Looking at an open sea via a generalized $(2+1)$-dimensional dispersive long-wave system for the shallow water: scaling transformations, hetero-Bäcklund transformations, bilinear forms and N solitons

Abstract

Oceanic water-wave studies are attractive. Hereby, with a view to modelling 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, we inquire into a generalized (2+1)-dimensional dispersive long-wave system, by virtue of the scaling transformations we hereby obtain, Bell polynomials and symbolic computation. As for the horizontal velocity and wave elevation above the undisturbed water surface, we find two branches of the hetero-Bäcklund transformations, two branches of the bilinear forms and two branches of the N-soliton solutions, with N as a positive integer. What we have obtained relies on the coefficients in the system.

1 Introduction

Oceanic wave dynamics has been rapidly developing [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].

To model 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, a generalized (2+1)-dimensional dispersive long-wave system [21, 22],

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

(1a)

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

(1b)

has appeared. Hereby, t and (x, y) represent the time and propagation plane, respectively, the horizontal velocity u(x, y, t) and the wave elevation above the undisturbed water surface v(x, y, t) denote the real differentiable functions as for x, y and t, the subscripts stand for the partial derivatives, while $\alpha \ne 0$, $\beta$ and $\delta \ne 0$ mean the real constants [21]. For System (1), Ref. [21] has worked out certain non-auto- and auto-Bäcklund transformations with solitons, while of current interest, Ref. [22] has constructed some similarity reductions.

In hydrodynamics, some special cases of System (1) have been investigated:

• with $\alpha =\beta =1$ and $\delta =\pm 1$, a (2+1)-dimensional dispersive long-wave system, which describes the nonlinear and dispersive long gravity waves in two horizontal directions on the shallow water of a wide channel or an open sea of finite depth [23,24,25,26],

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

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

  (2b)

  with u(x, y, t) representing the horizontal velocity, v(x, y, t) standing for the wave elevation above the undisturbed water surface and t and (x, y), respectively, being the time and propagation plane;

• with $\alpha =1$, $\beta =0$ and $\delta =\pm 1$, for the shallow water, a (2+1)-dimensional dispersive long-wave system [27],

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

  (3a)
  $$\begin{aligned}&v_{t}+ \left( \, u v+u_{xy} \, \right) _{x}=0 \;\; , \end{aligned}$$

  (3b)

  which is thought “helpful for coastal and civil engineers to apply the nonlinear water model to coastal harbour design” [27];

• with $\alpha =1$ and $\delta =\pm 1$, a (2+1)-dimensional dispersive long-wave system modelling the nonlinear and dispersive long gravity waves in two horizontal directions on the shallow water of a wide channel or an open sea of finite depth [28],

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

  (4a)
  $$\begin{aligned}&v_{t}+ \left( \, u v+\beta u+u_{xy} \, \right) _{x}=0\;\; ; \end{aligned}$$

  (4b)

• with $\beta =1$ and $\delta =\pm 1$, a (2+1)-dimensional dispersive long-wave system [29, 30],

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

  (5a)
  $$\begin{aligned}&v_{t}+\alpha \left( \, u v+u+u_{xy} \, \right) _{x}=0 \;\;; \end{aligned}$$

  (5b)

• with $\alpha =\beta =1$, a (2+1)-dimensional dispersive long-wave system for the water waves [31],

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

  (6a)
  $$\begin{aligned}&v_{t}+ \left( \, u v+u+\delta ^2 u_{xy} \, \right) _{x}=0 \;\;; \end{aligned}$$

  (6b)

• with $y=x$, $\alpha =\beta =1$ and $\delta =\pm 1$, a Broer–Kaup system which models the evolution of the horizontal velocity u(x, t) of water waves of height v(x, t) propagating in both directions in an infinitely-long narrow channel of finite constant depth [25, 32, 33],

  $$\begin{aligned}&u_{t}+v_{x}+u u_{x} =0 \;\; , \end{aligned}$$

  (7a)
  $$\begin{aligned}&v_{t}+u_{x}+ \left( \, u v \, \right) _{x}+u_{xxx}=0 \;\; ; \end{aligned}$$

  (7b)

• with $y=x$, $\alpha =1$, $\beta =0$ and $\delta =\pm 1$, a (1+1)-dimensional dispersive long-wave system modelling the shallow water waves [27],

  $$\begin{aligned}&u_{t}+v_{x}+u u_{x} =0 \;\; , \end{aligned}$$

  (8a)
  $$\begin{aligned}&v_{t}+ \left( \, u v+u_{xx} \, \right) _{x}=0 \;\; . \end{aligned}$$

  (8b)

However, to our knowledge, there has existed neither the bilinear-form work nor the N-soliton work on System (1) as yet, where N is a positive integer. Using the Bell polynomials and symbolic computation [34,35,36,37,38,39,40,41,42,43,44,45], in this Report, starting from our scaling transformations, we will construct some hetero-Bäcklund transformations, which differ from those published in Ref. [21]. Making use of the binary Bell polynomials and symbolic computation¹, in this Report, we will find out certain bilinear forms and N solitons for System (1).

2 Scaling transformations and hetero-Bäcklund transformations for System (1)

Brief review of the known knowledge: Soon to be used are the Bell polynomials [58,59,60], i.e.,

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

(9)

in which w(x, y, t) means a $C^{\infty }$ function of x, y and t, $w_{k,s,l}=\partial ^{k}_{x}\partial ^{s}_{y}\partial ^{l}_{t} w$, $k=0,\cdots ,m,\,s=0,\cdots ,r,\,l=0,\cdots ,n$, with m, r and n being the non-negative integers.

The scaling transformations which we hereby obtain,

$$\begin{aligned}& x \rightarrow \rho ^{1} x \;\;, \qquad y \rightarrow \rho ^{\eta } y \;\;, \qquad t \rightarrow \rho ^{\sigma } t \;\;, \qquad \alpha \rightarrow \rho ^{2-\sigma } \alpha \;\;, \qquad \qquad \end{aligned}$$

(10a)

$$\begin{aligned}& \delta \rightarrow \rho ^{0} \delta \;\;, \qquad u \rightarrow \rho ^{-1} u \;\;, \qquad v \rightarrow \rho ^{-1-\eta } v \;\;, \qquad \beta \rightarrow \rho ^{-1-\eta } \beta \;\;, \end{aligned}$$

(10b)

lead to our assumptions

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

(11)

in which $\eta$ and $\sigma$ imply the integers, while $\rho >0$, $\gamma _1 \ne 0$, $\gamma _2 \ne 0$ and $\gamma _3$ represent the real constants.

With symbolic computation, we integrate Eq. (1a) once, respectively, with respect to x and y with the integration functions vanishing, so that

$$\begin{aligned} \gamma _1 w_{t}+\alpha \left[ \gamma _2 w_{xx}+\frac{1}{2} \gamma _1^2 w_{x}^2 \right] =0 \;\;. \end{aligned}$$

(12)

With the choice of

$$\begin{aligned} \gamma _2=\frac{\gamma _1^2 }{2} \;\;, \end{aligned}$$

(13)

Bell polynomials and symbolic computation help us convert Eq. (12) into a Y-polynomial expression, i.e.,

$$\begin{aligned} Y_{t}(w)+\frac{\alpha \gamma _1}{2} Y_{2x}(w)=0\;\;, \end{aligned}$$

(14)

which, with the assumption that

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

(15)

further turns into

$$\begin{aligned} h_{t}(x,y,t) +\frac{\alpha \gamma _1}{2} h_{xx}(x,y,t)=0\;\;, \end{aligned}$$

(16)

where h(x, y, t) denotes a positive differentiable function.

Similarly, integrating Eq. (1b) once with respect to x with the integration function vanishing, we obtain

$$\begin{aligned} \gamma _2 w_{yt}+\frac{\alpha \gamma _1 \gamma _2}{2} \left( w_{x}^2 \right) _{y} +\alpha \gamma _1 \left( \gamma _3+\beta \right) w_{x}+\alpha \delta ^2 \gamma _1 w_{xxy}=0\;\;. \end{aligned}$$

(17)

With the choice of

$$\begin{aligned} \gamma _3=- \beta \;\;, \end{aligned}$$

(18)

we integrate Eq. (17) once with respect to y with the integration function vanishing, and then, with the choices of

$$\begin{aligned} \gamma _1=\pm 2 \delta \;\;, \end{aligned}$$

(19)

Bell polynomials and symbolic computation shift Eq. (17) into a Y-polynomial expression, i.e.,

$$\begin{aligned} Y_{t}(w)\pm \alpha \delta Y_{2x}(w)=0\;\;, \end{aligned}$$

(20)

which results in the same as Eq. (14).

Thinking over all the above together, we conclude with two branches of the hetero-Bäcklund transformations for System (1):

$$\begin{aligned}& u(x,y,t)=\pm \frac{2 \delta h_{x}(x,y,t)}{h(x,y,t)} \;\;, \end{aligned}$$

(21a)

$$\begin{aligned}& v(x,y,t)=2 \delta ^2 \left[ \frac{h_{xy}(x,y,t)}{h(x,y,t)} -\frac{h_{x}(x,y,t) h_{y}(x,y,t)}{h(x,y,t)^2}\right] -\beta \;\;, \end{aligned}$$

(21b)

$$\begin{aligned}& h_{t}(x,y,t) \pm \alpha \delta h_{xx}(x,y,t)=0\;\; . \end{aligned}$$

(21c)

In other words, if h(x, y, t) denotes a solution of Eq. (21c), a known linear partial differential equation [61, 62], Eqs. (21) represent two hetero-Bäcklund transformations which link h(x, y, t) and the set of the solutions u(x, y, t) and v(x, y, t) of System (1).

Two branches of Hetero-Bäcklund Transformations (21) exist, because of the appearance of the “±” signs.

Hetero-Bäcklund Transformations (21), relying on $\alpha$, $\beta$ and $\delta$, the constant coefficients in System (1), are different from those in Ref. [21].

3 Bilinear forms and N solitons for System (1)

Bilinear forms could be used to derive certain solutions, e.g., the solitary wave solutions, or to construct some bilinear Bäcklund transformations [63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81].

Brief review of the known knowledge: Soon to be used are the binary Bell polynomials [58,59,60], i.e.,

$$\begin{aligned}&{\mathscr {Y}}_{mx,ry,nt}(p,q)\equiv \nonumber \\&\quad Y_{mx,ry,nt}(\psi _{0,0,0},\cdots ,\psi _{0,0,n},\cdots \psi _{m,r,0},\cdots ,\psi _{m,r,n}) \bigg |_{\psi _{k,s,l}=\left\{ \begin{aligned}&p_{k,s,l},\,\text {if } k+s+l\text { is odd}\;\;, \\&q_{k,s,l},\,\text {if } k+s+l\text { is even}\;\;, \\ \end{aligned} \right. }\nonumber \\ \end{aligned}$$

(22)

in which $\psi _{k,s,l}(x,y,t)'s, p(x,y,t)$ and q(x, y, t) denote the $C^{\infty }$ functions of x, y and t, while $p_{k,s,l}=\partial ^{k}_{x}\partial ^{s}_{y}\partial ^{l}_{t} p$ and $q_{k,s,l}=\partial ^{k}_{x}\partial ^{s}_{y}\partial ^{l}_{t} q$. References [82,83,84] have linked the ${\mathscr {Y}}$ polynomials to the Hirota operators as

$$\begin{aligned} \begin{aligned} {\mathscr {Y}}_{mx,ry,nt}\left[ p=\ln \left( \frac{f}{g}\right) ,\,q=\ln \left( f g\right) \right] =(f g)^{-1}D^m_x D^r_y D^n_t f\cdot g\;\;, \\ \end{aligned} \end{aligned}$$

(23)

in which f(x, y, t) and g(x, y, t) represent the $C^{\infty }$ functions of x, y and t, while $D_{x}$, $D_{y}$ and $D_{t}$ imply the Hirota operators defined by

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

(24)

with $x'$, $y'$ and $t'$ being the formal variables.

Scaling Transformations (10) bring about our assumptions

$$\begin{aligned}& u(x,y,t)=\theta _1 p_{x}(x,y,t)\;\;, \end{aligned}$$

(25a)

$$\begin{aligned}& v(x,y,t) =\theta _2 p_{xy}(x,y,t)+ \theta _3 q_{xy}(x,y,t)+\theta _4\;\;, \end{aligned}$$

(25b)

in which $\theta _2$ and $\theta _4$ denote the real constants, while $\theta _1$ and $\theta _3$ mean the non-zero real constants.

With symbolic computation, we integrate Eq. (1a) once, respectively, with respect to x and y with the integration functions vanishing, to obtain

$$\begin{aligned} \theta _1 p_{t}+\alpha \left( \, \theta _2 p_{xx}+\theta _3 q_{xx} +\frac{\theta _1^2}{2} p_{x}^2 \,\right) =0 \;\;. \end{aligned}$$

(26)

With the choices of

$$\begin{aligned} \theta _2=0, \qquad \theta _3=\frac{\theta _1^2}{2} \;\;, \end{aligned}$$

(27)

binary Bell polynomials and symbolic computation help us convert Eq. (26) into a ${\mathscr {Y}}$-polynomial expression, i.e.,

$$\begin{aligned} {\mathscr {Y}}_{t}(p)+\frac{\alpha \theta _1}{2} {\mathscr {Y}}_{2x}(p,q)=0\;\;. \end{aligned}$$

(28)

Similarly, integrating Eq. (1b) once with respect to x with the integration function vanishing gives rise to

$$\begin{aligned} \theta _3 q_{yt}+\alpha \left[ \,\theta _1 \theta _3 p_{x} q_{xy} +\theta _1 \left( \theta _4+\beta \right) p_{x} +\delta ^2 \theta _1 p_{xxy}\,\right] =0\;\;. \end{aligned}$$

(29)

With the choices of

$$\begin{aligned} \theta _3=2 \delta ^2, \qquad \theta _1=\pm 2 \delta \;\;, \end{aligned}$$

(30)

and the integration of Eq. (29) once with respect to y with the integration function vanishing, binary Bell polynomials and symbolic computation help us convert Eq. (29) into a ${\mathscr {Y}}$-polynomial expression, i.e.,

$$\begin{aligned} {\mathscr {Y}}_{y, t}(p,q)+\frac{\alpha \theta _1}{2} {\mathscr {Y}}_{2x,y}(p,q) +\frac{\alpha \theta _1 \left( \theta _4+\beta \right) }{\theta _3} {\mathscr {Y}}_{x}(p)=0\;\;. \end{aligned}$$

(31)

Further, with

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

(32a)

$$\begin{aligned}& q(x,y,t)=\ln \left[ f(x,y,t) g(x,y,t)\right] \;\;, \end{aligned}$$

(32b)

through Eqs. (28) and (31), we are able to transform System (1) into the following two branches of the bilinear forms through the binary Bell polynomials for System (1) via Assumptions (25):

$$\begin{aligned}&\left( \,D_t \pm \alpha \delta D^2_x\,\right) f\cdot g=0\;\;, \end{aligned}$$

(33a)

$$\begin{aligned}&\left[ \,D_y D_t \pm \alpha \delta D^2_x D_y \pm \frac{\alpha }{\delta } \left( \theta _4+\beta \right) D_x \,\right] f\cdot g =0\;\;. \end{aligned}$$

(33b)

The reason for the existence of two branches of Bilinear Forms (33) is that there appear the “±” signs. Both of the branches are dependent on $\alpha$, $\beta$ and $\delta$, the constant coefficients in System (1).

Expanding f(x, y, t) and g(x, y, t) in Eqs. (33a) and (33b) with respect to a formal expansion parameter $\epsilon$ as

$$\begin{aligned}& f(x,y,t)=1+\sum _{\varsigma =1}^{N}\epsilon ^{\varsigma }f_\varsigma (x,y,t) \;\;, \end{aligned}$$

(34a)

$$\begin{aligned}& g(x,y,t)=1+\sum _{\varpi =1}^{N}\epsilon ^{\varpi }g_\varpi (x,y,t) \;\;, \end{aligned}$$

(34b)

and then setting $\epsilon =1$, we obtain two branches of the N-soliton solutions for System (1) as

$$\begin{aligned} \left. \begin{aligned} u(x,y,t)= & {} \\ v(x,y,t)= & {} \end{aligned} \!\!\!\! \right\} \text {from Expressions}~(25)\text { and }(32)\;\;, \end{aligned}$$

with

$$\begin{aligned}& f(x,y,t)=\sum _{\mu _i ,\mu _j=0,1}\mathrm {exp}\left[ \sum _{i=1}^{N}\mu _i (\lambda _{i} x+\phi _{i} y+\tau _{i} t+\omega _i) +\sum _{1\le i<j}^{(N)}\mu _i\mu _j\Upsilon _{ij}\right] \;\;, \end{aligned}$$

(35a)

$$\begin{aligned}& g(x,y,t)= \sum _{\mu _i ,\mu _j=0,1}\mathrm {exp}\left[ \sum _{i=1}^{N}\mu _i (\lambda _{i} x+\phi _{i} y+\tau _{i} t+\kappa _i ) +\sum _{1\le i<j}^{(N)}\mu _i\mu _j\Upsilon _{ij}\right] \;\;, \end{aligned}$$

(35b)

$$\begin{aligned}& e^{\omega _{i}}=b_i \;\;, \end{aligned}$$

(35c)

$$\begin{aligned}& e^{\kappa _{i}}=c_i \;\;, \end{aligned}$$

(35d)

$$\begin{aligned}& \tau _{i}=\pm \frac{\alpha \delta \left( c_i+b_i\right) \lambda _{i}^2}{c_i- b_i} \;\;, \end{aligned}$$

(35e)

$$\begin{aligned}& \phi _{i}=\frac{\left( \theta _4+\beta \right) \left( c_i-b_i\right) ^2}{4 \delta ^2 b_i c_i \lambda _{i}} \;\;, \end{aligned}$$

(35f)

$$\begin{aligned}& e^{\Upsilon _{ij}}=\frac{\left( b_i b_j \lambda _i-b_i c_j \lambda _i+b_j c_i \lambda _j-b_i b_j \lambda _j\right) \left( b_j c_i \lambda _i-b_i c_j \lambda _j-c_i c_j \lambda _i+c_i c_j \lambda _j\right) }{\left( b_j c_i \lambda _i+b_j c_i \lambda _j-b_i b_j \lambda _j-c_i c_j \lambda _i\right) \left( b_i b_j \lambda _i-b_i c_j \lambda _i-b_i c_j \lambda _j+c_i c_j \lambda _j\right) } \;\;, \end{aligned}$$

(35g)

$\varsigma$, $\varpi$, i and j being the positive integers with $\varsigma \le N$, $\varpi \le N$, $i\le N$ and $j\le N$, $\lambda _{i}$’s, $\omega _{i}$’s and $\kappa _{i}$’s denoting the real constants, $f_\varsigma (x,y,t)$’s and $g_\varpi (x,y,t)$’s being the real differentiable functions of x, y and t, $b_i$ and $c_i$ representing the parameters characterizing the ith soliton, the sum $\sum _{\mu _i ,\mu _j=0,1}$ taken over all the possible combinations of $\mu _j=0,1$, while $\sum _{1\le i<j}^{(N)}$ being the summation over all the possible pairs chosen from the N elements under the condition $i < j$.

It is noted that there exist two branches of the N-soliton solutions for System (1) because of the “±” signs. Both of the branches are dependent on $\alpha$, $\beta$ and $\delta$, the constant coefficients in System (1).

4 Conclusions

The story of oceans has been the story of life. In this Report, we have kept an eye on 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, and inquired into System (1), a generalized (2+1)-dimensional dispersive long-wave system. We have constructed Scaling Transformations (10), and used the Bell polynomials and symbolic computation. As for u(x, y, t) and v(x, y, t), the horizontal velocity and wave elevation above the undisturbed water surface, through the Bell polynomials, we have constructed two branches of the hetero-Bäcklund transformations, i.e., Hetero-Bäcklund Transformations (21), different from those in Ref. [21]. Also as for u(x, y, t) and v(x, y, t), through the binary Bell polynomials, we have derived two branches of the bilinear forms, i.e., Bilinear Forms (33), with two branches of the N-soliton solutions, i.e., N-Soliton Solutions (25), (32) and (35). Our results have been presented to rely on the coefficients in System (1).