1 Introduction

As yet, for the sake of studying the water waves (including the soliton considerations) [1,2,3,4,5,6,7,8,9,10,11,12,13,14], people have used such systems/equations as a variable-coefficient nonlinear dispersive-wave system describing the long gravity water waves in a shallow oceanic environment [9], a variable-coefficient generalized dispersive water-wave system describing the long weakly-nonlinear and weakly-dispersive surface waves of variable depth in the shallow water [10, 11], a generalized (2+1)-dimensional dispersive long-wave system describing the nonlinear and dispersive long gravity waves in two horizontal directions on the shallow water of an open sea or a wide channel of finite depth [12], types of the nonlocal Boussinesq equations in the water waves [15, 16], an extended Kadomtsev-Petviashvili equation in a fluid [17], a (3+1)-dimensional Kadomtsev-Petviashvili equation [18], (2+1)- and (3+1)-dimensional extended shallow water wave equations [19], (2+1)- and (3+1)-dimensional shallow water wave equations [20], shallow water equations from the generalized Camassa-Holm framework [21], coupled Ramani and Nizhnik-Novikov-Veselov systems [22] and nonlinear time fractional partial differential equations [23]. Other relevant systems and/or models in fluid mechanics have been reported, e.g., in Refs. [24,25,26,27,28].

Another example, which we purpose to investigate, is a (2+1)-dimensional generalized modified dispersive water-wave system describing the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth, i.e.,

$$\begin{aligned}&u_{yt}+\alpha u_{xxy}-2 \alpha v_{xx} -\beta u u_{xy} -\beta u_{x} u_{y} =0 ,\end{aligned}$$
(1a)
$$\begin{aligned}&v_{t}- \alpha v_{xx} -\beta \left( u v\right) _{x}=0 , \end{aligned}$$
(1b)

with u(x, y, t) meaning the height of the water surface, v(x, y, t) indicating the horizontal velocity of the water wave, both u(x, y, t) and v(x, y, t) being the real differentiable functions, \(\alpha \) and \(\beta \) implying the real non-zero constants, while the subscripts being the partial derivatives as for the scaled space variables x, y and time variable t.

There have existed some special cases of System (1), as follows:

  • when \(\alpha =-1\) and \(\beta =-2\), describing the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth, a (2+1)-dimensional Broer-Kaup-Kupershmidt system [29,30,31], i.e.,

    $$\begin{aligned}&u_{yt}- u_{xxy}+2 v_{xx} +2 u u_{xy} +2 u_{x} u_{y} =0 ,\end{aligned}$$
    (2a)
    $$\begin{aligned}&v_{t}+ v_{xx} +2 \left( u v\right) _{x}=0 , \end{aligned}$$
    (2b)

    with u(x, y, t) meaning the height of the water surface, v(x, y, t) indicating the horizontal velocity of the water wave, while x, y and t being the scaled space variables and time variable, separately [29];

  • when \(\alpha =1\) and \(\beta =2\), describing the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth, a (2+1)-dimensional modified dispersive water-wave system [32], i.e.,

    $$\begin{aligned}&u_{yt}+ u_{xxy}-2 v_{xx} -2 u u_{xy} -2 u_{x} u_{y} =0 ,\end{aligned}$$
    (3a)
    $$\begin{aligned}&v_{t}- v_{xx} -2 \left( u v\right) _{x}=0\;\; ; \end{aligned}$$
    (3b)

  • when \(\alpha =\frac{1}{2}\), \(\beta =1\) and \(y=x\), describing the long waves in the shallow water, a (1+1)-dimensional Broer-Kaup system [33, 34], i.e.,

    $$\begin{aligned}&u_{t}+\frac{1}{2} u_{xx}- v_{x} - u u_{x}=0 , \end{aligned}$$
    (4a)
    $$\begin{aligned}&v_{t}-\frac{1}{2} v_{xx} - \left( u v\right) _{x}=0 , \end{aligned}$$
    (4b)

    with u(x, t) standing for the scaled wave horizontal velocity, while v(x, t) related to the wave horizontal velocity and wave height [34];

  • when \(\alpha =-1\), \(\beta =-2\) and \(y=x\), describing the long waves in the shallow water, a (1+1)-dimensional Broer-Kaup-Kupershmidt system [29, 35], i.e.,

    $$\begin{aligned}&u_{t}- u_{xx}+2 v_{x} +2 u u_{x}=0 , \end{aligned}$$
    (5a)
    $$\begin{aligned}&v_{t}+ v_{xx} +2 \left( u v\right) _{x}=0 \;\; . \end{aligned}$$
    (5b)

Using symbolic computation [36,37,38,39,40,41,42,43], we aim to construct out a set of the scaling transformations, a set of the hetero-Bäcklund transformations as well as four sets of the similarity reductions for System (1). By the way, more symbolic-computation results can be seen, e.g., in Refs. [44,45,46,47,48,49,50,51,52,53,54].

2 Scaling and Hetero-Bäcklund Transformations for System (1)

Scaling transformations can help us find certain assumptions, so as to make us construct, e.g., some hetero-Bäcklund transformations [55, 56] or bilinear forms [9, 56, 57].

We work out a set of the scaling transformations

$$\begin{aligned} x \rightarrow \rho ^{1} x , \qquad y \rightarrow \rho ^{\xi } x , \qquad t \rightarrow \rho ^{2} t , \qquad u \rightarrow \rho ^{-1} u , \qquad v \rightarrow \rho ^{-1-\xi } v , \end{aligned}$$
(6)

and then make the assumptions that

$$\begin{aligned} u(x,y,t)=\eta _1 w_{x}(x,y,t)+\eta _2 ,\qquad v(x,y,t)=\eta _3 w_{xy}(x,y,t) , \end{aligned}$$
(7)

with \(\rho >0\) standing for a positive constant, \(\xi \) implying an integer, while \(\eta _1 \ne 0\), \(\eta _2\) and \(\eta _3 \ne 0\) indicating three real constants.

We employ symbolic computation and Assumptions (7), integrate Eq. (1b) once in relation to x and y, separately, with the integration functions vanishing and decide on

$$\begin{aligned} \alpha =\frac{1}{2} \beta \eta _1, \end{aligned}$$
(8)

to get the following Bell-polynomial expression:

$$\begin{aligned} Y_{t}(w)-\alpha Y_{2x}(w)-\beta \eta _2 Y_{x}(w)=0 , \end{aligned}$$
(9)

with the Bell polynomials defined as [58, 59]

$$\begin{aligned}&Y_{mx,ry,nt}(w)\equiv Y_{m,r,n}(w_{0,0,0},\cdots ,w_{0,0,n},\cdots ,w_{0,r,0},\cdots ,w_{0,r,n},\cdots ,\\&\qquad \qquad \qquad \qquad w_{m,r,0},\cdots ,w_{m,r,n}) =e^{-w}\partial ^{m}_{x}\partial ^{r}_{y}\partial ^{n}_{t}e^w, \end{aligned}$$

w(x, y, t) as a \(C^{\infty }\) function with respect of x, y and t, \(w_{k,g,l}=\partial ^{k}_{x}\partial ^{g}_{y}\partial ^{l}_{t} w\) (\(k=0,\cdots ,m,\,g=0,\cdots ,r,\,l=0,\cdots ,n\)), while m, r and n as three non-negative integers.

Similarly, making use of symbolic computation and Assumptions (7), integrating Eq. (1a) once in relation to x and y,  separately, with the integration functions vanishing and choosing that

$$\begin{aligned} \alpha \left( \eta _1-2 \eta _3\right) =-\frac{1}{2} \beta \eta _1^2 , \end{aligned}$$
(11)

help us find a Bell-polynomial expression, i.e.,

$$\begin{aligned} Y_{t}(w)-\alpha Y_{2x}(w)-\beta \eta _2 Y_{x}(w)=0, \end{aligned}$$
(12)

which is the same as Bell-Polynomial Expression (9).

Further, System (1) with the assumption

$$\begin{aligned} w(x,y,t)= \ln \left[ h(x,y,t)\right] , \end{aligned}$$
(13)

develops into

$$\begin{aligned} h_{t}(x,y,t)-\alpha \, h_{xx}(x,y,t)-\beta \eta _2 \, h_{x}(x,y,t)=0, \end{aligned}$$
(14)

in which h(x, y, t) means a positive differentiable function.

Taking into consideration all the above, with symbolic computation, we end up with the following set of the hetero-Bäcklund transformations for System (1):

$$\begin{aligned}&u(x,y,t)=\frac{2 \alpha }{\beta } \, \frac{h_{x}(x,y,t)}{h(x,y,t)}+\eta _2,\end{aligned}$$
(15a)
$$\begin{aligned}&v(x,y,t)=\frac{2 \alpha }{\beta } \left[ \frac{h_{xy}(x,y,t)}{h(x,y,t)} -\frac{h_{x}(x,y,t)}{h(x,y,t)} \frac{h_{y}(x,y,t)}{h(x,y,t)}\right] ,\end{aligned}$$
(15b)
$$\begin{aligned}&h_{t}(x,y,t)-\alpha \, h_{xx}(x,y,t)-\beta \eta _2 \, h_{x}(x,y,t)=0\;\; . \end{aligned}$$
(15c)

Eq. (15c) denotes a known linear partial differential equation, whose information has been reported [60, 61]. Moreover, with symbolic computation, we hereby present the following sample solutions for Eq. (15c):

$$\begin{aligned} h(x,y,t)=1+e^{\zeta _{1}x+\zeta _{2}(y)+(\alpha \zeta _{1}^2 +\beta \eta _{2}\zeta _{1})t}, \end{aligned}$$

where \(\zeta _{1}\) is a real non-zero constant and \(\zeta _{2}(y)\) is a real differentiable function of y.

Explanation with the relevant physics: Eqs. (15) stand for a set of the hetero-Bäcklund transformations, which can link the solutions h(x, y, t) of Eq. (15c) and the solutions u(x, y, t) and v(x, y, t) of System (1). As for the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth, with respect to u(x, y, t), the height of the water surface, and v(x, y, t), the horizontal velocity of the water wave, Hetero-Bäcklund Transformations (15) rely on \(\alpha \) and \(\beta \), the coefficients for System (1).

3 Four Sets of the Similarity Reductions for System (1)

Choosing the assumptions

$$\begin{aligned}&u(x,y,t)=\theta (x,y,t)+\omega (x,y,t) p[z(x,y,t)] , \end{aligned}$$
(16a)
$$\begin{aligned}&v(x,y,t)=\gamma (x,y,t)+\kappa (x,y,t) q[z(x,y,t)] , \end{aligned}$$
(16b)

which are similar to those in Refs. [62,63,64,65,66,67], thinking about the case of \(z_{x} \ne 0\) and \(z_{y} \ne 0\), and then, with symbolic computation, substituting Assumptions (16) into System (1), we find

$$\begin{aligned}&\delta _{0} p'''+\delta _{1} \left( p p''+p'^2\right) +\delta _{2} p p'+\delta _{3} p^2 +\delta _{4} q''+\delta _{5} q'+\delta _{6} q+\delta _{7} p''\nonumber \\&\quad \quad \quad +\delta _{8} p' +\delta _{9} p+\delta _{10}=0 , \end{aligned}$$
(17a)
$$\begin{aligned}&\psi _{0} q''+\psi _{1} \left( p q'+p' q\right) +\psi _{2} p q+\psi _{3} q'+\psi _{4} p' +\psi _{5} q+\psi _{6} p+\psi _{7}=0 , \end{aligned}$$
(17b)

in which

$$\begin{aligned}&\delta _{0}=\alpha \omega z_{x}^2 z_{y} , \end{aligned}$$
(18a)
$$\begin{aligned}&\delta _{1}=-\beta \omega ^2 z_{x} z_{y} , \end{aligned}$$
(18b)
$$\begin{aligned}&\delta _{2}=-\beta \omega \left( 2 \omega _{y} z_{x}+2 \omega _{x} z_{y} +\omega z_{xy}\right) , \end{aligned}$$
(18c)
$$\begin{aligned}&\delta _{3}=-\beta \left( \omega _{x} \omega _{y}+\omega \omega _{xy}\right) , \end{aligned}$$
(18d)
$$\begin{aligned}&\delta _{4}=-2 \alpha \kappa z_{x}^2 , \end{aligned}$$
(18e)
$$\begin{aligned}&\delta _{5}=-2 \alpha \left( 2 \kappa _{x} z_{x}+\kappa z_{xx}\right) , \end{aligned}$$
(18f)
$$\begin{aligned}&\delta _{6}=-2 \alpha \kappa _{xx} , \end{aligned}$$
(18g)
$$\begin{aligned}&\delta _{7}=\alpha z_{x} \left( \omega _{y} z_{x}+2 \omega _{x} z_{y}\right) +\omega \left( z_{y} z_{t}-\beta \theta z_{x} z_{y} +2 \alpha z_{x} z_{xy}+\alpha z_{y} z_{xx}\right) , \end{aligned}$$
(18h)
$$\begin{aligned}&\delta _{8}=\left( \omega _{t} z_{y}+\omega _{y} z_{t}+\omega z_{yt}\right) -\beta \left( \theta _{y} \omega z_{x}+\theta \omega _{y} z_{x} +\theta _{x} \omega z_{y}+\theta \omega _{x} z_{y}+\theta \omega z_{xy}\right) \nonumber \\&\quad \qquad +\alpha \left( 2 \omega _{x} z_{xy}+2 \omega _{xy} z_{x}+\omega _{y} z_{xx} +\omega _{xx} z_{y}+\omega z_{xxy}\right) , \end{aligned}$$
(18i)
$$\begin{aligned}&\delta _{9}=\omega _{yt}-\beta \left( \theta _{x} \omega _{y} +\theta _{y} \omega _{x}+\theta _{xy} \omega +\theta \omega _{xy}\right) +\alpha \omega _{xxy} , \end{aligned}$$
(18j)
$$\begin{aligned}&\delta _{10}=\theta _{yt}-\beta \left( \theta _{x} \theta _{y} +\theta \theta _{xy}\right) -2 \alpha \gamma _{xx}+\alpha \theta _{xxy} , \end{aligned}$$
(18k)
$$\begin{aligned}&\psi _{0}=-\alpha \kappa z_{x}^2 , \end{aligned}$$
(18l)
$$\begin{aligned}&\psi _{1}=-\beta \omega \kappa z_{x} , \end{aligned}$$
(18m)
$$\begin{aligned}&\psi _{2}=-\beta \left( \omega \kappa _{x}+\omega _{x} \kappa \right) , \end{aligned}$$
(18n)
$$\begin{aligned}&\psi _{3}=-2 \alpha \kappa _{x} z_{x}+\kappa \left( z_{t}-\beta \theta z_{x} -\alpha z_{xx}\right) , \end{aligned}$$
(18o)
$$\begin{aligned}&\psi _{4}=-\beta \gamma \omega z_{x} , \end{aligned}$$
(18p)
$$\begin{aligned}&\psi _{5}=\kappa _{t}-\beta \left( \theta \kappa _{x}+\theta _{x} \kappa \right) -\alpha \kappa _{xx} , \end{aligned}$$
(18q)
$$\begin{aligned}&\psi _{6}=-\beta \left( \gamma _{x} \omega +\gamma \omega _{x}\right) , \end{aligned}$$
(18r)
$$\begin{aligned}&\psi _{7}=\gamma _{t}-\beta \left( \theta \gamma _{x}+\theta _{x} \gamma \right) -\alpha \gamma _{xx} , \end{aligned}$$
(18s)

with \(\theta (x,y,t)\), \(\omega (x,y,t) \ne 0\), \(\gamma (x,y,t)\), \(\kappa (x,y,t) \ne 0\) and \(z(x,y,t) \ne 0\) as the real to-be-determined differentiable functions, p(z) and q(z) as the real differentiable functions, while the prime sign as d/dz.

Seeing that Eqs. (17) with Expressions (18) stand for a set of the ordinary differential equations (ODEs) as for p(z) and q(z), we require the ratios of different derivatives and powers of p(z) and q(z) to represent the functions of z only, i.e.,

$$\begin{aligned} \delta _{i}=\Omega _{i}(z) \delta _{0} , \qquad \psi _{j}=\Gamma _{j}(z) \psi _{0} , \end{aligned}$$
(19)

with \(\Omega _{i}(z)\)’s (\(i=0,...,10\)) and \(\Gamma _{j}(z)\)’s (\(j=0,...,7\)) as some real to-be-determined functions of z only. Hence, a set of the conditions for \(\theta (x,y,t)\), \(\omega (x,y,t) \ne 0\), \(\gamma (x,y,t)\), \(\kappa (x,y,t) \ne 0\) and \(z(x,y,t) \ne 0\) are built up, for which any set of the solutions could develop into, at least, a similarity reduction.

On account of the second freedom in Remark 3 in Ref. [62], Eqs. (19) with \(i=1,4\) and \(j=1\) come to

$$\begin{aligned}&\omega (x,y,t)=\pm \frac{\alpha }{\beta } z_{x}, \qquad \kappa (x,y,t)=\mp \frac{\alpha }{2 \beta } z_{x} z_{y}, \qquad \end{aligned}$$
(20a)
$$\begin{aligned}&\Omega _{1}(z)=\mp 1, \qquad \Omega _{4}(z)=1, \qquad \Gamma _{1}(z)=\pm 1 \;\;. \end{aligned}$$
(20b)

Since the first freedom in Remark 3 in Ref. [62] helps us transform Eqs. (19) with \(i=2\) into

$$\begin{aligned} z(x,y,t)=\lambda _1 x+\lambda _2 y+\lambda _3 t+\lambda _4 , \qquad \Omega _{2}(z)=0 , \end{aligned}$$
(21)

Eqs. (19) with \(i=3,5,6\) and \(j=2\) bring about

$$\begin{aligned} \Omega _{3}(z)=\Omega _{5}(z)=\Omega _{6}(z)=\Gamma _{2}(z)=0 , \end{aligned}$$
(22)

with \(\lambda _1 \ne 0\), \(\lambda _2 \ne 0\), \(\lambda _3 \ne 0\) and \(\lambda _4\) denoting the real constants.

Because the first freedom in Remark 3 in Ref. [62] makes us simplify Eqs. (19) with \(j=4\) to

$$\begin{aligned} \gamma (x,y,t)=0 , \qquad \Gamma _{4}(z)=0 , \end{aligned}$$
(23)

Eqs. (19) with \(j=6,7\) indicate

$$\begin{aligned} \Gamma _{6}(z)=\Gamma _{7}(z)=0 \;. \end{aligned}$$
(24)

Until now, with symbolic computation, there exist two choices of \(\theta (x,y,t)\) making \(\Omega _{7}(z)\) represent a constant:

Choice 1:

According to the first freedom in Remark 3 in Ref. [62], Eqs. (19) with \(j=7\) make for

$$\begin{aligned} \theta (x,y,t)=\frac{\lambda _3}{\beta \lambda _1} , \qquad \Omega _{7}(z)=0 , \end{aligned}$$
(25)

as a result that Eqs. (19) with \(i=8,9,10\) and \(j=3,5\) can be simplified into

$$\begin{aligned} \Omega _{8}(z)=\Omega _{9}(z)=\Omega _{10}(z)=\Gamma _{3}(z)=\Gamma _{5}(z)=0 \;. \end{aligned}$$
(26)

So far, System (1) could be transformed into the following ODEs:

$$\begin{aligned}&p''' \mp \left( p p''+p'^2\right) +q''=0 , \end{aligned}$$
(27a)
$$\begin{aligned}&q'' \pm \left( p' q+p q'\right) =0 \;\;. \end{aligned}$$
(27b)

Thus, simplifying each of two sets of ODEs (27) into an ODE makes us find

$$\begin{aligned} q=-p' \pm \frac{1}{2} p^2+\phi _1 z+\phi _2, \end{aligned}$$
(28)

and considering ODEs (27)-(28) helps us obtain

$$\begin{aligned} p''-\frac{1}{2} p^3 \mp \left( \phi _1 z+\phi _2\right) p +\left( \phi _3-\phi _1\right) =0, \end{aligned}$$
(29)

with \(\phi _1\), \(\phi _2\) and \(\phi _3\) as the real constants of integration.

With symbolic computation, we end up with two sets of the similarity reductions for System (1), i.e.,

$$\begin{aligned}&u(x,y,t)=\frac{\lambda _3}{\beta \lambda _1} \pm \frac{\alpha }{\beta } \lambda _1 p[z(x,y,t)] , \end{aligned}$$
(30a)
$$\begin{aligned}&v(x,y,t)=\mp \frac{\alpha }{2 \beta } \lambda _1 \lambda _2 \left\{ -p'[z(x,y,t)] \pm \frac{1}{2} p[z(x,y,t)]^2+\phi _1 z+\phi _2\right\} , \end{aligned}$$
(30b)
$$\begin{aligned}&z(x,y,t)=\lambda _1 x+\lambda _2 y+\lambda _3 t+\lambda _4 , \end{aligned}$$
(30c)
$$\begin{aligned}&p''-\frac{1}{2} p^3 \mp \left( \phi _1 z+\phi _2\right) p +\left( \phi _3-\phi _1\right) =0 \;\;. \end{aligned}$$
(30d)

ODEs (30d) stand for two known ODEs, each of which has been investigated in Refs. [68, 69]. Right now, with symbolic computation, choosing \(\phi _1=\phi _2=\phi _3=0\), we can get the following sample solutions for ODEs (30d):

$$\begin{aligned} p(z)=\pm \frac{2}{z+\zeta _{3}}, \end{aligned}$$

where \(\zeta _{3}\) is a real constant.

Explanation with the relevant physics: As for the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth, with respect to u(x, y, t), the height of the water surface, and v(x, y, t), the horizontal velocity of the water wave, Similarity Reductions (30) rely on \(\alpha \) and \(\beta \), the coefficients for System (1). Two sets of Similarity Reductions (30) appear, as a result of the existence of the “±” signs.

Choice 2:

Based on the second freedom in Remark 3 in Ref. [62], Eqs. (19) with \(j=7\) give rise to

$$\begin{aligned} \theta (x,y,t)=\frac{\lambda _3-\alpha \lambda _1^2}{\beta \lambda _1} , \qquad \Omega _{7}(z)=1 , \end{aligned}$$
(31)

so that Eqs. (19) with \(i=8,9,10\) and \(j=3,5\) can be transformed into

$$\begin{aligned} \Omega _{8}(z)=\Omega _{9}(z)=\Omega _{10}(z)=\Gamma _{5}(z)=0 , \qquad \Gamma _{3}(z)=-1 \;. \end{aligned}$$
(32)

Until now, System (1) could be simplified to the following ODEs:

$$\begin{aligned}&p''' \mp \left( p p''+p'^2\right) +q''+p''=0 , \end{aligned}$$
(33a)
$$\begin{aligned}&q'' \pm \left( p' q+p q'\right) -q'=0 \;\;. \end{aligned}$$
(33b)

Therefore, simplifying each of two sets of ODEs (33) into an ODE helps us work out

$$\begin{aligned} q=-p' \pm \frac{1}{2} p^2-p+\phi _4 z+\phi _5, \end{aligned}$$
(34)

and taking into consideration ODEs (33)-(34) makes us discover

$$\begin{aligned} p''-\frac{1}{2} p^3 \pm \frac{3}{2} p^2 \mp \left( \phi _4 z+\phi _5 \pm 1\right) p +\left( \phi _4 z-\phi _4+\phi _5+\phi _6\right) =0, \end{aligned}$$
(35)

with \(\phi _4\), \(\phi _5\) and \(\phi _6\) as the real constants of integration.

With symbolic computation, we conclude with two more sets of the similarity reductions for System (1), i.e.,

$$\begin{aligned}&u(x,y,t)=\frac{\lambda _3-\alpha \lambda _1^2}{\beta \lambda _1} \pm \frac{\alpha }{\beta } \lambda _1 p[z(x,y,t)] , \end{aligned}$$
(36a)
$$\begin{aligned}&v(x,y,t)=\mp \frac{\alpha }{2 \beta } \lambda _1 \lambda _2 \left\{ -p'[z(x,y,t)] \pm \frac{1}{2} p[z(x,y,t)]^2-p[z(x,y,t)]+\phi _4 z+\phi _5\right\} , \end{aligned}$$
(36b)
$$\begin{aligned}&z(x,y,t)=\lambda _1 x+\lambda _2 y+\lambda _3 t+\lambda _4 , \end{aligned}$$
(36c)
$$\begin{aligned}&p''-\frac{1}{2} p^3 \pm \frac{3}{2} p^2 \mp \left( \phi _4 z+\phi _5 \pm 1\right) p +\left( \phi _4 z-\phi _4+\phi _5+\phi _6\right) =0 \;\;. \end{aligned}$$
(36d)

ODEs (36d) indicate two known ODEs, each of which has been investigated in Refs. [68, 69]. This time, with symbolic computation, selecting \(\phi _4=\phi _6=0\) and \(\phi _5=\pm \frac{1}{2}\), we are able to obtain the following sample solutions for ODEs (36d):

$$\begin{aligned} p(z)=\frac{2}{z+\zeta _{4}} \pm 1 \qquad \textrm{or} \qquad p(z)=-\frac{2}{z+\zeta _{5}} \pm 1, \end{aligned}$$

where both \(\zeta _{4}\) and \(\zeta _{5}\) are the real constants.

Explanation with the relevant physics: As for the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth, with respect to u(x, y, t), the height of the water surface, and v(x, y, t), the horizontal velocity of the water wave, Similarity Reductions (36) rely on \(\alpha \) and \(\beta \), the coefficients for System (1). Two sets of Similarity Reductions (36) appear, as a result of the existence of the “±” signs.

4 Conclusions

To sum up, we have briefly reviewed the recent developments on the shallow water and soliton consideration with analytic solutions, and with symbolic computation, studied System (1), i.e., a (2+1)-dimensional generalized modified dispersive water-wave system describing the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth. With respect to u(x, y, t), the height of the water surface, and v(x, y, t), the horizontal velocity of the water wave, we have symbolically computed out Scaling Transformations (6), Hetero-Bäcklund Transformations (15), from System (1) to Eq. (15c), Similarity Reductions (30), from System (1) to ODEs (30d), and Similarity Reductions (36), from System (1) to ODEs (36d). We have paid attention that Eq. (15c) stands for a known linear partial differential equation, while each of ODEs (30d) and ODEs (36d), a known ODE. Hetero-Bäcklund Transformations (15), Similarity Reductions (30) and Similarity Reductions (36) have been obtained to rely on \(\alpha \) and \(\beta \), the coefficients for System (1).

We hope that Hetero-Bäcklund Transformations (15), Similarity Reductions (30) and Similarity Reductions (36), with their sample solutions and other solutions, could help the future shallow-water studies. Information about the future work for System (1) includes the Darboux transformation, inverse scattering transformation, etc.