnarray}$
Case (vi) By considering {x, z}-independent u and v, we have
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{1}{2}{u}^{2}{v}^{2}{u}_{\xi \xi }+{u}^{2}{{vu}}_{\xi y}+\displaystyle \frac{1}{2}{u}^{2}{u}_{{yy}}+{u}^{2}{{vu}}_{\xi }{v}_{\xi }+{u}^{2}{v}_{y}{u}_{\xi }-\displaystyle \frac{1}{4}{u}^{2}{u}_{\xi }+2{u}^{2}{{vv}}_{\xi }+2{u}^{2}{v}_{y},\\ {v}_{t}=-\displaystyle \frac{1}{2}{u}^{2}{v}^{2}{v}_{\xi \xi }-{u}^{2}{{vv}}_{\xi y}-\displaystyle \frac{1}{2}{u}^{2}{v}_{{yy}}-{{uv}}^{2}{u}_{\xi }{v}_{\xi }-{{uvv}}_{y}{u}_{\xi }+\displaystyle \frac{1}{2}{{uvu}}_{\xi }-{{uvu}}_{y}{v}_{\xi }\\ -{{uu}}_{y}{v}_{y}+\displaystyle \frac{1}{2}{{uu}}_{y}-\displaystyle \frac{1}{4}{u}^{2}{v}_{\xi }.\end{array}\right.\end{eqnarray}$
Remark 3.1 Since (2+1)-dimensional systems (14 )–(19 ) are all reduced systems of system (2 ), their Lax pairs can be obtained by directly removing the derivatives of two independent variables from the Lax pair (4 ) of system (2 ).
3.3. The (3+1)-dimensional reduced systems of system (2 )
Here, we reduce system (2 ) to a variety of (3+1)-dimensional systems. For brevity, we only consider the reduction in the case of β = 0.
Case (i) Provided that uξ = vξ = 0, system (2 ) reduces to
$\begin{eqnarray}\begin{array}{c}\left\{\begin{array}{c}{u}_{t}=2{{uv}}_{x}+2{{vu}}_{x}+\left({v}_{x}+{{uv}}_{y}+{{vv}}_{z}+{v}^{2}-\displaystyle \frac{1}{2}u\right){u}_{z}+2{u}^{2}{v}_{y}+\displaystyle \frac{1}{2}{u}_{{xx}}+\displaystyle \frac{1}{2}{u}^{2}{u}_{{yy}}+{{uvu}}_{{yz}}\\ +{{uu}}_{{xy}}+{{vu}}_{{xz}}+\displaystyle \frac{1}{2}{v}^{2}{u}_{{zz}}+2{{uvv}}_{z},\\ {v}_{t}=-\displaystyle \frac{1}{2}{v}_{{xx}}-{{uv}}_{{xy}}-{{vv}}_{{xz}}-\displaystyle \frac{1}{2}{u}^{2}{v}_{{yy}}-{{uvv}}_{{yz}}-\displaystyle \frac{1}{2}{v}^{2}{v}_{{zz}}+\left(\displaystyle \frac{1}{2}-{v}_{y}\right){u}_{x}+\left(\displaystyle \frac{1}{2}u-{{uv}}_{y}\right){u}_{y}\\ +\left(\displaystyle \frac{1}{2}v-{{vv}}_{y}\right){u}_{z}+2{{vv}}_{x}+\left({v}^{2}-\displaystyle \frac{1}{2}u\right){v}_{z}.\end{array}\right.\end{array}\end{eqnarray}$
Case (ii) If u and v are both z-independent, we have
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{1}{2}{u}^{2}{u}_{{yy}}+{{uvu}}_{x\xi }+{{uu}}_{{xy}}+\displaystyle \frac{1}{2}{u}^{2}{v}^{2}{u}_{\xi \xi }+{u}^{2}{{vu}}_{\xi y}+2{{uv}}_{x}+2{u}^{2}{{vv}}_{\xi }+2{{vu}}_{x}\\ +\left({{uv}}_{x}+{u}^{2}{{vv}}_{\xi }+{u}^{2}{v}_{y}-\displaystyle \frac{1}{4}{u}^{2}\right){u}_{\xi }+2{u}^{2}{v}_{y}+\displaystyle \frac{1}{2}{u}_{{xx}},\\ {v}_{t}=-\displaystyle \frac{1}{2}{v}_{{xx}}-{{uvv}}_{x\xi }-{{uv}}_{{xy}}-\displaystyle \frac{1}{2}{u}^{2}{v}^{2}{v}_{\xi \xi }-{u}^{2}{{vv}}_{\xi y}-\displaystyle \frac{1}{2}{u}^{2}{v}_{{yy}}+\left(\displaystyle \frac{1}{2}-{{vv}}_{\xi }-{v}_{y}\right){u}_{x}\\ +\left(\displaystyle \frac{1}{2}{uv}-{{uv}}^{2}{v}_{\xi }-{{uvv}}_{y}\right){u}_{\xi }+\left(\displaystyle \frac{1}{2}u-{{uvv}}_{\xi }-{{uv}}_{y}\right){u}_{y}+2{{vv}}_{x}-\displaystyle \frac{1}{4}{u}^{2}{v}_{\xi }.\end{array}\right.\end{eqnarray}$
Case (iii) When u and v are y-independent functions, we obtain
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=2{{uvv}}_{z}+2{{uv}}_{x}+2{u}^{2}{{vv}}_{\xi }+2{{vu}}_{x}+\left({{uv}}_{x}+{u}^{2}{{vv}}_{\xi }+{{uvv}}_{z}-\displaystyle \frac{1}{4}{u}^{2}\right){u}_{\xi }+\displaystyle \frac{1}{2}{u}^{2}{v}^{2}{u}_{\xi \xi }\\ +{{uv}}^{2}{u}_{\xi z}+\left({v}_{x}+{{uvv}}_{\xi }+{{vv}}_{z}+{v}^{2}-\displaystyle \frac{1}{2}u\right){u}_{z}+\displaystyle \frac{1}{2}{u}_{{xx}}+{{uvu}}_{x\xi }+{{vu}}_{{xz}}+\displaystyle \frac{1}{2}{v}^{2}{u}_{{zz}},\\ {v}_{t}=-\displaystyle \frac{1}{2}{v}_{{xx}}-{{uvv}}_{x\xi }-{{vv}}_{{xz}}-\displaystyle \frac{1}{2}{u}^{2}{v}^{2}{v}_{\xi \xi }-{{uv}}^{2}{v}_{\xi z}-\displaystyle \frac{1}{2}{v}^{2}{v}_{{zz}}+\left(\displaystyle \frac{1}{2}-{{vv}}_{\xi }\right){u}_{x}\\ +\left(\displaystyle \frac{1}{2}{uv}-{{uv}}^{2}{v}_{\xi }\right){u}_{\xi }+\left(\displaystyle \frac{1}{2}v-{v}^{2}{v}_{\xi }\right){u}_{z}+2{{vv}}_{x}-\displaystyle \frac{1}{4}{u}^{2}{v}_{\xi }+\left({v}^{2}-\displaystyle \frac{1}{2}u\right){v}_{z}.\end{array}\right.\end{eqnarray}$
Case (iv) On the condition that u and v are x-independent, system (2 ) gives rise to
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=2{{uvv}}_{z}+{{uvu}}_{{yz}}+\displaystyle \frac{1}{2}{v}^{2}{u}_{{zz}}+\displaystyle \frac{1}{2}{u}^{2}{v}^{2}{u}_{\xi \xi }+{u}^{2}{{vu}}_{\xi y}+{{uv}}^{2}{u}_{\xi z}+2{u}^{2}{{vv}}_{\xi }+2{u}^{2}{v}_{y}+\displaystyle \frac{1}{2}{u}^{2}{u}_{{yy}}\\ +\left({u}^{2}{{vv}}_{\xi }+{u}^{2}{v}_{y}+{{uvv}}_{z}-\displaystyle \frac{1}{4}{u}^{2}\right){u}_{\xi }+\left({{uvv}}_{\xi }+{{uv}}_{y}+{{vv}}_{z}+{v}^{2}-\displaystyle \frac{1}{2}u\right){u}_{z},\\ {v}_{t}=-\displaystyle \frac{1}{2}{u}^{2}{v}^{2}{v}_{\xi \xi }-{u}^{2}{{vv}}_{\xi y}-{{uv}}^{2}{v}_{\xi z}-\displaystyle \frac{1}{2}{u}^{2}{v}_{{yy}}-{{uvv}}_{{yz}}-\displaystyle \frac{1}{2}{v}^{2}{v}_{{zz}}+\left(\displaystyle \frac{1}{2}{uv}-{{uv}}^{2}{v}_{\xi }-{{uvv}}_{y}\right){u}_{\xi }\\ +\left(\displaystyle \frac{1}{2}u-{{uvv}}_{\xi }-{{uv}}_{y}\right){u}_{y}+\left(\displaystyle \frac{1}{2}v-{v}^{2}{v}_{\xi }-{{vv}}_{y}\right){u}_{z}-\displaystyle \frac{1}{4}{u}^{2}{v}_{\xi }+\left({v}^{2}-\displaystyle \frac{1}{2}u\right){v}_{z}.\end{array}\right.\end{eqnarray}$
4. Exact solutions of a reduced system
It is well known that exact solutions often play crucial roles in the study of asymptotic behavior, blow up (or extinction) and geometric properties of invariant geometric flows related to the PDEs under study. The purpose of this section is to construct exact solutions of (1+1)-dimensional CBB system (7 ) with β = 0, which takes the form of
$\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=\displaystyle \frac{1}{2}{u}^{2}{u}_{{yy}}+2{u}^{2}{v}_{y},\\ {v}_{t}=-\displaystyle \frac{1}{2}{u}^{2}{v}_{{yy}}-{{uu}}_{y}{v}_{y}+\displaystyle \frac{1}{2}{{uu}}_{y}.\end{array}\right.\end{eqnarray}$
In recent decades, a large number of approaches have been developed to construct the exact solutions of integrable PDEs, such as Darboux transformation [16], Hirota's direct method [17], inverse scattering transformation [18] and Lie symmetry analysis [19, 20]. Whereas, it is very difficult to find exact solutions for system (24 ) since it is related to the CBB equation with a reciprocal transformation. We now construct exact solutions of system (24 ) based on Lie symmetry analysis and the power series method.
The Lie group method of infinitesimal transformations, introduced by Sophus Lie, is a powerful and universal method for constructing exact solutions and performing symmetry reductions of differential equations (DEs). More information about the symmetry group method can be found in [19, 20]. To apply the classical method to system (24 ), we consider the one-parameter Lie group of infinitesimal transformations in (y, t, u, v) given by24 ) with respect to the Lie symmetry (25 ) requires that the prolonged field annihilates system (24 ) on its solution manifold, namely,25 ), keeping only the necessary terms26 ) yields an overdetermined system of PDEs27 ) gives the Lie symmetries of system (24 )
$\begin{eqnarray*}\begin{array}{l}y\to y+\varepsilon \zeta (y,t,u,v),\,\,t\to t+\varepsilon \tau (y,t,u,v),\\ u\to u\,+\,\varepsilon \eta (y,t,u,v),\,\,v\to v+\varepsilon \phi (y,t,u,v),\end{array}\end{eqnarray*}$
where ϵ is the group parameter. The associated Lie algebra is realized by vector fields $\begin{eqnarray}\begin{array}{l}V=\zeta (y,t,u,v){\partial }_{y}+\tau (y,t,u,v){\partial }_{t}\\ +\eta (y,t,u,v){\partial }_{u}+\phi (y,t,u,v){\partial }_{v}.\end{array}\end{eqnarray}$
The infinitesimal invariance criterion of system ( $\begin{eqnarray}\begin{array}{l}{\Pr }^{(2)}V\left({u}_{t}-\displaystyle \frac{1}{2}{u}^{2}{u}_{{yy}}-2{u}^{2}{v}_{y}\right){| }_{(24)}=0,\\ {\Pr }^{(2)}V\left({v}_{t}+\displaystyle \frac{1}{2}{u}^{2}{v}_{{yy}}+{{uu}}_{y}{v}_{y}-\displaystyle \frac{1}{2}{{uu}}_{y}\right){| }_{(24)}=0,\end{array}\end{eqnarray}$
where Pr(2)V is the standard second-order prolongation operator of the vector field ( $\begin{eqnarray*}\begin{array}{l}{\Pr }^{(2)}V=V+{\eta }^{y}{\partial }_{{u}_{y}}+{\phi }^{y}{\partial }_{{v}_{y}}+{\eta }^{t}{\partial }_{{u}_{t}}+{\phi }^{t}{\partial }_{{v}_{t}}+{\eta }^{{yy}}{\partial }_{{u}_{{yy}}}+{\phi }^{{yy}}{\partial }_{{v}_{{yy}}},\end{array}\end{eqnarray*}$
where ηy, φy, ηt, φt, ηyy and φyy are given explicitly in terms of the infinitesimals [19]. Setting its coefficients of the linearly dependent derivatives uy, vy, uyy, vyy, ⋯ to be zero, system ( $\begin{eqnarray}\begin{array}{l}{\eta }_{t}-\displaystyle \frac{1}{2}{u}^{2}{\eta }_{{yy}}-2{u}^{2}{\phi }_{y}=0,\\ \displaystyle \frac{1}{2}u{\tau }_{t}+\eta -u{\zeta }_{y}=0,\\ u({\eta }_{u}-{\tau }_{t})-2\eta -u({\phi }_{v}-{\zeta }_{y})=0,\\ {\zeta }_{t}+\displaystyle \frac{1}{2}{u}^{2}(2{\eta }_{{uy}}-{\zeta }_{{yy}})=0,\\ {\phi }_{t}+\displaystyle \frac{1}{2}{u}^{2}{\phi }_{{yy}}-\displaystyle \frac{1}{2}u{\eta }_{y}=0,\\ {\zeta }_{t}-\displaystyle \frac{1}{2}{u}^{2}(2{\phi }_{{yv}}-{\zeta }_{{yy}})-u{\eta }_{y}=0,\\ \eta +u{\tau }_{t}+u({\eta }_{u}-2{\zeta }_{y})=0,\\ u({\phi }_{v}-{\tau }_{t})+2u{\phi }_{y}-\eta -u({\eta }_{u}-{\zeta }_{y})=0,\\ {\zeta }_{u}={\zeta }_{v}={\phi }_{u}={\eta }_{v}={\phi }_{{vv}}\\ ={\eta }_{{uu}}={\tau }_{u}={\tau }_{v}={\tau }_{y}=0.\end{array}\end{eqnarray}$
Solving system ( $\begin{eqnarray*}\zeta ={c}_{1}+{c}_{2}y,\,\,\tau =-2{c}_{2}t+{c}_{3},\,\,\eta =2{c}_{2}u,\,\,\phi ={c}_{2}v+{c}_{4},\end{eqnarray*}$
where ci are arbitrary constants. Hence, we have the following assertion.Theorem 4.1. System (24 ) admits four-dimensional Lie algebra generated by the vector fields
$\begin{eqnarray*}\begin{array}{l}{V}_{1}={\partial }_{y},\,\,\,\,{V}_{2}={\partial }_{t},\\ {V}_{3}={\partial }_{v},\,\,\,\,{V}_{4}=y{\partial }_{y}-2t{\partial }_{t}+2u{\partial }_{u}+v{\partial }_{v}.\end{array}\end{eqnarray*}$
We now perform similarity reduction for system (24 ). Having determined the infinitesimals, the symmetry variables are found by solving the characteristic equation
$\begin{eqnarray*}\displaystyle \frac{{\rm{d}}y}{\zeta }=\displaystyle \frac{{\rm{d}}t}{\tau }=\displaystyle \frac{{\rm{d}}u}{\eta }=\displaystyle \frac{{\rm{d}}v}{\phi },\end{eqnarray*}$
or the corresponding invariant surface condition.To construct a non-trivial invariant solution of system (24 ), here we just consider the vector field V4. By integrating the characteristic equation
$\begin{eqnarray*}\displaystyle \frac{{\rm{d}}y}{y}=\displaystyle \frac{{\rm{d}}t}{-2t}=\displaystyle \frac{{\rm{d}}u}{2u}=\displaystyle \frac{{\rm{d}}v}{v},\end{eqnarray*}$
we have the similarity transformation $\begin{eqnarray}u(y,t)=\displaystyle \frac{1}{t}f(\omega ),\,\,\,\,v(y,t)=\displaystyle \frac{1}{\sqrt{t}}g(\omega ),\end{eqnarray}$
with the reduced variable $\omega =y\sqrt{t}$.Substituting system (28 ) into system (24 ) gives
$\begin{eqnarray}\left\{\begin{array}{l}-f+\displaystyle \frac{1}{2}\omega f^{\prime} -\displaystyle \frac{1}{2}{f}^{2}f^{\prime\prime} -2{f}^{2}g^{\prime} =0,\\ -\displaystyle \frac{1}{2}g+\displaystyle \frac{1}{2}\omega g^{\prime} +\displaystyle \frac{1}{2}{f}^{2}g^{\prime\prime} +{ff}^{\prime} g^{\prime} -\displaystyle \frac{1}{2}{ff}^{\prime} =0.\end{array}\right.\end{eqnarray}$
As explained before, we need the solutions of system (29 ) to construct the solutions of system (24 ). In what follows, we derive the solution of system (29 ) in a power series of the form
$\begin{eqnarray}f(\omega )=\displaystyle \sum _{n=0}^{\infty }{a}_{n}{\omega }^{n},\,\,\,\,g(\omega )=\displaystyle \sum _{n=0}^{\infty }{b}_{n}{\omega }^{n}.\end{eqnarray}$
Inserting system (30 ) into system (29 ), we have
$\begin{eqnarray}\begin{array}{l}-\displaystyle \sum _{n=0}^{\infty }{a}_{n}{\omega }^{n}+\displaystyle \frac{1}{2}\displaystyle \sum _{n=1}^{\infty }{{na}}_{n}{\omega }^{n}\\ -\displaystyle \frac{1}{2}\displaystyle \sum _{n=0}^{\infty }\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{k=0}^{m}{n}^{* }({n}^{* }+1){a}_{k}{a}_{{m}^{* }-1}{a}_{{n}^{* }+1}{\omega }^{n}\\ -2\displaystyle \sum _{n=0}^{\infty }\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{k=0}^{m}{n}^{* }{a}_{k}{a}_{{m}^{* }-1}{b}_{{n}^{* }}{\omega }^{n}=0,\\ -\displaystyle \frac{1}{2}\displaystyle \sum _{n=0}^{\infty }{b}_{n}{\omega }^{n}\\ +\displaystyle \frac{1}{2}\displaystyle \sum _{n=1}^{\infty }{{nb}}_{n}{\omega }^{n}+\displaystyle \frac{1}{2}\displaystyle \sum _{n=0}^{\infty }\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{k=0}^{m}{n}^{* }({n}^{* }+1){a}_{k}{a}_{{m}^{* }-1}{b}_{{n}^{* }+1}{\omega }^{n}\\ +\displaystyle \sum _{n=0}^{\infty }\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{k=0}^{m}{n}^{* }{m}^{* }{a}_{k}{a}_{{m}^{* }}{b}_{{n}^{* }}\\ -\displaystyle \frac{1}{2}\displaystyle \sum _{n=0}^{\infty }\displaystyle \sum _{m=0}^{n}{n}^{* }{a}_{m}{a}_{{n}^{* }}{\omega }^{n}=0,\end{array}\end{eqnarray}$
where n* = n − m + 1, m* = m − k + 1.In view of system (31 ), by comparing coefficients for n = 0, we obtain31 ), we can determine the coefficients recursively30 ), with the coefficients recursively given by systems (32 ) and (33 ). In view of system (33 ), we have the following estimation34 ) is proved, thus the power series (30 ) are convergent.
$\begin{eqnarray}\begin{array}{l}{a}_{2}=-\displaystyle \frac{1}{{a}_{0}}-2{b}_{1},\\ {b}_{2}=\displaystyle \frac{{a}_{1}}{2{a}_{0}}-\displaystyle \frac{{a}_{1}{b}_{1}}{{a}_{0}}+\displaystyle \frac{{b}_{0}}{2{a}_{0}^{2}},\end{array}\end{eqnarray}$
where a0 ≠ 0, a1, b0 and b1 are arbitrary constants. Similarly, when n ≥ 1, by comparing the coefficients of ωn in system ( $\begin{eqnarray}\begin{array}{l}{a}_{n+2}=\displaystyle \frac{1}{(n+1)(n+2){a}_{0}^{2}}\left((n-2){a}_{n}\right.\\ -\displaystyle \sum _{m=1}^{n}\displaystyle \sum _{k=0}^{m}{n}^{* }({n}^{* }+1){a}_{k}{a}_{{m}^{* }-1}{a}_{{n}^{* }+1}\\ \left.-4\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{k=0}^{m}{n}^{* }{a}_{k}{a}_{{m}^{* }-1}{b}_{{n}^{* }}\right),\\ {b}_{n+2}=\displaystyle \frac{1}{(n+1)(n+2){a}_{0}^{2}}\left((1-n){b}_{n}\right.\\ -\displaystyle \sum _{m=1}^{n}\displaystyle \sum _{k=0}^{m}{n}^{* }({n}^{* }+1){a}_{k}{a}_{{m}^{* }-1}{b}_{{n}^{* }+1}\\ \left.-2\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{k=0}^{m}{n}^{* }{m}^{* }{a}_{k}{a}_{{m}^{* }}{b}_{{n}^{* }}+\displaystyle \sum _{m=0}^{n}{n}^{* }{a}_{m}{a}_{{n}^{* }}\right).\end{array}\end{eqnarray}$
Now, we prove the convergence of the power series solution ( $\begin{eqnarray*}\begin{array}{l}| {a}_{n+2}| \leqslant P\left(| {a}_{n}| \right.\\ +\displaystyle \sum _{m=1}^{n}\displaystyle \sum _{k=0}^{m}| {a}_{k}| | {a}_{{m}^{* }-1}| | {a}_{{n}^{* }+1}| \\ \left.+\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{k=0}^{m}| {a}_{k}| | {a}_{{m}^{* }-1}| | {b}_{{n}^{* }}| \right),\\ | {b}_{n+2}| \leqslant P\left(| {b}_{n}| \right.\\ +\displaystyle \sum _{m=1}^{n}\displaystyle \sum _{k=0}^{m}| {a}_{k}| | {a}_{{m}^{* }-1}| | {b}_{{n}^{* }+1}| \\ +\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{k=0}^{m}| {a}_{k}| | {a}_{{m}^{* }}| | {b}_{{n}^{* }}| \\ \left.+\displaystyle \sum _{m=0}^{n}| {a}_{m}| | {a}_{{n}^{* }}| \right),\end{array}\end{eqnarray*}$
where $P=\tfrac{1}{| {a}_{0}^{2}| }$. Next, we define two new power series $\begin{eqnarray}\begin{array}{l}R=R(\omega )=\displaystyle \sum _{n=0}^{\infty }{r}_{n}{\omega }^{n},\\ S=S(\omega )=\displaystyle \sum _{n=0}^{\infty }{s}_{n}{\omega }^{n},\end{array}\end{eqnarray}$
with rn = ∣an∣, sn = ∣bn∣, n = 0, 1, 2 and $\begin{eqnarray*}\begin{array}{l}{r}_{n+2}=P\left({r}_{n}\right.\\ +\displaystyle \sum _{m=1}^{n}\displaystyle \sum _{k=0}^{m}{r}_{k}{r}_{{m}^{* }-1}{r}_{{n}^{* }+1}\\ \left.+\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{k=0}^{m}{r}_{k}{r}_{{m}^{* }-1}{s}_{{n}^{* }}\right),\\ {s}_{n+2}=P\left({s}_{n}\right.\\ +\displaystyle \sum _{m=1}^{n}\displaystyle \sum _{k=0}^{m}{r}_{k}{r}_{{m}^{* }-1}{s}_{{n}^{* }+1}\\ \left.+\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{k=0}^{m}{r}_{k}{r}_{{m}^{* }}{s}_{{n}^{* }}+\displaystyle \sum _{m=0}^{n}{r}_{m}{r}_{{n}^{* }}\right),\end{array}\end{eqnarray*}$
for n = 1, 2, ⋯ . We derive that $\begin{eqnarray*}\begin{array}{l}| {a}_{n}| \leqslant {r}_{n},\,\,\,\,| {b}_{n}| \leqslant {s}_{n},\\ n=0,1,2,\cdots .\end{array}\end{eqnarray*}$
Thus, R(ω) and S(ω) are majorant series of f(ω) and g(ω), respectively, and we obtain $\begin{eqnarray*}\begin{array}{c}R(\omega )={r}_{0}+{r}_{1}\omega +{r}_{2}{\omega }^{2}\\ +P(R-{r}_{0}){\omega }^{2}+{PR}({R}^{2}-{r}_{0}^{2})(R-{r}_{0}-{r}_{1}\omega )\\ +P({R}^{2}(S-{s}_{0})-{r}_{0}^{2}{s}_{1}\omega )\omega ,\\ S(\omega )={s}_{0}+{s}_{1}\omega +{s}_{2}{\omega }^{2}+P(S-{s}_{0}){\omega }^{2}\\ +{PR}({R}^{2}-{r}_{0}^{2})(S-{s}_{0}-{s}_{1}\omega )\\ +P(R(R-{r}_{0})(S-{S}_{0})-{r}_{0}{r}_{1}{s}_{0}{\omega }^{2})\omega \\ +P(R(R-{r}_{0})-{r}_{0}{r}_{1}\omega )\omega .\end{array}\end{eqnarray*}$
Now, we consider the following implicit functional system $\begin{eqnarray*}\begin{array}{c}{ \mathcal F }(\omega ,R,S)=R-{r}_{0}-{r}_{1}\omega -{r}_{2}{\omega }^{2}\\ -{PR}({R}^{2}-{r}_{0}^{2})(R-{r}_{0}-{r}_{1}\omega )\\ -P({R}^{2}(S-{s}_{0})-{r}_{0}^{2}{s}_{1}\omega )\omega ,\\ { \mathcal G }(\omega ,R,S)=S-{s}_{0}-{s}_{1}\omega -{s}_{2}{\omega }^{2}\\ -{PR}({R}^{2}-{r}_{0}^{2})(S-{s}_{0}-{s}_{1}\omega )\\ -P(S-{s}_{0}){\omega }^{2}\\ -P(R(R-{r}_{0})(S-{S}_{0})-{r}_{0}{r}_{1}{s}_{1}{\omega }^{2})\omega \\ -P(R(R-{r}_{0})-{r}_{0}{r}_{1}\omega )\omega .\end{array}\end{eqnarray*}$
Since ${ \mathcal F }$ and ${ \mathcal G }$ are analytic in the neighborhood of (0, r0, s0), ${ \mathcal F }(0,{r}_{0},{s}_{0})={ \mathcal G }(0,{r}_{0},{s}_{0})=0$, $\tfrac{\partial ({ \mathcal F },{ \mathcal G })}{\partial (R,S)}{| }_{(0,{r}_{0},{s}_{0})}=1\ne 0,$ according to the implicit function theorem, the convergence of power series (Therefore, the exact power series solution for system (24 ) reads as
$\begin{eqnarray*}\begin{array}{c}u(y,t)=\displaystyle \frac{{a}_{0}}{t}+\displaystyle \frac{{a}_{1}y}{\sqrt{t}}-\left(\displaystyle \frac{1}{{a}_{0}}+2{b}_{1}\right){y}^{2}\\ +\displaystyle \sum _{n=1}^{\infty }\displaystyle \frac{1}{\left(n+1)(n+2){a}_{0}^{2}t\right.}\left(\left(n-2\right){a}_{n}\right.\\ -\displaystyle \sum _{m=1}^{n}\displaystyle \sum _{k=0}^{m}(n-m+1)(n-m+2){a}_{k}{a}_{m-k}{a}_{n-m+2}\\ \left.-4\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{k=0}^{m}(n-m+1){a}_{k}{a}_{m-k}{b}_{n-m+1}\right){\left(y\sqrt{t}\right)}^{n+2},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{c}v(y,t)=\displaystyle \frac{{b}_{0}}{\sqrt{t}}+{b}_{1}y\\ +\left(\displaystyle \frac{{a}_{1}}{2{a}_{0}}-\displaystyle \frac{{a}_{1}{b}_{1}}{{a}_{0}}+\displaystyle \frac{{b}_{0}}{2{a}_{0}^{2}}\right){y}^{2}\sqrt{t}\\ +\displaystyle \sum _{n=1}^{\infty }\displaystyle \frac{1}{\left(n+1)(n+2){a}_{0}^{2}\sqrt{t}\right.}\left(\left(1-n\right){b}_{n}\right.\\ -\displaystyle \sum _{m=1}^{n}\displaystyle \sum _{k=0}^{m}(n-m+1)\\ \times (n-m+2){a}_{k}{a}_{m-k}{b}_{n-m+2}\\ -2\displaystyle \sum _{m=0}^{n}\displaystyle \sum _{k=0}^{m}(m-k+1)\\ \times (n-m+1){a}_{k}{a}_{m-k+1}{b}_{n-m+1}\\ \left.+\displaystyle \sum _{m=0}^{n}(n-m+1){a}_{m}{a}_{n-m+1}\right){\left(y\sqrt{t}\right)}^{n+2}.\end{array}\end{eqnarray*}$
5. Conclusion
In this paper, the deformation operators of CBB system (1 ) were constructed using three conservation laws of system (1 ) satisfying the deformation theorem; then, the (4+1)-dimensional system (2 ) was obtained. Moreover, following the Lax pair and higher-order flow system of the former low-dimensional system (1 ), the Lax integrability and symmetry integrability of the (4+1)-dimensional CBB system (2 ) were also established. We also obtained a large number of reduced integrable systems of (4+1)-dimensional system (2 ) along the coordinate axes, which included three (1+1)-dimensional systems (7 ), (10 ) and (12 ), six (2+1)-dimensional systems (14 )–(19 ), and four (3+1)-dimensional systems (20 )–(23 ). Three kinds of different deformation operators were applied to the reduced (1+1)-dimensional systems (7 ), (10 ) and (12 ), respectively, and derived the corresponding (4+1)-dimensional systems (9 ), (11 ) and (13 ), which are equivalent systems of system (2 ). For illustration, we constructed the exact solution of a (1+1)-dimensional system using Lie symmetry analysis and the power series method.