Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Abstract

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

1. Introduction

It is widely known that most physical phenomena of the natural world are governed by nonlinear partial differential equations (NPDEs). Such equations arise in several physical and other problems such as in the study of electromagnetism, electrodynamics, fluid flow, elasticity, propagation of heat or sound, quantum mechanics, meteorology, and oceanography, just to mention a few. Some of these models include the modified Zakharov–Kuznetsov model [1] that recounts the ion-acoustic drift solitary waves existing in a magnetoplasma with electron–positron ions, which are found in a primordial universe. A generalized system of three-dimensional modified Kadomtsev–Petviashvili–Burgers-type equation, which describes ion-acoustic and dust-magneto-acoustic waves in one of cosmic or laboratory dusty plasmas, was studied in [2]. Moreover, in [3], the vector bright solitons of the Fokas–Lenells system, which models the femtosecond pulses in an optical fibre, was investigated. In addition, the Boussinesq–Burgers-type system of equations, which delineates shallow water waves appearing close to lakes or ocean beaches, was studied in [4]. The list continues; see also [5,6,7,8,9,10,11,12,13,14,15].

To obtain a good understanding of physical phenomena that are governed by NPDEs, it is imperative to look for their closed-form solutions. However, this is an arduous exercise, and for several years, researchers and scientists have been working on developing methods for finding closed-form solutions.

A good number of methods for obtaining closed-form solutions of NPDEs have been proposed. We have, amongst others, the following various special methods for finding closed-form solutions of NPDEs: the inverse scattering transform technique [16], the bifurcation method [17], the simplest equation technique [18], the extended simplest equation technique [19], Kudryashov’s technique [20], Hirota’s technique [21], Bäcklund transformation [22], Darboux transformation [23], the homogeneous balance method [24], $(G^{\prime }/G)$—expansion technique [25], Lie symmetry method [26,27,28], etc.

It was during the halfway point of the nineteenth century that Sophus Lie, a Norwegian mathematician, endowed the study of continuous transformation groups, presently called Lie groups. These groups have had an intense impact on pure and applied mathematics in addition to engineering, physics, and other applied sciences. It provides the most effective and powerful techniques for obtaining closed-form solutions to NPDEs. For example, see [26,27,28,29,30,31,32,33]. It is worth noting here that the notion of a Galois group had influenced Lie’s work on differential equations (DEs).

In the study of DEs, conservation laws play a crucial part. Conservation laws can be used to check whether a given differential equation is integrable. The accuracy of numerical solution methods can be checked by invoking conservation laws. They can be utilized in constructing solutions of partial differential equations (PDEs) by reducing their order. Conservation laws refer to certain physical properties that do not change subsequently within an isolated physical system. For example, conservation of energy, linear momentum, angular momentum, and electric charge. Recently, exact solutions were obtained for some PDEs using conservation laws. For differential equations that arise from the variational principle, there exists a connection between the symmetries of differential equations and conservation laws. This fact was established by Emmy Noether, a German mathematician, in 1918 and is stated in Noether’s theorem. For more details, see for example [34,35,36,37,38,39,40,41,42,43,44,45] and the references therein.

In this work, we study the generalized two-dimensional generalized shallow water wave equation, given by [46]

$$\begin{array}{c}\hfill u_{tx}+\alpha u_y{u}_{xx}+2\alpha u_x{u}_{xy}+\beta u_{xy}+\nu u_{xxxy}=0,\end{array}$$

(1)

where $\alpha$, $\beta$, and $\nu$ are nonzero constants. Here, $t,x$, and y represent, respectively, time and two scaled space coordinates whereas $u(t,x,y)$ represents the height of the Riemann wave.

In [46], the authors presented painlévé analysis, nonlocal symmetries, and exact solutions of a variable coefficients form of (1), whereas in [47], the binary Bell polynomials technique was employed to a variable coefficient form of Equation (1). We note that, in the study of fluids, this equation decomposes to well-known equations when taking special cases of constants $\alpha ,\beta$, and $\nu$. For example, when $\alpha =-2$, $\beta =0$, and $\nu =-1$, it becomes a $(2+1)$-dimensional breaking soliton [48]:

$$\begin{array}{c}\hfill u_{tx}-2u_y{u}_{xx}-4u_x{u}_{xy}-u_{xxxy}=0,\end{array}$$

(2)

which is one of the models used in describing the two-dimensional interplay of a Riemann wave distributed along the y-axis with a long wave generated along the x-axis. Wazwaz [49] obtained solutions of this equation using the modified form of Hirota’s bilinear method. The authors in [48] presented solutions of Equation (2) using the symmetry method. In the case where $\alpha =2$, $\beta =0$, and $\nu =1$, Equation (1) transforms to a ($2+1$)-dimensional Calogero–Bogoyavlenskii–Schiff [50]

$$\begin{array}{c}\hfill u_{tx}+2u_y{u}_{xx}+4u_x{u}_{xy}+u_{xxxy}=0,\end{array}$$

(3)

which describes the exchange of a long generating wave in the x direction with a Riemann generating wave in the y direction. Moreover, the researcher indicated that Equation (3) possesses soliton as well as N-soliton solutions, which are smooth in one coordinate. In [51], Wazwaz invoked Hirota’s bilinear approach to secure solutions of (3) and further utilized the tanh-coth method to gain its travelling wave solutions in [52]. The authors in [53] derived a Bäcklund transformation alongside a Lax pair of the form in the Calogero–Bogoyavlenskii–Schiff Equation (3) via singular manifold method. Furthermore, in [54], the authors invoked a tanh function as well as improved the $(G^{\prime }/G)$—expansion methods to construct travelling wave solutions of (3). It was also revealed that some analytic solutions secured via these methods are analogous. However, a generalized version of (3), namely

$$\begin{array}{c}\hfill u_{tx}+a{u}_x{u}_{xy}+b{u}_y{u}_{xx}+u_{xxxy}=0,\end{array}$$

(4)

has been studied by many researchers by simply replacing 4 by a and 2 by b in (3). For instance, the $(G^{\prime }/G)$—expansion technique was invoked to achieve some travelling wave solutions of (4) in [55]. Moreover, in [56], the authors obtained breather-type and analytic soliton solutions of (4) by utilizing Hirota’s bilinear method and the extended homoclinic test technique. Using the idea of a tanh-coth approach, some closed-form solutions of (4) were obtained in [57]. Furthermore, sine-cosine and complex techniques were engaged to secure, respectively, analytic and meromorphic exact solutions of (4) in [58,59].

The formation of this paper is as follows: In Section 2, first, we employ Lie’s theory to reduce (1) to an ordinary differential equation (ODE). Closed-form solutions are then derived by an integration of the ODE and by invoking Kudryashov’s method. Thereafter, in Section 3, we construct conserved densities and spatial fluxes for (1) by invoking the multiplier method and Noether’s theorem. Finally, we present concluding remarks in Section 4.

2. Exact Solutions

In this section, we derive exact solutions of the two-dimensional generalized shallow water wave Equation (1). We begin by deriving symmetries of (1) and thereafter use them to obtain an ODE. We then employ the direct integration procedure together with Kudryashov’s method to construct closed-form solutions of (1).

2.1. Symmetries and Symmetry Reductions

Consider the symmetry group of (1) brought about by

$$\begin{array}{c}\hfill U=T\frac{\partial }{\partial t}+X\frac{\partial }{\partial x}+Y\frac{\partial }{\partial y}+\eta \frac{\partial }{\partial u},\end{array}$$

where all T, X, Y, and $\eta$ are functions of variables t, x, y, and u. Using the fourth prolongation pr${}^{\left(4\right)}U$ in (1) and with the help of Maple, we obtain an overdetermined system of linear homogeneous PDEs:

$$\begin{array}{cc}\hfill & T_u=0,T_x=0,T_y=0,X_y=0,Y_u=0,Y_x=0,Y_{tt}=0,T_{ttt}=0,X_u=0,Y_{yy}=0,\hfill \\ \hfill & 2{\eta }_u+T_t-Y_y=0,2\alpha {\eta }_x+\beta T_t-\beta Y_y-Y_t=0,\alpha {\eta }_y+X_t=0,2Y_{ty}-T_{tt}=0,\hfill \\ \hfill & 2X_x-T_t+Y_y=0.\hfill \end{array}$$

The above linear PDEs may be easily solved and yield the following eight Lie point symmetries:

$$\begin{array}{cc}\hfill & U_1=\frac{\partial }{\partial t},U_2=\frac{\partial }{\partial x},U_3=\frac{\partial }{\partial y},U_4=\frac{\partial }{\partial u},\hfill \\ \hfill & U_5=2\alpha t\frac{\partial }{\partial y}+x\frac{\partial }{\partial u},U_6=2\alpha t\frac{\partial }{\partial t}+\alpha x\frac{\partial }{\partial x}-\left(\beta x+\alpha u\right)\frac{\partial }{\partial u},\hfill \\ \hfill & U_7=\alpha x\frac{\partial }{\partial x}-2\alpha y\frac{\partial }{\partial y}-\left(\alpha u+\beta x\right)\frac{\partial }{\partial u},\hfill \\ \hfill & U_8=2\alpha t^2\frac{\partial }{\partial t}+\alpha tx\frac{\partial }{\partial x}+2\alpha ty\frac{\partial }{\partial y}-\left(\beta tx-xy+\alpha tu\right)\frac{\partial }{\partial u}.\hfill \end{array}$$

(5)

Consider the symmetry $U=U_1+a{U}_2+b{U}_3$, where a and b are constants. Using this symmetry U, we reduce (1) to a PDE with one less independent variable, that is, two independent variables. The symmetry U gives three invariants:

$$\begin{array}{c}\hfill f=x-at,g=y-bt,\theta =u.\end{array}$$

(6)

Using these invariants, Equation (1) reduces to

$$\begin{array}{cc}\hfill & \nu {\theta }_{fffg}+\beta {\theta }_{fg}+\alpha {\theta }_g{\theta }_{ff}+2\alpha {\theta }_f{\theta }_{fg}-b{\theta }_{fg}-a{\theta }_{ff}=0,\hfill \end{array}$$

(7)

which has

$$\begin{array}{cc}\hfill & R_1=\frac{\partial }{\partial f},R_2=\frac{\partial }{\partial g},R_3=\alpha H\left(g\right)\frac{\partial }{\partial g}+aH\left(g\right)\frac{\partial }{\partial \theta },\hfill \\ \hfill & R_4=\alpha f\frac{\partial }{\partial f}-\alpha g\frac{\partial }{\partial g}+\left(bf-\beta f-\alpha \theta \right)\frac{\partial }{\partial \theta }\hfill \end{array}$$

as its symmetries. The symmetry $R=R_1+c{R}_2$, with c a constant, provides two invariants $q=g-cf$ and $\mathcal{U}=\theta$. These invariants transform (7) into a nonlinear ordinary differential equation (NLODE)

$$\begin{array}{c}\hfill k_1{\mathcal{U}}^{″″}\left(q\right)-k_2{\mathcal{U}}^{\prime }\left(q\right){\mathcal{U}}^{″}\left(q\right)+k_3{\mathcal{U}}^{″}\left(q\right)=0,\end{array}$$

(8)

where $k_1=b-a{c}^2-\beta$, $k_2=-3\alpha c$, $k_3=-\nu c^2$ and $q=\left(ac-b\right)t-cx+y$.

2.2. Solution Using Direct Integration

In this subsection, we derive a solution of the two-dimensional generalized shallow water Equation (1) by direct integration of the ODE (8). Taking ${\mathcal{U}}^{\prime }\left(q\right)=\mathcal{V}\left(q\right)$, Equation (8) becomes

$$\begin{array}{c}\hfill k_1{\mathcal{V}}^{\prime ″}-k_2\mathcal{V}{\mathcal{V}}^{\prime }+k_3{\mathcal{V}}^{\prime }=0.\end{array}$$

(9)

Twice integration of (9) with respect to q gives

$$\begin{array}{c}\hfill \frac{1}{2}{k}_1{{\mathcal{V}}^{\prime }}^2-\frac{1}{6}{k}_2{\mathcal{V}}^3+\frac{1}{2}{k}_3{\mathcal{V}}^2+r_1\mathcal{V}+r_2=0\end{array}$$

(10)

with $r_1$ and $r_2$ constants. Equation (10) can be rewritten as

$${\mathcal{V}}^{\prime 2}=\frac{k_2}{3k_1}{\mathcal{V}}^3-\frac{k_3}{k_1}{\mathcal{V}}^2-\frac{2r_1}{k_1}\mathcal{V}-\frac{2r_2}{k_1}.$$

(11)

Assume that $n_1$, $n_2$, and $n_3$ are roots of

$${\mathcal{V}}^3-\frac{3k_3}{k_2}{\mathcal{V}}^2-\frac{6r_1}{k_2}\mathcal{V}-\frac{6r_2}{k_2}=0$$

with $n_1>n_2>n_3$. Equation (11) now becomes

$${\mathcal{V}}^{\prime 2}=\frac{k_2}{3k_1}(\mathcal{V}-n_1)(\mathcal{V}-n_2)(\mathcal{V}-n_3)$$

and its solution could be written in terms of the Jacobi elliptic function [60,61,62]

$$\mathcal{V}\left(q\right)=n_2+(n_1-n_2){\mathrm{cn}}^2\left\{\sqrt{\frac{k_2(n_1-n_3)}{12k_1}}q,R^2\right\},R^2=\frac{n_1-n_2}{n_1-n_3},$$

(12)

where cn is the elliptic cosine function. Integrating Equation (12) with respect to q and returning to the original variables, we accomplish the solution of (1) as

$$\begin{array}{cc}\hfill u(t,x,y)=& \sqrt{\frac{12k_1{\left(n_1-n_2\right)}^2}{k_2(n_1-n_3)R^8}}\left\{\mathrm{EllipticE}\left[\mathrm{sn}\left(\sqrt{\frac{k_2(n_1-n_3)}{12k_1}}q,R^2\right),R^2\right]\right\}\hfill \\ \hfill & +\left\{n_2-\left(n_1-n_2\right)\frac{1-R^4}{R^4}\right\}q+K,\hfill \end{array}$$

(13)

with $q=\left(ac-b\right)t-cx+y$, K being a constant, and $\mathrm{EllipticE}[g,k]$ being the incomplete elliptic integral given as [60]

$$\mathrm{EllipticE}[g,k]={\int }_0^g\sqrt{\frac{1-k^2{s}^2}{1-s^2}}ds.$$

(14)

The wave profile of the periodic solution (13) for parametric values $k_1=70,k_2=10,a=-4,b=0.2,c=0.6,K=1,n_1=100,n_2=50.05,$ and $n_3=-60$ at $t=-14$, can be seen in Figure 1.

2.3. Solution via Kudryashov’s Method

In this subsection, we present the solution of (1), which is obtained by employing Kudryashov’s method [20]. This method is one of the most popular approaches for determining closed-form solutions of NPDEs. The first step is to reduce the NPDE (1) to a nonlinear ODE, which we already performed using the Lie symmetries in the previous section. Thus, we work with the ODE (8). We suppose that a solution of (8) can be expressed as

$$\begin{array}{c}\hfill \mathcal{U}\left(q\right)=\sum _{n=0}^N{A}_n{Y}^n\left(q\right),\end{array}$$

(15)

where $Y\left(q\right)$ satisfies the first-order nonlinear ODE

$$\begin{array}{c}\hfill Y^{\prime }\left(q\right)=Y^2\left(q\right)-Y\left(q\right).\end{array}$$

(16)

We note that the solution of (16) is

$$\begin{array}{c}\hfill Y\left(q\right)=\frac{1}{1+expq}.\end{array}$$

(17)

For Equation (8), the balancing procedure yields $N=1$. Thus, from (15), we have

$$\begin{array}{c}\hfill \mathcal{U}\left(q\right)=A_0+A_{1}Y\left(q\right).\end{array}$$

(18)

Now substituting (18) into (8) and using (16), we obtain

$$\begin{array}{cc}\hfill & 2b{A}_{1}c{Y}^3\left(q\right)-2a{A}_1{c}^2{Y}^3\left(q\right)+3a{A}_1{c}^2{Y}^2\left(q\right)-3b{A}_{1}c{Y}^2\left(q\right)\hfill \\ \hfill & -A_1{c}^{2}Y\left(q\right)a+b{A}_{1}cY\left(q\right)+6\alpha {A_1}^2{c}^2{Y}^5\left(q\right)-15\alpha {A_1}^2{c}^2{Y}^4\left(q\right)\hfill \\ \hfill & +12\alpha {A_1}^2{c}^2{Y}^3\left(q\right)-3\alpha {A_1}^2{c}^2{Y}^2\left(q\right)-2\beta A_{1}c{Y}^3\left(q\right)+3\beta A_{1}c{Y}^2\left(q\right)\hfill \\ \hfill & -\beta A_{1}cY\left(q\right)-24\nu A_1{c}^3{Y}^5\left(q\right)+60\nu A_1{c}^3{Y}^4\left(q\right)-50\nu A_1{c}^3{Y}^3\left(q\right)\hfill \\ \hfill & +15\nu A_1{c}^3{Y}^2\left(q\right)-\nu A_1{c}^{3}Y\left(q\right)=0.\hfill \end{array}$$

Separating on the powers of $Y\left(q\right)$ gives the algebraic equations for the coefficients $A_0$ and $A_1$ as

$$\begin{array}{cc}\hfill & \alpha {A_1}^2{c}^2-4\nu A_1{c}^3=0,\hfill \\ \hfill & bc{A}_1-\nu A_1{c}^3-a{c}^2{A}_1-\beta A_{1}c=0,\hfill \\ \hfill & \beta A_{1}c-\alpha {A_1}^2{c}^2+5\nu A_1{c}^3+a{c}^2{A}_1-bc{A}_1=0,\hfill \\ \hfill & 6\alpha {A_1}^2{c}^2-25\nu A_1{c}^3-a{c}^2{A}_1+bc{A}_1-\beta A_{1}c=0.\hfill \end{array}$$

The solution of these equations is

$$\begin{array}{c}\hfill A_0=A_0,A_1=\frac{4\nu c}{\alpha },a=\frac{b-\beta -c^2\nu }{c}.\end{array}$$

Thus, the solution of (1) reads

$$\begin{array}{c}\hfill u(t,x,y)=A_0+\frac{4\nu c}{\alpha \left(1+exp\left\{\left(-\beta -c^2\nu \right)t-cx+y\right\}\right)}.\end{array}$$

(19)

The wave profile of the solution (19) for parametric values $A_0=0,\alpha =1,\beta =0.05,$ $c=0.01,\nu =1,t=0$ is presented in Figure 2.

Invariant solution using $U_5$

The Lie point symmetry $U_5$ produces the invariants $I_1=t$, $I_2=x$, and $I_3=u-xy/\left(2\alpha t\right)$. The use of these invariants furnishes us with the group invariant solution of (1) given by

$$\begin{array}{c}\hfill u(t,x,y)=\frac{xy}{2\alpha t}-\frac{\beta }{2\alpha }x-\frac{C}{x}+h\left(t\right),\end{array}$$

(20)

where C is a constant and h is a function of t.

Invariant solution using $U_6$

The point symmetry $U_6$ of (1) yields the group invariant solution:

$$\begin{array}{c}\hfill u(t,x,y)=\frac{1}{\sqrt{t}}G(f,g)-\frac{\beta }{2\alpha }x,f=\frac{x}{\sqrt{t}},g=y.\end{array}$$

(21)

The substitution of this expression for u into (1) gives the PDE

$$\begin{array}{c}\hfill 2\nu G_{fffg}+4\alpha G_f{G}_{fg}+2\alpha G_g{G}_{ff}-f{G}_{ff}-2G_f=0,\end{array}$$

(22)

for which the symmetries are

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_1=\frac{\partial }{\partial g},{\mathrm{\Gamma }}_2=\frac{\partial }{\partial G},{\mathrm{\Gamma }}_3=2\alpha \frac{\partial }{\partial f}+g\frac{\partial }{\partial g},{\mathrm{\Gamma }}_4=f\frac{\partial }{\partial f}-2g\frac{\partial }{\partial g}-G\frac{\partial }{\partial G}.\end{array}$$

(23)

The symmetry ${\mathrm{\Gamma }}_3$ gives the invariant solution $G(f,g)=U\left(z\right)+fg/\left(2\alpha \right)$, where $z=g$. This expression for G satisfies the Equation (22) for arbitrary function U. Thus, the group-invariant solution under $U_6$ of (1) is

$$\begin{array}{c}\hfill u(t,x,y)=\frac{xy}{2\alpha t}-\frac{\beta }{2\alpha }x-\frac{U\left(y\right)}{\sqrt{t}},\end{array}$$

(24)

where $U\left(y\right)$ is an arbitrary function of y.

Invariant solution using $U_7$

Here, we use the symmetry operator $U_7$ to obtain the invariant solution associated with this symmetry. The usual procedure gives

$$\begin{array}{c}\hfill u(t,x,y)=\frac{1}{x}U\left(\xi \right)-\frac{\beta }{2\alpha }x,\xi =x^{2}y.\end{array}$$

(25)

Substituting this expression of u into Equation (1) gives the nonlinear ODE

$$\begin{array}{c}\hfill \alpha U^{\prime 2}+6\alpha \xi U^{\prime }{U}^{″}-2\alpha U{U}^{″}+3\nu U^{″}+4\nu {\xi }^2{U}^{{}^{″″}}+12\nu \xi U^{{}^{\prime ″}}=0.\end{array}$$

(26)

Invariant solution using $U_8$

Finally, the use of symmetry $U_8$ gives the invariant solution

$$\begin{array}{c}\hfill u(t,x,y)=\frac{1}{\sqrt{t}}\mathrm{\Phi }(f,g)+\frac{xy}{2\alpha t}-\frac{\beta x}{2\alpha },\end{array}$$

(27)

with $f=x/t^{1/2}$ and $g=y/t$, which when substituted in (1) gives the reduced PDE

$$\begin{array}{c}\hfill \alpha {\mathrm{\Phi }}_{ff}{\mathrm{\Phi }}_g+2\alpha {\mathrm{\Phi }}_f{\mathrm{\Phi }}_{fg}+\nu {\varphi }_{fffg}=0.\end{array}$$

(28)

The above PDE (28) has three Lie symmetries

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_1=\frac{\partial }{\partial f}+F_1\left(g\right)\frac{\partial }{\partial g},{\mathrm{\Gamma }}_2=F_2\left(g\right)\frac{\partial }{\partial g}+\frac{\partial }{\partial \Phi },{\mathrm{\Gamma }}_3=f\frac{\partial }{\partial f}+F_3\left(g\right)\frac{\partial }{\partial g}-\mathrm{\Phi }\frac{\partial }{\partial \mathrm{\Phi }}.\end{array}$$

(29)

Now utilizing ${\mathrm{\Gamma }}_1$ with $F_1\left(g\right)=1$, we perform a reduction of Equation (28). We obtain the invariant solution $\mathrm{\Phi }(f,g)=U\left(z\right)$, where $z=g-f$. This reduces Equation (28) to an ODE

$$\begin{array}{c}\hfill \nu U^{″″}-3\alpha U^{\prime }{U}^{″}=0,\end{array}$$

(30)

for which the solution is

$$\begin{array}{c}\hfill U\left(z\right)=-\sqrt{\frac{2^{8/3}\nu }{{\alpha }^{4/3}}}\zeta \left\{\sqrt{\frac{{\alpha }^{2/3}}{2^{4/3}\nu }}z+C_0;s_1,s_2\right\},s_1=\frac{2^{5/2}}{\sqrt[3]{\alpha }}{C}_0,s_2=2C_1,\end{array}$$

where $C_0,C_1$ are arbitrary constants and $\zeta$ is the Weierstrass zeta function [60]. Thus, the solution of (1) is

$$\begin{array}{c}\hfill u(t,x,y)=-\sqrt{\frac{2^{8/3}\nu }{{\alpha }^{4/3}t}}\zeta \left\{\sqrt{\frac{{\alpha }^{2/3}}{2^{4/3}\nu }}\left(\frac{y}{t}-\frac{x}{\sqrt{t}}\right)+C_0;s_1,s_2\right\}+\frac{xy}{2\alpha t}-\frac{\beta x}{2\alpha }.\end{array}$$

(31)

For the operator ${\mathrm{\Gamma }}_2$ with $F_2\left(g\right)=1$, we obtain the invariant solution $\mathrm{\Phi }(f,g)=U\left(f\right)+g$, which reduces (28) to the ODE $U_{ff}=0$, for which the solution is $U=C_{1}f+C_2$, where $C_1,C_2$ are constants. Reverting to the original variables, we attain the solution of (1) given by

$$\begin{array}{c}\hfill u(t,x,y)=\frac{C_{1}x}{t}+\frac{C_2}{\sqrt{t}}+\frac{y}{t^{3/2}}+\frac{xy}{2\alpha t}-\frac{\beta x}{2\alpha }.\end{array}$$

(32)

The dynamic behaviour of the solution (32), for parametric values $\alpha =10,\beta =5,$ $C_1=0.3,C_2=0.01,$ and $x=0.5$, is sketched in Figure 3.

3. Conservation Laws

We now construct conservation laws for the two-dimensional generalized constant-coefficient shallow water Equation (1). To achieve this task, we utilize two methods; the first one is the multiplier method, and the second is classical Noether’s theorem.

3.1. Conservation Laws Utilizing the Multiplier Method

We determine the conserved vectors of Equation (1) by employing the multiplier method [26]. Here, we seek first-order multipliers $\mathcal{Q}=\mathcal{Q}(t,x,y,u,u_t,u_x,u_y)$, which are established from

$$\begin{array}{c}\hfill \frac{\delta }{\delta u}\left[\mathcal{Q}\left(u_{tx}+\alpha u_y{u}_{xx}+2\alpha u_x{u}_{xy}+\beta u_{xy}+\nu u_{xxxy}\right)\right]=0,\end{array}$$

(33)

where $\delta /\delta u$ is the Euler operator [37]

$$\begin{array}{c}\hfill \frac{\delta }{\delta u}=\frac{\partial }{\partial u}+\sum _{s=1}^{\infty }{(-1)}^s{D}_{i{}_1}\cdots D_{i{}_s}\frac{\partial }{\partial u_{i{}_1\cdots i{}_s}}\end{array}$$

and for our purpose

$$\begin{array}{c}\hfill \frac{\delta }{\delta u}={\partial }_u-D_x{\partial }_{u_x}-D_y{\partial }_{u_y}+D_t{D}_x{\partial }_{u_{tx}}+D_x^2{\partial }_{u_{xx}}+D_x{D}_y{\partial }_{u_{xy}}+D_x^3{D}_y{\partial }_{u_{xxxy}},\end{array}$$

(34)

with $D_t$, $D_x$, and $D_y$ being the total derivative operators given by

$$\begin{array}{cc}\hfill & D_t={\partial }_t+u_t{\partial }_u+u_{tt}{\partial }_{u_t}+u_{tx}{\partial }_{u_x}+u_{ty}{\partial }_{u_y}+\cdots .\hfill \\ \hfill & D_x={\partial }_x+u_x{\partial }_u+u_{tx}{\partial }_{u_t}+u_{xx}{\partial }_{u_x}+u_{xy}{\partial }_{u_y}+\cdots .\hfill \\ \hfill & D_y={\partial }_y+u_y{\partial }_u+u_{ty}{\partial }_{u_t}+u_{xy}{\partial }_{u_x}+u_{yy}{\partial }_{u_y}+\cdots .\hfill \end{array}$$

(35)

Expanding (33) and separating on the appropriate derivatives of u, we obtain twenty-six multiplier-determining equations:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{tx}+\alpha u_x{\mathcal{Q}}_{xy}+\beta {\mathcal{Q}}_{xy}=0,{\mathcal{Q}}_{tu}+\alpha {\mathcal{Q}}_{xy}=0,{\mathcal{Q}}_{t{u}_t}+4{\mathcal{Q}}_u=0,{\mathcal{Q}}_{x{u}_t}=0,{\mathcal{Q}}_{xy}=0,\hfill \\ \hfill & {\mathcal{Q}}_{x{u}_x}-{\mathcal{Q}}_u=0,{\mathcal{Q}}_{t{u}_x}-2\alpha u_y{\mathcal{Q}}_u+\alpha {\mathcal{Q}}_y=0,{\mathcal{Q}}_{t{u}_y}-2\alpha u_x{\mathcal{Q}}_u+2\alpha {\mathcal{Q}}_x-2\beta {\mathcal{Q}}_u=0,\hfill \\ \hfill & {\mathcal{Q}}_{xx}=0,{\mathcal{Q}}_{ux}=0,{\mathcal{Q}}_{yy}=0,{\mathcal{Q}}_{uy}=0,{\mathcal{Q}}_{y{u}_t}=0,{\mathcal{Q}}_{y{u}_x}=0,{\mathcal{Q}}_{y{u}_y}-2{\mathcal{Q}}_u=0,\hfill \\ \hfill & {\mathcal{Q}}_{uu}=0,{\mathcal{Q}}_{u{u}_t}=0,{\mathcal{Q}}_{u{u}_x}=0,{\mathcal{Q}}_{u{u}_y}=0,{\mathcal{Q}}_{u_t{u}_t}=0,{\mathcal{Q}}_{u_t{u}_x}=0,{\mathcal{Q}}_{u_t{u}_y}=0,\hfill \\ \hfill & {\mathcal{Q}}_{u_x{u}_x}=0,{\mathcal{Q}}_{u_x{u}_y}=0,{\mathcal{Q}}_{u_y{u}_y}=0,{\mathcal{Q}}_{x{u}_y}=0,\hfill \end{array}$$

for which the solution is

$$\begin{array}{cc}\hfill \mathcal{Q}=& y{F}^{\prime }\left(t\right)+G\left(t\right)-\alpha u_{x}F\left(t\right)-2K_5\alpha t^2{u}_t-K_5\alpha tx{u}_x-2K_5\alpha ty{u}_y\hfill \\ \hfill & -K_5\alpha tu+2K_6\beta t{u}_y-2K_1\alpha t{u}_y-K_5\beta tx+4K_{6}t{u}_t+K_{6}x{u}_x\hfill \\ \hfill & +2K_{6}y{u}_y+K_{5}xy+K_{6}u+K_2{u}_t+K_3{u}_x+K_4{u}_y+K_{1}x,\hfill \end{array}$$

where F and G are arbitrary functions of t and $K_i$, where $i=1,\cdots ,6$, are constants of integration. The conserved quantities of Equation (1) are constructed using the divergence identity

$$\begin{array}{c}\hfill D_t{T}^t+D_x{T}^x+D_y{T}^y=\mathcal{Q}\left(u_{tx}+\alpha u_y{u}_{xx}+2\alpha u_x{u}_{xy}+\beta u_{xy}+\nu u_{xxxy}\right),\end{array}$$

(36)

where $T^t$ represents the conserved density and $T^x$ and $T^y$ are spatial fluxes. Thus, after some reckoning, we obtain low-order conservation laws corresponding to the eight multipliers and these are given below.

Case 1. For ${\mathcal{Q}}_1=x-2\alpha t{u}_y$, we have the conserved vector

$$\begin{array}{cc}\hfill T_1^t=& \frac{1}{2}\alpha tu{u}_{xy}-\frac{1}{2}\alpha t{u}_x{u}_y+\frac{1}{2}x{u}_x-\frac{1}{2}u,\hfill \\ \hfill T_1^x=& \frac{2}{3}{\alpha }^{2}tu{u}_x{u}_{yy}+\frac{2}{3}{\alpha }^{2}tu{u}_{xy}{u}_y-\frac{4}{3}{\alpha }^{2}t{u}_x{u_y}^2+\frac{3}{4}\alpha \nu tu{u}_{xxyy}+\frac{1}{2}\alpha \beta tu{u}_{yy}-\frac{1}{2}\alpha \nu t{u}_x{u}_{xyy}\hfill \\ \hfill & +\frac{1}{4}\alpha \nu t{u}_{xx}{u}_{yy}-\alpha \nu t{u}_{xxy}{u}_y+\frac{1}{2}\alpha \nu t{u_{xy}}^2-\frac{1}{2}\alpha \beta t{u_y}^2+\frac{1}{2}\alpha tu{u}_{ty}-\frac{1}{2}\alpha t{u}_t{u}_y+\alpha x{u}_x{u}_y\hfill \\ \hfill & +\frac{1}{2}\beta x{u}_y+\frac{3}{4}\nu x{u}_{xxy}-\frac{1}{2}\nu u_{xy}+\frac{1}{2}x{u}_t,\hfill \\ \hfill T_1^y=& -\frac{2}{3}{\alpha }^{2}tu{u}_x{u}_{xy}-\frac{2}{3}{\alpha }^{2}tu{u}_{xx}{u}_y-\frac{2}{3}{\alpha }^{2}t{u_x}^2{u}_y-\frac{3}{4}\alpha \nu tu{u}_{xxxy}-\frac{1}{2}\alpha \beta tu{u}_{xy}-\frac{1}{4}\alpha \nu t{u}_x{u}_{xxy}\hfill \\ \hfill & -\frac{1}{2}\alpha \beta t{u}_x{u}_y+\frac{1}{4}\alpha \nu t{u}_{xx}{u}_{xy}-\frac{1}{4}\alpha \nu t{u}_{xxx}{u}_y-\alpha tu{u}_{tx}+\frac{1}{2}\alpha x{u_x}^2-\frac{1}{2}\alpha u{u}_x+\frac{1}{2}\beta x{u}_x\hfill \\ \hfill & -\frac{1}{2}\beta u+\frac{1}{4}\nu x{u}_{xxx}-\frac{1}{4}\nu u_{xx};\hfill \end{array}$$

Case 2. For ${\mathcal{Q}}_2=u_t$, we have the conserved vector

$$\begin{array}{cc}\hfill T_2^t=& \frac{2}{3}\alpha u{u}_x{u}_{xy}+\frac{1}{3}\alpha u{u}_{xx}{u}_y+\frac{1}{2}\nu u{u}_{xxxy}+\frac{1}{2}\beta u{u}_{xy}+\frac{1}{4}u{u}_{tx}+\frac{1}{4}{u}_t{u}_x,\hfill \\ \hfill T_2^x=& -\frac{3}{8}\nu u{u}_{txxy}+\frac{3}{8}\nu u_t{u}_{xxy}-\frac{1}{4}\nu u_{tx}{u}_{xy}+\frac{1}{8}\nu u_{txx}{u}_y+\frac{1}{4}\nu u_{txy}{u}_x-\frac{1}{8}\nu u_{ty}{u}_{xx}-\frac{1}{3}\alpha u{u}_{tx}{u}_y\hfill \\ \hfill & -\frac{1}{3}\alpha u{u}_{ty}{u}_x-\frac{1}{4}\beta u{u}_{ty}-\frac{1}{4}u{u}_{tt}+\frac{2}{3}\alpha u_t{u}_y{u}_x+\frac{1}{4}\beta u_t{u}_y+\frac{1}{4}{u_t}^2,\hfill \\ \hfill T_2^y=& \frac{1}{3}\alpha u_t{u_x}^2-\frac{1}{3}\alpha u{u}_{tx}{u}_x+\frac{1}{4}\beta u_t{u}_x+\frac{1}{8}\nu u_{txx}{u}_x-\frac{1}{8}\nu u{u}_{txxx}+\frac{1}{8}\nu u_t{u}_{xxx}-\frac{1}{8}\nu u_{tx}{u}_{xx}\hfill \\ \hfill & -\frac{1}{4}\beta u{u}_{tx};\hfill \end{array}$$

Case 3. For ${\mathcal{Q}}_3=u_x$, we have the conserved vector

$$\begin{array}{cc}\hfill T_3^t=& \frac{1}{4}{u_x}^2-\frac{1}{4}u{u}_{xx},\hfill \\ \hfill T_3^x=& \frac{2}{3}\alpha {u_x}^2{u}_y+\frac{1}{3}\alpha u{u}_x{u}_{xy}+\frac{5}{8}\nu u_x{u}_{xxy}+\frac{1}{4}\beta u_x{u}_y+\frac{1}{4}{u}_t{u}_x+\frac{1}{8}\nu u{u}_{xxxy}-\frac{3}{8}\nu u_{xx}{u}_{xy}\hfill \\ \hfill & +\frac{1}{8}\nu u_{xxx}{u}_y+\frac{1}{4}\beta u{u}_{xy}+\frac{1}{4}u{u}_{tx},\hfill \\ \hfill T_3^y=& \frac{1}{3}\alpha {u_x}^3+\frac{1}{4}\beta {u_x}^2-\frac{1}{3}\alpha u{u}_x{u}_{xx}+\frac{1}{4}\nu u_x{u}_{xxx}-\frac{1}{8}\nu u{u}_{xxxx}-\frac{1}{4}\beta u{u}_{xx}-\frac{1}{8}\nu {u_{xx}}^2;\hfill \end{array}$$

Case 4. For ${\mathcal{Q}}_4=u_y$, we have the conserved vector

$$\begin{array}{cc}\hfill T_4^t=& \frac{1}{4}{u}_x{u}_y-\frac{1}{4}u{u}_{xy},\hfill \\ \hfill T_4^x=& \frac{2}{3}\alpha u_x{u_y}^2+\frac{1}{4}\beta {u_y}^2-\frac{1}{3}\alpha u{u}_{xy}{u}_y+\frac{1}{2}\nu u_{xxy}{u}_y+\frac{1}{4}{u}_t{u}_y-\frac{3}{8}\nu u{u}_{xxyy}+\frac{1}{4}\nu u_x{u}_{xyy}\hfill \\ \hfill & -\frac{1}{8}\nu u_{xx}{u}_{yy}-\frac{1}{4}\nu {u_{xy}}^2-\frac{1}{3}\alpha u{u}_x{u}_{yy}-\frac{1}{4}\beta u{u}_{yy}-\frac{1}{4}u{u}_{ty},\hfill \\ \hfill T_4^y=& \frac{1}{3}\alpha u{u}_x{u}_{xy}+\frac{1}{3}\alpha u{u}_{xx}{u}_y+\frac{3}{8}\nu u{u}_{xxxy}+\frac{1}{4}\beta u{u}_{xy}+\frac{1}{2}u{u}_{tx}+\frac{1}{3}\alpha {u_x}^2{u}_y+\frac{1}{8}\nu u_x{u}_{xxy}\hfill \\ \hfill & +\frac{1}{4}\beta u_x{u}_y-\frac{1}{8}\nu u_{xx}{u}_{xy}+\frac{1}{8}\nu u_{xxx}{u}_y;\hfill \end{array}$$

Case 5. For ${\mathcal{Q}}_5=-2\alpha t^2{u}_t-\alpha tx{u}_x-2\alpha ty{u}_y-\alpha tu-\beta tx+xy$, we have the conserved vector

$$\begin{array}{cc}\hfill T_5^t=& -\frac{4}{3}{\alpha }^2{t}^{2}u{u}_x{u}_{xy}-\frac{2}{3}{\alpha }^2{t}^{2}u{u}_y{u}_{xx}-\alpha \beta t^{2}u{u}_{xy}-\alpha \nu t^{2}u{u}_{xxxy}-\frac{1}{2}\alpha t^{2}u{u}_{tx}-\frac{1}{2}\alpha t^2{u}_t{u}_x\hfill \\ \hfill & +\frac{1}{4}\alpha txu{u}_{xx}+\frac{1}{2}\alpha tyu{u}_{xy}-\frac{1}{4}\alpha tx{u}_x^2-\frac{1}{2}\alpha ty{u}_x{u}_y+\frac{1}{4}\alpha tu{u}_x-\frac{1}{2}\beta tx{u}_x+\frac{1}{2}\beta tu\hfill \\ \hfill & +\frac{1}{2}xy{u}_x-\frac{1}{2}{u}_y,\hfill \\ \hfill T_5^x=& \frac{1}{2}\alpha t^{2}u{u}_{tt}-\frac{1}{2}{\beta }^{2}tx{u}_y+\alpha tu{u}_t-\frac{1}{2}\beta tx{u}_t+\frac{1}{2}\beta \nu t{u}_{xy}-\frac{1}{4}\alpha xu{u}_x+\frac{3}{4}\nu xy{u}_{xxy}\hfill \\ \hfill & +\frac{1}{2}\beta xy{u}_y-\frac{1}{3}{\alpha }^{2}txu{u}_x{u}_{xy}+\frac{2}{3}{\alpha }^{2}tyu{u}_x{u}_{yy}+\frac{2}{3}{\alpha }^{2}tyu{u}_{xy}{u}_y-\frac{1}{8}\alpha \nu txu{u}_{xxxy}\hfill \\ \hfill & +\frac{3}{4}\alpha \nu tyu{u}_{xxyy}-\frac{1}{4}\alpha \beta txu{u}_{xy}+\frac{1}{2}\alpha \beta tyu{u}_{yy}-\frac{5}{8}\alpha \nu tx{u}_x{u}_{xxy}-\frac{1}{2}\alpha \nu ty{u}_x{u}_{xyy}\hfill \\ \hfill & -\frac{5}{4}\alpha \beta tx{u}_x{u}_y+\frac{3}{8}\alpha \nu tx{u}_{xx}{u}_{xy}+\frac{1}{4}\alpha \nu ty{u}_{xx}{u}_{yy}-\frac{1}{8}\alpha \nu tx{u}_{xxx}{u}_y-\alpha \nu ty{u}_{xxy}{u}_y\hfill \\ \hfill & -\frac{4}{3}{\alpha }^2{t}^2{u}_t{u}_x{u}_y+\frac{3}{4}\alpha t^2\nu u{u}_{txxy}+\frac{1}{2}\alpha t^2\nu u_{tx}{u}_{xy}+\frac{1}{4}\alpha t^2\nu u_{ty}{u}_{xx}-\frac{1}{4}\alpha t^2\nu u_{txx}{u}_y\hfill \\ \hfill & -\frac{1}{2}\alpha t^2\nu u_{txy}{u}_x-\frac{3}{4}\alpha t^2\nu u_{xxy}{u}_t+\frac{1}{2}\alpha \beta t^{2}u{u}_{ty}-\frac{1}{2}\alpha \beta t^2{u}_t{u}_y-\frac{2}{3}{\alpha }^{2}tx{u_x}^2{u}_y\hfill \\ \hfill & -\frac{4}{3}{\alpha }^{2}ty{u}_x{u_y}^2+{\alpha }^{2}tu{u}_x{u}_y+\frac{1}{2}\alpha \nu ty{u_{xy}}^2-\frac{1}{2}\alpha \beta ty{u_y}^2-\frac{1}{4}\alpha txu{u}_{tx}+\frac{1}{2}\alpha tyu{u}_{ty}\hfill \\ \hfill & +\frac{3}{2}\alpha \nu tu{u}_{xxy}+\alpha \beta tu{u}_y-\frac{1}{4}\alpha tx{u}_t{u}_x-\frac{1}{2}\alpha ty{u}_t{u}_y-\frac{1}{2}\alpha \nu t{u}_x{u}_{xy}-\frac{3}{4}\beta \nu tx{u}_{xxy}\hfill \\ \hfill & +\alpha xy{u}_x{u}_y+\frac{2}{3}{\alpha }^2{t}^{2}u{u}_{tx}{u}_y+\frac{2}{3}{\alpha }^2{t}^{2}u{u}_{ty}{u}_x+\frac{1}{4}{u}^2\alpha +\frac{1}{2}\nu u_x-\frac{1}{2}\alpha t^2{u_t}^2-\frac{1}{4}\nu x{u}_{xx}\hfill \\ \hfill & -\frac{1}{2}\nu y{u}_{xy}+\frac{1}{2}{u}_{t}xy,\hfill \\ \hfill T_5^y=& \frac{1}{3}{\alpha }^{2}txu{u}_x{u}_{xx}-\frac{2}{3}{\alpha }^{2}tyu{u}_x{u}_{xy}-\frac{2}{3}{\alpha }^{2}tyu{u}_{xx}{u}_y+\frac{1}{4}\alpha \beta txu{u}_{xx}+\frac{1}{8}\alpha \nu txu{u}_{xxxx}\hfill \\ \hfill & -\frac{3}{4}\alpha \nu tyu{u}_{xxxy}-\frac{1}{2}\alpha \beta tyu{u}_{xy}-\frac{1}{4}\alpha \nu tx{u}_x{u}_{xxx}-\frac{1}{4}\alpha \nu ty{u}_x{u}_{xxy}-\frac{1}{2}\alpha \beta ty{u}_x{u}_y\hfill \\ \hfill & +\frac{1}{4}\alpha \nu ty{u}_{xx}{u}_{xy}-\frac{1}{4}\alpha \nu ty{u}_{xxx}{u}_y+\frac{2}{3}{\alpha }^2{t}^{2}u{u}_{tx}{u}_x-\frac{1}{4}\alpha \nu t^2{u}_x{u}_{txx}-\frac{1}{2}\alpha \beta t^2{u}_x{u}_t\hfill \\ \hfill & +\frac{1}{4}\alpha t^2\nu u{u}_{txxx}+\frac{1}{4}\alpha t^2\nu u_{xx}{u}_{tx}-\frac{1}{4}\alpha t^2\nu u_{xxx}{u}_t+\frac{1}{2}\alpha \beta u{t}^2{u}_{tx}-\frac{2}{3}{\alpha }^{2}ty{u_x}^2{u}_y\hfill \\ \hfill & -\frac{3}{4}\alpha \beta tx{u_x}^2+\frac{1}{8}\alpha \nu tx{u_{xx}}^2-\alpha tyu{u}_{tx}+\frac{3}{4}\alpha \beta tu{u}_x+\frac{3}{8}\alpha \nu tu{u}_{xxx}-\frac{1}{8}\alpha \nu t{u}_x{u}_{xx}\hfill \\ \hfill & -\frac{1}{4}\beta \nu tx{u}_{xxx}-\frac{1}{4}y\nu u_{xx}-\frac{1}{2}\beta yu+\frac{1}{2}{\beta }^{2}tu-\frac{2}{3}{\alpha }^2{t}^2{u}_t{u_x}^2+\frac{1}{2}\alpha xy{u_x}^2-\frac{1}{2}\alpha yu{u}_x\hfill \\ \hfill & +\frac{1}{2}y\beta x{u}_x+\frac{1}{4}y\nu u_{xxx}x-\frac{1}{3}{\alpha }^{2}tx{u_x}^3+\frac{1}{3}{\alpha }^{2}tu{u_x}^2-\frac{1}{2}{\beta }^{2}tx{u}_x+\frac{1}{4}\beta \nu t{u}_{xx};\hfill \end{array}$$

Case 6. For ${\mathcal{Q}}_6=2\beta t{u}_y+2y{u}_y+4t{u}_t+x{u}_x+u$, we have the conserved vector

$$\begin{array}{cc}\hfill T_6^t=& \frac{8}{3}u{u}_x{u}_{xy}\alpha t+\frac{4}{3}u{u}_{xx}{u}_y\alpha t+2u{u}_{xxxy}\nu t+\frac{3}{2}u{u}_{xy}\beta t+u{u}_{tx}t-\frac{1}{4}u{u}_{xx}x-\frac{1}{2}u{u}_{xy}y\hfill \\ \hfill & -\frac{1}{4}u{u}_x+\frac{1}{2}{u}_x{u}_y\beta t+u_x{u}_{t}t+\frac{1}{4}{u_x}^{2}x+\frac{1}{2}{u}_x{u}_{y}y,\hfill \\ \hfill T_6^x=& \frac{1}{2}\beta y{u_y}^2+\frac{1}{4}{u}_t{u}_{x}x+\frac{1}{2}{u}_t{u}_{y}y+\frac{1}{2}{\beta }^{2}t{u_y}^2-u{u}_{tt}t-\frac{1}{2}\nu y{u_{xy}}^2-\frac{3}{2}\nu u{u}_{xxy}+\frac{1}{2}\nu u_x{u}_{xy}\hfill \\ \hfill & +\frac{1}{4}xu{u}_{tx}-\frac{1}{2}yu{u}_{ty}-\beta u{u}_y+{u_t}^{2}t-u{u}_t+\frac{4}{3}\alpha \beta t{u}_x{u_y}^2-\frac{4}{3}\alpha tu{u}_{tx}{u}_y-\frac{4}{3}\alpha tu{u}_{ty}{u}_x\hfill \\ \hfill & -\frac{3}{4}\beta \nu tu{u}_{xxyy}+\frac{8}{3}\alpha t{u}_t{u}_x{u}_y+\frac{1}{2}\beta \nu t{u}_x{u}_{xyy}-\frac{1}{4}\beta \nu t{u}_{xx}{u}_{yy}+\beta \nu t{u}_{xxy}{u}_y+\frac{1}{3}\alpha xu{u}_x{u}_{xy}\hfill \\ \hfill & -\frac{2}{3}\alpha yu{u}_x{u}_{yy}-\frac{2}{3}\alpha yu{u}_{xy}{u}_y-\frac{2}{3}\alpha \beta tu{u}_x{u}_{yy}-\frac{2}{3}\alpha \beta tu{u}_{xy}{u}_y+\frac{2}{3}\alpha x{u_x}^2{u}_y+\frac{4}{3}\alpha y{u}_x{u_y}^2\hfill \\ \hfill & +\frac{1}{4}\beta x{u}_x{u}_y-\frac{1}{2}{\beta }^{2}tu{u}_{yy}-\frac{1}{2}\beta \nu t{u_{xy}}^2-\frac{3}{2}\nu tu{u}_{txxy}-\frac{3}{2}\beta tu{u}_{ty}+\frac{3}{2}\nu t{u}_t{u}_{xxy}+\frac{3}{2}\beta t{u}_t{u}_y\hfill \\ \hfill & -\nu t{u}_{tx}{u}_{xy}+\frac{1}{2}\nu t{u}_{txx}{u}_y+\nu t{u}_{txy}{u}_x-\frac{1}{2}\nu t{u}_{ty}{u}_{xx}+\frac{1}{8}\nu xu{u}_{xxxy}-\frac{3}{4}\nu yu{u}_{xxyy}\hfill \\ \hfill & +\frac{5}{8}\nu x{u}_x{u}_{xxy}+\frac{1}{2}\nu y{u}_x{u}_{xyy}-\frac{3}{8}\nu x{u}_{xx}{u}_{xy}-\frac{1}{4}\nu y{u}_{xx}{u}_{yy}+\frac{1}{8}\nu x{u}_{xxx}{u}_y+\nu y{u}_{xxy}{u}_y\hfill \\ \hfill & -\alpha u{u}_x{u}_y+\frac{1}{4}\beta xu{u}_{xy}-\frac{1}{2}\beta yu{u}_{yy},\hfill \\ \hfill T_6^y=& \frac{2}{3}\alpha yu{u}_y{u}_{xx}+\frac{2}{3}\alpha \beta t{u_x}^2{u}_y-\frac{4}{3}\alpha tu{u}_{tx}{u}_x-\frac{1}{3}\alpha xu{u}_x{u}_{xx}+\frac{2}{3}\alpha yu{u}_x{u}_{xy}+\frac{1}{4}\beta \nu t{u}_x{u}_{xxy}\hfill \\ \hfill & +\frac{3}{4}\beta \nu tu{u}_{xxxy}-\frac{1}{4}\beta \nu t{u}_{xx}{u}_{xy}+\frac{1}{4}\beta \nu t{u}_{xxx}{u}_y+\frac{2}{3}\alpha \beta tu{u}_y{u}_{xx}+\frac{2}{3}\alpha \beta tu{u}_x{u}_{xy}\hfill \\ \hfill & +\frac{1}{4}\nu y{u}_{xxx}{u}_y+\frac{1}{2}u{\beta }^{2}t{u}_{xy}-\frac{1}{4}u\beta u_{xx}x+\frac{1}{2}u\beta u_{xy}y+\frac{4}{3}\alpha t{u}_t{u_x}^2+\frac{2}{3}\alpha y{u_x}^2{u}_y\hfill \\ \hfill & +\frac{1}{2}{\beta }^{2}t{u}_x{u}_y+\beta t{u}_t{u}_x+\frac{1}{2}\nu t{u}_{txx}{u}_x+\frac{1}{4}\nu x{u}_x{u}_{xxx}+\frac{1}{4}\nu y{u}_x{u}_{xxy}+\frac{1}{2}\beta y{u}_x{u}_y\hfill \\ \hfill & -\frac{1}{2}\nu tu{u}_{txxx}-\frac{1}{8}\nu xu{u}_{xxxx}+\frac{3}{4}\nu yu{u}_{xxxy}+\frac{1}{2}\nu t{u}_t{u}_{xxx}-\frac{1}{2}\nu t{u}_{tx}{u}_{xx}-\frac{1}{4}\nu y{u}_{xx}{u}_{xy}\hfill \\ \hfill & +uy{u}_{tx}-\frac{1}{3}\alpha u{u_x}^2+\frac{1}{4}\beta x{u_x}^2-\frac{1}{4}\beta u{u}_x+\frac{1}{8}\nu u_x{u}_{xx}-\frac{1}{8}\nu x{u_{xx}}^2-\frac{3}{8}\nu u{u}_{xxx}\hfill \\ \hfill & +\frac{1}{3}\alpha x{u_x}^3;\hfill \end{array}$$

Case 7. For ${\mathcal{Q}}_7=y{F}^{\prime }\left(t\right)-\alpha u_{x}F\left(t\right)$, we have the conserved vector

$$\begin{array}{cc}\hfill T_7^t=& -\frac{1}{2}y{u}_x{F}^{\prime }\left(t\right)+\left(\frac{1}{4}\alpha u{u}_{xx}-\frac{1}{4}\alpha u_x^2\right)F\left(t\right),\hfill \\ \hfill T_7^x=& -\frac{1}{2}yu{F}^{″}\left(t\right)+\left(\alpha y{u}_x{u}_y-\frac{1}{4}\alpha u{u}_x+\frac{3}{4}\nu y{u}_{xxy}+\frac{1}{2}\beta y{u}_y-\frac{1}{2}u\beta +\frac{1}{2}{u}_{t}y-\frac{1}{4}\nu u_{xx}\right)F^{\prime }\left(t\right)\hfill \\ \hfill & -\left(\frac{1}{3}{\alpha }^{2}u{u}_x{u}_{xy}+\frac{2}{3}{\alpha }^2{u}_x^2{u}_y+\frac{5}{8}\alpha \nu u_x{u}_{xxy}+\frac{1}{4}\alpha \beta u_x{u}_y+\frac{1}{4}\alpha u_t{u}_x+\frac{1}{4}\alpha \beta u{u}_{xy}\right.\hfill \\ \hfill & \left.+\frac{1}{8}\alpha \nu u{u}_{xxxy}+\frac{1}{4}\alpha u{u}_{tx}-\frac{3}{8}\alpha \nu u_{xx}{u}_{xy}+\frac{1}{8}\alpha \nu u_{xxx}{u}_y\right)F\left(t\right)\hfill \\ \hfill T_7^y=& \left(\frac{1}{2}\alpha y{u}_x^2+\frac{1}{2}\beta y{u}_x+\frac{1}{4}\nu y{u}_{xxxx}\right)F^{\prime }\left(t\right)+\left(\frac{1}{3}{\alpha }^{3}u{u}_x{u}_{xx}-\frac{1}{3}{\alpha }^2{u}_x^3-\frac{1}{4}\alpha \beta u_x^2+\frac{1}{8}\alpha \nu u_{xx}^2\right.\hfill \\ \hfill & \left.-\frac{1}{4}\alpha \nu u_x{u}_{xxx}+\frac{1}{8}\alpha \nu u{u}_{xxxx}+\frac{1}{4}\alpha \beta u{u}_{xx}\right)F\left(t\right);\hfill \end{array}$$

Case 8. For ${\mathcal{Q}}_8=G\left(t\right)$, we have the conserved vector

$$\begin{array}{cc}\hfill T_8^t=& \frac{1}{2}{u}_{x}G\left(t\right),\hfill \\ \hfill T_8^x=& -\frac{1}{2}u{G}^{\prime }\left(t\right)+\left(\alpha u_x{u}_y+\frac{1}{2}\beta u_y+\frac{3}{4}\nu u_{xxy}+\frac{1}{2}{u}_t\right)G\left(t\right),\hfill \\ \hfill T_8^y=& \left(\frac{1}{2}\alpha u_x^2+\frac{1}{2}\beta u_x+\frac{1}{2}\nu u_{xxx}\right)G\left(t\right).\hfill \end{array}$$

3.2. Conservation Laws Utilizing Noether’s Theorem

We now construct conservation laws for Equation (1). We utilize the classical Noether’s theorem [34,43] to achieve this task.

Equation (1) has a second-order Lagrangian

$$\begin{array}{cc}\hfill \mathcal{L}=& \frac{1}{2}\left(\nu u_{xx}{u}_{xy}-u_t{u}_x-\alpha u_x^2{u}_y-\beta u_x{u}_y\right),\hfill \end{array}$$

(37)

since $\delta \mathcal{L}/\delta u=0$ on (1). Here, $\delta /\delta u$ is the Euler operator. Thus, we can determine the Noether symmetries of (1) by employing the Lagrangian (37) on the determining equation

$$\begin{array}{c}\hfill :X^{\left[2\right]}\mathcal{L}+\mathcal{L}\left\{D_t\left(\tau \right)+D_x\left(\xi \right)+D_y\left(\varphi \right)\right\}-D_t\left(G^t\right)-D_x\left(G^x\right)-D_y\left(G^y\right)=0,\end{array}$$

(38)

with gauge functions $G^t$, $G^x$, and $G^y$. Expanding (38) and solving the resultant linear PDEs yield the following Noether symmetries and gauge functions:

$$\begin{array}{ccc}& & X_1=\frac{\partial }{\partial t},G^t=0,G^x=0,G^y=0,\hfill \\ & & X_2=\frac{\partial }{\partial y},G^t=0,G^x=0,G^y=0,\hfill \\ & & X_3=g\left(t\right)\frac{\partial }{\partial u},G^t=0,G^x=-\frac{1}{2}u{g}^{\prime }\left(t\right),G^y=0,\hfill \\ & & X_4=2\alpha t\frac{\partial }{\partial y}+x\frac{\partial }{\partial u},G^t=-\frac{1}{2}u,G^x=0,G^y=-\frac{1}{2}\beta u,\hfill \\ & & X_5=\alpha f\left(t\right)\frac{\partial }{\partial x}+y{f}^{\prime }\left(t\right)\frac{\partial }{\partial u},G^t=0,G^x=-\frac{1}{2}uy{f}^{″}\left(t\right)-\frac{1}{2}\beta u{f}^{\prime }\left(t\right),G^y=0,\hfill \\ & & X_6=4\alpha t\frac{\partial }{\partial t}+\alpha x\frac{\partial }{\partial x}+2\alpha y\frac{\partial }{\partial y}-(\beta x+\alpha u)\frac{\partial }{\partial u},G^t=\frac{1}{2}\beta u,G^x=0,G^y=\frac{1}{2}{\beta }^{2}u.\hfill \end{array}$$

Thus, the conserved vectors corresponding to the above obtained Noether point symmetries, using [41]

$$T^k=\mathcal{L}{\xi }^k+\left({\eta }^{\alpha }-u_{x^j}^{\alpha }{\xi }^j\right)\left(\frac{\partial \mathcal{L}}{\partial u_{x^k}^{\alpha }}-\sum _{l=1}^k{D}_{x^l}\left(\frac{\partial \mathcal{L}}{\partial u_{x^l{x}^k}^{\alpha }}\right)\right)+\sum _{l=k}^n\left({\zeta }_l^{\alpha }-u_{x^l{x}^j}^{\alpha }{\xi }^j\right)\frac{\partial \mathcal{L}}{\partial u_{x^k{x}^l}^{\alpha }}$$

are given by, respectively,

$$\begin{array}{cc}\hfill T_1^t=& -\frac{1}{2}\alpha u_x^2{u}_y-\frac{1}{2}\beta u_x{u}_y+\frac{1}{2}\nu u_{xx}{u}_{xy},\hfill \\ \hfill T_1^x=& \alpha u_t{u}_x{u}_y+\frac{3}{4}\nu u_t{u}_{xxy}-\frac{1}{4}\nu u_{xx}{u}_{ty}-\frac{1}{2}\nu u_{tx}{u}_{xy}+\frac{1}{2}\beta u_t{u}_y+\frac{1}{2}{u}_t^2,\hfill \\ \hfill T_1^y=& \frac{1}{2}\alpha u_t{u}_x^2+\frac{1}{2}\beta u_t{u}_x+\frac{1}{4}\nu u_t{u}_{xxx}-\frac{1}{4}\nu u_{xx}{u}_{tx};\hfill \\ \end{array}$$

$$\begin{array}{cc}\hfill T_2^t=& \frac{1}{2}{u}_x{u}_y,\hfill \\ \hfill T_2^x=& \frac{1}{2}{u}_t{u}_y+\alpha u_x{u}_y^2+\frac{3}{4}\nu u_y{u}_{xxy}-\frac{1}{2}\nu u_{xy}^2-\frac{1}{4}\nu u_{xx}{u}_{yy}+\frac{1}{2}\beta u_y^2,\hfill \\ \hfill T_2^y=& \frac{1}{4}\nu u_{xx}{u}_{xy}+\frac{1}{4}\nu u_{xxx}{u}_y-\frac{1}{2}{u}_t{u}_x;\hfill \\ \\ \hfill T_3^t=& -\frac{1}{2}{u}_{x}g\left(t\right),\hfill \\ \hfill T_3^x=& \left(-\alpha u_x{u}_y-\frac{3}{4}\nu u_{xxy}-\frac{1}{2}\beta u_y-\frac{1}{2}{u}_t\right)g\left(t\right)+\frac{1}{2}u{g}^{\prime }\left(t\right),\hfill \\ \hfill T_3^y=& \left(-\frac{1}{2}\alpha u_x^2-\frac{1}{2}\beta u_x-\frac{1}{4}\nu u_{xxx}\right)g\left(t\right);\hfill \\ \\ \hfill T_4^t=& \frac{1}{2}u-\frac{1}{2}x{u}_x+\alpha t{u}_x{u}_y,\hfill \\ \hfill T_4^x=& \frac{1}{2}\nu u_{xy}-\frac{1}{2}x{u}_t-\alpha x{u}_x{u}_y-\frac{1}{2}\beta x{u}_y-\frac{1}{2}\nu x{u}_{xxy}+\alpha t{u}_t{u}_y+2{\alpha }^{2}t{u}_x{u}_y^2+\alpha \nu t{u}_y{u}_{xxy}\hfill \\ \hfill & -\alpha \nu t{u}_{xy}^2-\alpha \nu t{u}_{xx}{u}_{yy}+\alpha \beta t{u}_y^2,\hfill \\ \hfill T_4^y=& \frac{1}{2}\beta u+\alpha \nu t{u}_y{u}_{xxx}-\frac{1}{2}\nu x{u}_{xxx}-\frac{1}{2}\beta x{u}_x-\frac{1}{2}\alpha x{u}_x^2-\alpha t{u}_t{u}_x+\alpha \nu t{u}_{xx}{u}_{xy};\hfill \\ \\ \hfill T_5^t=& \frac{1}{2}\alpha u_x^{2}f\left(t\right)-\frac{1}{2}y{u}_x{f}^{\prime }\left(t\right),\hfill \\ \hfill T_5^x=& \left(\frac{1}{2}{\alpha }^2{u}_x^2{u}_y+\frac{3}{4}\alpha \nu u_x{u}_{xxy}-\frac{1}{4}\alpha \nu u_{xx}{u}_{xy}\right)f\left(t\right)+\left(-\frac{1}{2}y{u}_t-\alpha y{u}_x{u}_y-\frac{3}{4}\nu y{u}_{xxy}\right.\hfill \\ \hfill & \left.+\frac{1}{4}\nu u_{xx}-\frac{1}{2}\beta y{u}_y+\frac{1}{2}\beta u\right)f^{\prime }\left(t\right)+\frac{1}{2}yu{f}^{\prime \prime }\left(t\right),\hfill \\ \hfill T_5^y=& \left(\frac{1}{2}{\alpha }^2{u}_x^3+\frac{1}{2}\alpha \beta u_x^2+\frac{1}{4}\alpha \nu u_{xxx}{u}_x-\frac{1}{4}\alpha \nu u_{xx}^2\right)f\left(t\right)-\left(\frac{1}{2}\alpha y{u}_x^2+\frac{1}{2}\beta y{u}_x\right.\hfill \\ \hfill & \left.+\frac{1}{4}\nu y{u}_{xxx}\right)f^{\prime }\left(t\right);\hfill \\ \\ \hfill T_6^t=& \frac{1}{2}x\alpha u_x^2-2t{\alpha }^2{u}_y{u}_x^2+\frac{1}{2}x\beta u_x+\frac{1}{2}\alpha u{u}_x+y\alpha u_y{u}_x-2t\alpha \beta u_y{u}_x+2t\alpha \nu u_{xy}{u}_{xx}\hfill \\ \hfill & -\frac{1}{2}\beta u,\hfill \\ \hfill T_6^x=& \frac{1}{2}x{u}_y{u}_x^2{\alpha }^2+2y{u}_y^2{u}_x{\alpha }^2+u{u}_y{u}_x{\alpha }^2+4t{u}_y{u}_x{u}_t{\alpha }^2+y\beta u_y^2\alpha -y\nu u_{xy}^2\alpha +2t{u}_t^2\alpha \hfill \\ \hfill & +\frac{1}{2}\beta u{u}_y\alpha +x\beta u_y{u}_x\alpha -\nu u_x{u}_{xy}\alpha -\frac{3}{4}\nu u_y{u}_{xx}\alpha -\frac{1}{2}y\nu u_{yy}{u}_{xx}\alpha -\frac{1}{4}x\nu u_{xy}{u}_{xx}\alpha \hfill \\ \hfill & +\frac{3}{4}\nu u{u}_{xxy}\alpha +\frac{3}{2}y\nu u_y{u}_{xxy}\alpha +\frac{3}{4}x\nu u_x{u}_{xxy}\alpha +\frac{1}{2}u{u}_t\alpha +y{u}_y{u}_t\alpha +2t\beta u_y{u}_t\alpha \hfill \\ \hfill & +3t\nu u_{xxy}{u}_t\alpha -t\nu u_{xx}{u}_{ty}\alpha -2t\nu u_{xy}{u}_{tx}\alpha +\frac{1}{2}x{\beta }^2{u}_y-\frac{1}{2}\beta \nu u_{xy}+\frac{3}{4}x\beta \nu u_{xxy}\hfill \\ \hfill & +\frac{1}{2}x\beta u_t,\hfill \end{array}$$

$$\begin{array}{cc}\hfill T_6^y=& \frac{1}{2}x{\alpha }^2{u}_x^3+x\alpha \beta u_x^2+\frac{1}{2}{\alpha }^{2}u{u}_x^2+2t{\alpha }^2{u}_t{u}_x^2+\frac{1}{2}x{\beta }^2{u}_x+\frac{1}{2}\alpha \beta u{u}_x-\frac{1}{2}\alpha \nu u_{xx}{u}_x\hfill \\ \hfill & +\frac{1}{4}x\alpha \nu u_{xxx}{u}_x-y\alpha u_t{u}_x+2t\alpha \beta u_t{u}_x-\frac{1}{4}x\alpha \nu u_{xx}^2-\frac{1}{4}\beta \nu u_{xx}+\frac{1}{2}y\alpha \nu u_{xy}{u}_{xx}\hfill \\ \hfill & +\frac{1}{4}x\beta \nu u_{xxx}+\frac{1}{4}\alpha \nu u{u}_{xxx}+\frac{1}{2}y\alpha \nu u_y{u}_{xxx}+t\alpha \nu u_{xxx}{u}_t-t\alpha \nu u_{xx}{u}_{tx}-\frac{1}{2}{\beta }^{2}u.\hfill \end{array}$$

It should be noted that we derived eight conservation laws of (1) by invoking the multiplier method whereas Noether’s theorem yielded six conservation laws. However, we see that only one conservation law is common when using both methods. The conservation laws obtained with the Lagrange multiplier method could be obtained by means of Noether’s theorem using contact transformations. Thus, we conclude that there are benefits and constraints of the different methods used when deriving the conservation laws of nonlinear partial differential equations.

4. Concluding Remarks

In this paper, we investigated the generalized two-dimensional generalized shallow water wave Equation (1). We determined the Lie point symmetries of this equation, performed symmetry reductions, and reduced it to an NLODE (8). This NLODE (8) was integrated directly and its solution was obtained in the form of an incomplete elliptic integral. Moreover, Kudrayshov’s method was employed to obtain the solution of the NLODE (8). These solutions were presented graphically. Finally, we used the multiplier method and Noether’s theorem to derive the conservation laws of (1). The multiplier method yielded eight multipliers, which gave eight local conservation laws for Equation (1), whereas Noether’s theorem provided us with six local conservation laws, which included energy and linear momentum conservation laws.