Considering the Shallow Water of a Wide Channel or an Open Sea Through a Generalized (2+1)-dimensional Dispersive Long-wave System

Abstract

Under investigation in this paper is a generalized (2+1)-dimensional dispersive long-wave system, describing the nonlinear and dispersive long gravity waves in two horizontal directions in the shallow water of a wide channel of finite depth or an open sea. Via symbolic computation, we derive the same bilinear forms as those reported, but through a different method. Four sets of the similarity reductions are obtained, each of which leads to a known ordinary differential equation. The results rely on the coefficients in the original system, with respect to the horizontal velocity and wave elevation above the undisturbed water surface.

1 Introduction

Studies on fluids have been reported [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. For investigating the nonlinear and dispersive long gravity waves in two horizontal directions, especially those in the shallow water of a wide channel or an open sea with finite depth, Ref. [15] has proposed the following generalized (2+1)-dimensional dispersive long-wave system:

$$\begin{aligned}&u_{yt}+\alpha \left[ \, v_{xx}+\frac{1}{2} \left( u^2\right) _{xy} \,\right] =0 \;\; , \end{aligned}$$

(1a)

$$\begin{aligned}&v_{t}+\alpha \left( \, u v+\beta u+\delta ^2 u_{xy}\, \right) _{x}=0 \;\;, \end{aligned}$$

(1b)

with u(x, y, t) as the horizontal velocity, v(x, y, t) as the wave elevation above the undisturbed water surface, u(x, y, t) and v(x, y, t) as the real differentiable functions in respect of the variables x, y and t, the subscripts as the partial derivatives, $\alpha \ne 0$, $\beta$ and $\delta \ne 0$ implying the real constants, while t and (x, y) denoting the time and propagation plane, separately. Also in Ref. [15], some special cases which can report the applications of System (1) have been listed.

Ref. [15] has derived two sets of the bilinear forms of System (1), i.e.,

$$\begin{aligned}&( D_t \pm \alpha \delta D_{x}^{2} ) f \cdot g=0 , \end{aligned}$$

(2a)

$$\begin{aligned}&\left[ D_y D_t \pm \alpha \delta D_{x}^{2} D_y \pm \frac{\alpha }{\delta } (\theta _{4} +\beta ) D_x \right] f \cdot g =0 , \end{aligned}$$

(2b)

in which $\theta _4$ indicates a real constant, f(x, y, t) and g(x, y, t) imply the $C^{\infty }$ functions of x, y and t, while $D_{x}$, $D_{y}$ and $D_{t}$ represent the Hirota operators defined as [16]

$$\begin{aligned}&D_{x}^{m}D_{y}^{r}D_{t}^{n} f(x,y,t)\cdot g(x,y,t) \\&\quad \equiv \left( \frac{\partial }{\partial x}-\frac{\partial }{\partial x'}\right) ^{m}\,\left( \frac{\partial }{\partial y}-\frac{\partial }{\partial y'}\right) ^{r}\, \left( \frac{\partial }{\partial t}-\frac{\partial }{\partial t'}\right) ^{n}\,f(x,y,t)\,g(x',y',t') \bigg |_{x'=x,\,y'=y,\,t'=t}\;\;, \end{aligned}$$

with $x'$, $y'$ and $t'$ denoting the formal variables, while m, r and n meaning three non-negative integers. Besides, Ref. [15] has also obtained certain scaling transformations, hetero-Bäcklund transformations and N-soliton solutions for System (1), where N is a positive integer. For System (1), Ref. [17] has constructed certain hetero- and auto-Bäcklund transformations with some soliton solutions, while Ref. [18] has given out some similarity reductions.¹

To System (1), contributions of this paper could be introduced in the following aspects:

• Background: Nowadays, many nonlinear evolution equations/systems have been put into use in some physical studies, e.g., optical fibers, fluids and plasmas [17,18,19,20,21,22,23,24,25,26,27,28].

• Motivations: On the one hand, we plan to construct the same bilinear forms as Bilinear Forms (2) with a different method, to confirm the correctness of Bilinear Forms (2). On the other hand, we would like to find out more similarity reductions, which link System (1) to some ordinary differential equations (ODEs), to complement the existing results.

• Novelty and outlines: Bäcklund transformations and solutions of System (1) could be derived via the bilinear forms [15]. In comparison with the Bell polynomials in Ref. [15], the Hirota method may give rise to more potential bilinear forms [29]. Besides, similarity reductions in this paper, which are different from those in Ref. [18], might fit some other situations.

• Originality: To date, for System (1), similarity reductions different from those in Ref. [18] have not been investigated. In Sect. 2, we will derive two sets of the bilinear forms, which are the same as those in Ref. [15], but through a different method, i.e., the Hirota method [16, 30,31,32,33]. In Sect. 3, with symbolic computation² [34,35,36,37,38], we will obtain four sets of the similarity reductions for System (1), which are different from those in Ref. [18]. Conclusions will be given in Sect. 4.

• Significance and potential applications: This paper could be of some use for the future studies on the nonlinear and dispersive long gravity waves in two horizontal directions, especially those in the shallow water of a wide channel or an open sea with finite depth.

2 Two Sets of the Bilinear Forms for System (1) through the Hirota Method

Since our goal is to construct some bilinear forms for System (1) in respect of f(x, y, t) and g(x, y, t), the Hirota method brings about the assumptions

$$\begin{aligned}&u(x,y,t)=\zeta _1\left[ \ln \left( \frac{f}{g} \right) \right] _x\;\;, \end{aligned}$$

(3a)

$$\begin{aligned}&v(x,y,t) =\zeta _2\left[ \ln \left( \frac{f}{g} \right) \right] _{xy} +\zeta _3 [\ln \left( fg \right) ]_{xy}+\zeta _4\;\;, \end{aligned}$$

(3b)

where $\zeta _2$ and $\zeta _4$ are two real constants, while $\zeta _1$ and $\zeta _3$ imply two real non-zero constants.

Integrating Eq. (1a) once in respect of x and y, respectively, with the integration function vanishing, we get

$$\begin{aligned} \zeta _1\left[ \ln \left( \frac{f}{g} \right) \right] _{t}+ \alpha \zeta _2 \left[ \ln \left( \frac{f}{g} \right) \right] _{xx}+ \alpha \zeta _3 \left[ \ln \left( fg \right) \right] _{xx} + \frac{1}{2} \alpha \zeta _{1}^{2} \left\{ \left[ \ln \left( \frac{f}{g} \right) \right] _{x}\right\} ^{2} = 0 \;\;.\nonumber \\ \end{aligned}$$

(4)

To bring in the Hirota operators, based on the following formulae  [16]:

$$\begin{aligned}&\left[ \ln \left( \frac{f}{g} \right) \right] _x=\frac{D_x{f\cdot g}}{fg} \;\;, \end{aligned}$$

(5a)

$$\begin{aligned}&\left[ \ln \left( \frac{f}{g} \right) \right] _t=\frac{D_t{f\cdot g}}{fg} \;\;, \end{aligned}$$

(5b)

$$\begin{aligned}&\left[ \ln \left( fg \right) \right] _{xx}=\frac{D_{x}^{2}{f\cdot g}}{fg}-\left( \frac{D_x{f\cdot g}}{fg}\right) ^{2} \;\;, \end{aligned}$$

(5c)

with the assumption that

$$\begin{aligned} \zeta _2=0\;\;, \qquad \zeta _3=\frac{1}{2} \zeta _{1}^{2} \;\;, \end{aligned}$$

(6)

we convert Eq. (1a) into

$$\begin{aligned} \left( D_t+\frac{1}{2} \alpha \zeta _1 D_{x}^{2} \right) {f \cdot g}=0 \;\;. \end{aligned}$$

(7)

Similarly, we integrate Eq. (1b) once in respect of x with the integration function vanishing, to find

$$\begin{aligned}&\zeta _2 \left[ \ln \left( \frac{f}{g} \right) \right] _{yt}+\zeta _3 \left[ \ln \left( fg \right) \right] _{yt}+\alpha \zeta _1 \zeta _2 \left[ \ln \left( \frac{f}{g} \right) \right] _{x} \left[ \ln \left( \frac{f}{g} \right) \right] _{xy}\nonumber \\&\quad +\alpha \zeta _1 \zeta _3 \left[ \ln \left( \frac{f}{g} \right) \right] _{x} \left[ \ln \left( fg \right) \right] _{xy} +\alpha \zeta _1 \zeta _4 \left[ \ln \left( \frac{f}{g} \right) \right] _{x} \nonumber \\&\quad +\alpha \beta \zeta _1 \left[ \ln \left( \frac{f}{g} \right) \right] _{x}+\alpha \zeta _1 \delta ^{2} \left[ \ln \left( \frac{f}{g} \right) \right] _{xxy}=0 \;\;. \end{aligned}$$

(8)

According to Formulae (5) and the following formulae  [16]:

$$\begin{aligned}&\left[ \ln \left( fg \right) \right] _{xy} =\frac{D_{x} D_{y}{f\cdot g}}{fg}-\frac{D_x{f\cdot g}}{fg}\frac{D_y{f\cdot g}}{fg} \;\;, \end{aligned}$$

(9a)

$$\begin{aligned}&\left[ \ln \left( fg \right) \right] _{yt} =\frac{D_{y} D_{t}{f\cdot g}}{fg}-\frac{D_y{f\cdot g}}{fg}\frac{D_t{f\cdot g}}{fg} \;\;, \end{aligned}$$

(9b)

$$\begin{aligned}&\left[ \ln \left( \frac{f}{g} \right) \right] _{xxy} =\frac{D_{x}^{2} D_y {f\cdot g}}{fg}- 2 \frac{D_x D_y {f\cdot g}}{fg} \frac{D_x{f\cdot g}}{fg} -\frac{D_{x}^{2} {f\cdot g}}{fg}\frac{D_y{f\cdot g}}{fg} \nonumber \\&\qquad \qquad \qquad \qquad \quad +2 \left( \frac{D_x{f\cdot g}}{fg} \right) ^{2}\frac{D_y{f\cdot g}}{fg} \;\;, \end{aligned}$$

(9c)

Eqs. (7) and (8) give rise to

$$\begin{aligned} \zeta _1 =\pm 2 \delta \;\;, \qquad \left[ D_y D_t \pm \alpha \delta D_{x}^{2} D_y \pm \frac{\alpha }{\delta } (\zeta _{4} +\beta ) D_x \right] {f \cdot g}=0\;\;. \end{aligned}$$

(10)

Based on the above derivation, we are able to come up with the theorem:

Theorem 2.1

In brief, via Assumptions (3), we construct the following bilinear forms for System (1) via the Hirota method:

$$\begin{aligned}&( D_t \pm \alpha \delta D_{x}^{2} ){f \cdot g}=0\;\;, \end{aligned}$$

(11a)

$$\begin{aligned}&\left[ D_y D_t \pm \alpha \delta D_{x}^{2} D_y \pm \frac{\alpha }{\delta } (\zeta _{4} +\beta ) D_x \right] {f \cdot g}=0\;\;, \end{aligned}$$

(11b)

which are the same as Bilinear Forms (2) when $\zeta _{4}=\theta _{4}$.

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

For obtaining some similarity reductions, we give rise to the assumptions³

$$\begin{aligned}&u\left( x,y,t \right) =\theta \left( x,y,t \right) +\omega \left( x,y,t \right) p\left[ z\left( x,y,t \right) \right] , \end{aligned}$$

(12a)

$$\begin{aligned}&v\left( x,y,t \right) =\gamma \left( x,y,t \right) +\kappa \left( x,y,t \right) q\left[ z\left( x,y,t \right) \right] , \end{aligned}$$

(12b)

where $\theta (x,y,t)$, $\omega (x,y,t) \ne 0$, $\gamma (x,y,t)$, $\kappa (x,y,t) \ne 0$ and $z(x,y,t) \ne 0$ imply some real differentiable functions to be determined, while p[z(x, y, t)] and q[z(x, y, t)] are two real differentiable functions of z.

Making use of symbolic computation and inserting Assumptions (12) into System (1), we obtain that

$$\begin{aligned}&\chi _{0} p p''+\chi _{0} p'^2+\chi _{1} p''+\chi _{2} p p'+\chi _{3} p'+\chi _{4} p +\chi _{5} p^2\nonumber \\&\quad +\chi _{6} q''+\chi _{7} q'+\chi _{8} q+\chi _{9}=0 \;\;, \end{aligned}$$

(13a)

$$\begin{aligned}&\tau _{0} p'''+\tau _{1} p''+\tau _{2} p'+\tau _{3} p+\tau _{4} q'+\tau _{5} q \nonumber \\&\quad +\tau _{6} p' q +\tau _{6} p q'+\tau _{7} p q+\tau _{8}=0\;\;, \end{aligned}$$

(13b)

in which

$$\begin{aligned} \chi _{0}&=\alpha \omega ^2 z_x z_y \;\;, \end{aligned}$$

(14a)

$$\begin{aligned} \chi _{1}&=\omega z_t z_y + \alpha \theta \omega z_x z_y \;\; , \end{aligned}$$

(14b)

$$\begin{aligned} \chi _{2}&=2\alpha \omega \omega _y z_x + 2\alpha \omega \omega _x z_y + \alpha {\omega ^2} z_{xy} \;\;, \end{aligned}$$

(14c)

$$\begin{aligned} \chi _{3}&=\omega _t z_y+z_t \omega _y+\omega z_{yt} +\alpha \omega \theta _y z_x+\alpha \omega z_y \theta _x\nonumber \\&\quad + \alpha \theta \omega _y z_x+\alpha \theta \omega _x z_y+\alpha \theta \omega z_{xy} , \end{aligned}$$

(14d)

$$\begin{aligned} \chi _{4}&=\omega _{yt}+\alpha \omega _y \theta _x +\alpha \theta _y \omega _x +\alpha \omega \theta _{xy}+\alpha \theta \omega _{xy} \;\;, \end{aligned}$$

(14e)

$$\begin{aligned} \chi _{5}&=\alpha \omega _y \omega _x + \alpha \omega \omega _{xy} \;\;, \end{aligned}$$

(14f)

$$\begin{aligned} \chi _{6}&=2\alpha \kappa z^{2}_x \;\;, \end{aligned}$$

(14g)

$$\begin{aligned} \chi _{7}&=2\alpha z_x \kappa _x+\alpha \kappa z_{xx} \;\;, \end{aligned}$$

(14h)

$$\begin{aligned} \chi _{8}&=\alpha \kappa _{xx} \;\;, \end{aligned}$$

(14i)

$$\begin{aligned} \chi _{9}&=\theta _{yt}+\alpha \theta _y \theta _x+\alpha \theta \theta _{xy}+\alpha \gamma _{xx} \;\;, \end{aligned}$$

(14j)

$$\begin{aligned} \tau _{0}&=\alpha \delta ^2 \omega z_y z^2_x \;\;, \end{aligned}$$

(14k)

$$\begin{aligned} \tau _{1}&=\alpha \delta ^2 \omega _y z^2_x+2\alpha \delta ^2 z_y z_x \omega _x+2\alpha \delta ^2 \omega z_x z_{xy} +\alpha \delta ^2 \omega z_y z_{xx} \;\;, \end{aligned}$$

(14l)

$$\begin{aligned} \tau _{2}&=\alpha \beta \omega z_x +\alpha \gamma \omega z_x+2\alpha \delta ^2 \omega _x z_{xy}+2\alpha \delta ^2 z_x \omega _{xy} +\alpha \delta ^2 \omega _y z_{xx}\nonumber \\&\quad +\alpha \delta ^2 z_y \omega _{xx} +\alpha \delta ^2 \omega z_{xxy} \;\;, \end{aligned}$$

(14m)

$$\begin{aligned} \tau _{3}&=\alpha \omega \gamma _x+\alpha \beta \omega _x+\alpha \gamma \omega _x+\alpha \delta ^2 \omega _{xxy} \;\;, \end{aligned}$$

(14n)

$$\begin{aligned} \tau _{4}&=\kappa z_t+\alpha \theta \kappa z_x \;\;, \end{aligned}$$

(14o)

$$\begin{aligned} \tau _{5}&=\kappa _t+\alpha \kappa \theta _x+\alpha \theta \kappa _x \;\;, \end{aligned}$$

(14p)

$$\begin{aligned} \tau _{6}&=\alpha \kappa \omega z_x \;\;, \end{aligned}$$

(14q)

$$\begin{aligned} \tau _{7}&=\alpha \omega \kappa _x+\alpha \kappa \omega _x \;\;, \end{aligned}$$

(14r)

$$\begin{aligned} \tau _{8}&=\gamma _t+\alpha \theta \gamma _x+\alpha \beta \theta _x+\alpha \gamma \theta _x+\alpha \delta ^2 \theta _{xxy} , \end{aligned}$$

(14s)

$\chi _{i}$’s ($i=0,...,9$) and $\tau _{j}$’s ($j=0,...,8$) are some real differentiable functions with respect to x, y and t, while the prime sign means d/dz. Because p(z) and q(z) are the functions of z only, we are able to convert Eq. (13) into a set of the ODEs in respect of p(z) and q(z). Each set of $\theta (x,y,t)$, $\omega (x,y,t)$, $\gamma (x,y,t)$, $\kappa (x,y,t)$ and z(x, y, t) could lead to, at least, a similarity reduction of System (1). In this paper, we consider the case of $z_{x} z_{y} \ne 0$, so that $\chi _{0} \ne 0$ and $\tau _{0} \ne 0$, to obtain that

$$\begin{aligned} \chi _{i}=\Omega _{i}(z) \chi _{0} \;\;, \qquad \tau _{j}=\Gamma _{j}(z) \tau _{0} \;\;, \end{aligned}$$

(15)

with $\Omega _{i}(z)$’s and $\Gamma _{j}(z)$’s as some real to-be-determined functions of z only.

For the sake of simplicity, we give out the assumption that⁴

$$\begin{aligned} z(x,y,t)=\lambda _1 x+\lambda _2 y+\lambda _3 t+\lambda _4 \;\;, \end{aligned}$$

(16)

with $\lambda _1$, $\lambda _2$ and $\lambda _3$ as the real non-zero constants, while $\lambda _4$ as a real constant. Substituting Eqs. (14q), (14k) and (16) into Eqs. (15) turns to

$$\begin{aligned} \kappa (x,y,t)=\delta ^2 \lambda _1 \lambda _2 \;\;, \qquad \Gamma _{6}(z)=1\;\;. \end{aligned}$$

(17)

According to the second freedom of Remark 3 in Ref. [62], Eq. (14g) results in

$$\begin{aligned} \omega (x,y,t)=\pm \delta \lambda _1 \;\;, \qquad \Omega _{6}(z)=1\;\;. \end{aligned}$$

(18)

With the first freedom of Remark 3 in Ref. [62], Eq. (14b) leads to

$$\begin{aligned} \theta (x,y,t)=-\frac{\lambda _3}{\alpha \lambda _1}\;\;, \; \qquad \Omega _{1}(z)=0 \;\;, \end{aligned}$$

(19)

and Eq. (14m) helps us derive

$$\begin{aligned} \Gamma _{2}(z)=\frac{\beta +\gamma }{\delta ^2 \lambda _1 \lambda _2}\;\;. \end{aligned}$$

(20)

Based on the first and the second freedom of Remark 3 in Ref. [62], respectively, we will obtain two branches of the results.

Branch 1: $\gamma (x,y,t)=-\beta \;, \;\; \Gamma _{2}(z)=0$

Inserting $\gamma (x,y,t)=-\beta$ into Eqs. (14) brings about

$$\begin{aligned} \Omega _{2}(z)&=\Omega _{3}(z)=\Omega _{4}(z)=\Omega _{5}(z)=\Omega _{7}(z)=\Omega _{8}(z)=\Omega _{9}(z)=0\;\;, \end{aligned}$$

(21a)

$$\begin{aligned} \Gamma _{1}(z)&=\Gamma _{3}(z)=\Gamma _{4}(z)=\Gamma _{5}(z)=\Gamma _{7}(z)=\Gamma _{8}(z)=0\;\;. \end{aligned}$$

(21b)

Eqs. (13) can turn into

$$\begin{aligned}&p p''+p'^2+q''=0\;\;, \end{aligned}$$

(22a)

$$\begin{aligned}&p'''+p' q+p q'=0\;\; . \end{aligned}$$

(22b)

Then we integrate ODE (22a) twice about z, to obtain

$$\begin{aligned} q=-\frac{1}{2}p^2+\phi _1 z+\phi _2\;\;, \end{aligned}$$

(23)

with $\phi _1$ and $\phi _2$ being two real constants of integration. Integrating ODE (22b) once in respect of z and considering ODE (23), we can transfer ODEs (22) to a simple ODE, written as

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

(24)

where $\phi _3$ denotes a real constant of integration.

Thus, we derive two sets of the similarity reductions for System (1), i.e.,

$$\begin{aligned}&u(x,y,t)=-\frac{\lambda _3}{\alpha \lambda _1} \pm \delta \lambda _1 p[z(x,y,t)] \;\; , \end{aligned}$$

(25a)

$$\begin{aligned}&v(x,y,t)=-\beta -\delta ^2 \lambda _1 \lambda _2 \left\{ \frac{1}{2}p^2[z(x,y,t)]-\phi _1 z-\phi _2 \right\} \;\; , \end{aligned}$$

(25b)

$$\begin{aligned}&z(x,y,t)=\lambda _1 x+\lambda _2 y+\lambda _3 t+\lambda _4 \;\;, \end{aligned}$$

(25c)

$$\begin{aligned}&p''-\frac{1}{2}p^3+(\phi _1 z+\phi _2)p+\phi _3=0\;\;. \end{aligned}$$

(25d)

ODE (25d) is a known ODE, reported in Ref. [63].

Branch 2: $\gamma (x,y,t)=\delta ^2 \lambda _1 \lambda _2 -\beta \;, \;\; \Gamma _{2}(z)=1$

When $\gamma (x,y,t)=\delta ^2 \lambda _1 \lambda _2 -\beta$, we propose to derive

$$\begin{aligned} \Omega _{2}(z)&=\Omega _{3}(z)=\Omega _{4}(z)=\Omega _{5}(z)=\Omega _{7}(z)=\Omega _{8}(z)=\Omega _{9}(z)=0\;\;, \end{aligned}$$

(26a)

$$\begin{aligned} \Gamma _{1}(z)&=\Gamma _{3}(z)=\Gamma _{4}(z)=\Gamma _{5}(z)=\Gamma _{7}(z)=\Gamma _{8}(z)=0\;\;. \end{aligned}$$

(26b)

Eqs. (13) are converted into

$$\begin{aligned}&p p''+p'^2+q''=0\;\;, \end{aligned}$$

(27a)

$$\begin{aligned}&p'''+p'+p' q+p q'=0\;\; . \end{aligned}$$

(27b)

Similarly, we integrate ODE (27a) twice about z to find

$$\begin{aligned} q=-\frac{1}{2}p^2+\phi _4 z+\phi _5 \;\;, \end{aligned}$$

(28)

with $\phi _4$ and $\phi _5$ as two real constants of integration. Integrating ODE (27b) once about z and considering ODE (28) could develop into

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

(29)

with $\phi _6$ as a real constants of integration.

Thus, we require into another two sets of the similarity reductions for System (1), i.e.,

$$\begin{aligned}&u(x,y,t)=-\frac{\lambda _3}{\alpha \lambda _1} \pm \delta \lambda _1 p[z(x,y,t)] \;\; , \end{aligned}$$

(30a)

$$\begin{aligned}&v(x,y,t)=\delta ^2 \lambda _1 \lambda _2 -\beta -\delta ^2 \lambda _1 \lambda _2 \left\{ \frac{1}{2}p^2[z(x,y,t)]-\phi _4 z-\phi _5 \right\} \;\; , \end{aligned}$$

(30b)

$$\begin{aligned}&z(x,y,t)=\lambda _1 x+\lambda _2 y+\lambda _3 t+\lambda _4 , \end{aligned}$$

(30c)

$$\begin{aligned}&p''-\frac{1}{2}p^3+(\phi _4 z+\phi _5+1)p+\phi _6=0\;\;. \end{aligned}$$

(30d)

ODE (30d) is a known ODE, reported in Ref. [63].

With respect to the horizontal velocity and the wave elevation above the undisturbed water surface, we derive the following theorem about System (1), describing the nonlinear and dispersive long gravity waves in two horizontal directions in the shallow water of a wide channel of finite depth or an open sea.

Theorem 3.1

Similarity Reductions (25) and Similarity Reductions (30), both of which are different from those in Ref. [18], depend on all the constant coefficients in System (1), i.e., $\alpha$, $\beta$ and $\delta$. The reason why there are two sets of Similarity Reductions (25)/Similarity Reductions (30) is the existence of “±" sign.

4 Discussions

We have noticed that both Similarity Reductions (25) and Similarity Reductions (30) are different from those in Ref. [18], while both ODE (25d) and ODE (30d) are the known ODEs. Our results have been shown to depend on $\alpha$, $\beta$ and $\delta$, all the constant coefficients in System (1), and might be of some use in the studies on the nonlinear and dispersive long gravity waves in two horizontal directions in the shallow water of a wide channel of finite depth or an open sea.

5 Conclusions

As for a generalized (2+1)-dimensional dispersive long-wave system in respect of the horizontal velocity and the wave elevation above the undisturbed water surface, i.e., System (1), we have obtained the following:

• Two sets of the bilinear forms, i.e., Bilinear Forms (11), which are the same as Bilinear Forms (2), but through a different method, i.e., the Hirota method. Thus, the correctness of Bilinear Forms (2) can be confirmed.

• Four sets of the similarity reductions for System (1), i.e., Similarity Reductions (25), from System (1) to ODE (25d), and Similarity Reductions (30), from System (1) to ODE (30d).