Symmetry Reductions of the (1 + 1)-Dimensional Broer–Kaup System Using the Generalized Double Reduction Method

Abstract

The generalized theory of the double reduction of systems of partial differential equations (PDEs) based on the association of conservation laws with Lie–Bäcklund symmetries is one of the most effective algorithms for performing symmetry reductions of PDEs. In this article, we apply the theory to a (1 + 1)-dimensional Broer–Kaup (BK) system, which is a pair of nonlinear PDEs that arise in the modeling of the propagation of long waves in shallow water. We find symmetries and construct six local conservation laws of the BK system arising from low-order multipliers. We establish associations between the Lie point symmetries and conservation laws and exploit the association to perform double reductions of the system, reducing it to first-order ordinary differential equations or algebraic equations. Our paper contributes to the broader understanding and application of the generalized double reduction method in the analysis of nonlinear PDEs.

1. Introduction

The phenomenon of shallow water waves is observed in various scenarios, particularly in the study of tsunami wave dynamics. These waves are generally represented through nonlinear PDEs. Numerous mathematical models have been developed to characterize shallow water waves. These include the Korteweg–de Vries (KdV) equation [1], the Boussinesq equation [2], the Degasperis–Procesi equation [3], the Benjamin–Bona–Mahony (BBM) equation [4], and the Kadomtsev–Petviashvili (K-P) equation [5]. The focus of this article is on the BK system, which is a significant model used to analyze the propagation of long waves in shallow water [6]. The BK system we study is a (1 + 1)-dimensional system of two nonlinear PDEs [7,8],

$$\begin{array}{ccc}\hfill & u_t+{\displaystyle \frac{1}{2}}{u}_{xx}-v_x-u{u}_x=0,\hfill & \hfill v_t-{\displaystyle \frac{1}{2}}{v}_{xx}-{\left(uv\right)}_x=0.\end{array}$$

(1)

In the literature, variants of the BK system have been analyzed in several studies using various approaches. For example, in [9], the authors investigate exact solutions of a (2 + 1)-dimensional Broer–Kaup system by applying a substitution that reduces the problem to solving a (2 + 1)-dimensional scalar PDE. They specifically focus on deriving a particular class of solutions, namely traveling wave solutions. In [10], traveling wave solutions to the Broer–Kaup equations are derived using the $\left({\displaystyle \frac{G^{\prime }}{G}}\right)$-expansion method. Then, in [11], the double Wronskian solutions of the Whitham–Broer–Kaup system, a (1 + 1)-dimensional coupled system, are investigated, providing conditions for the generation of non-singular, non-trivial, and irreducible soliton solutions from two double Wronskians. Further, in [12], the classical Whitham–Broer–Kaup equations are studied using the generalized projective Riccati equation method, yielding 20 sets of solutions.

The theory of double reduction for PDEs, which is based on the association of conservation laws with symmetries, is well established [13,14,15,16,17,18]. Building on the foundational work of Kara et al. [19,20,21], Sjöberg [22,23] demonstrated that associating conservation laws with symmetries offers a method for obtaining invariant solutions of PDEs.

For a PDE (or system of PDEs) of order q with two independent variables and m dependent variables, admitting a non-trivial conservation law with at least one associated symmetry, Sjöberg [22,23] proposed a double reduction technique that not only reduces the order of the PDE (or system) to $(q-1)$ but also reduces the number of independent variables by one. More recently, Bokhari et al. [13], as well as Anco and Gandarias [18], extended the double reduction theory to systems involving multiple independent variables.

In this paper, we apply the generalized double reduction theory [13,14] to perform symmetry reductions of the BK system (1). We find six local conservation laws for the system arising from low-order multipliers and establish that each conservation law has associated Lie point symmetries.

The structure of the paper is as follows: Section 2 presents the preliminaries and key components of the generalized double reduction algorithm. In Section 3, we compute the Lie point symmetries and conservation laws of the BK system using the multiplier method, and we establish associations between the symmetries and conservation laws. Section 4 addresses the double reduction of the BK system, analyzing four distinct cases, each corresponding to one of the four symmetries. Finally, Section 5 offers concluding remarks.

2. Preliminaries

Consider a system of N PDEs, where $N>1$, involving n independent variables $x=\left(x_1,x_2,\dots ,x_n\right)$ and m dependent variables $u=\left(u^1,u^2,\dots ,u^m\right)$, which is represented by

$$F^{\mu }\left(x,u,u_{\left(1\right)},\dots ,u_{\left(k\right)}\right)=0,\mu =1,2,\dots ,N,$$

(2)

where $u_{\left(1\right)},u_{\left(2\right)},\dots ,u_{\left(k\right)}$ denotes the collections of all first-, second-, $\dots ,k$ th-order derivatives.

Let

$$X={\xi }_i(x,u){\displaystyle \frac{\partial }{\partial x_i}}+{\eta }^{\nu }(x,u){\displaystyle \frac{\partial }{\partial u^{\nu }}}$$

(3)

be the infinitesimal generator of a Lie group of point transformations for the system (2). The kth extended infinitesimal generator of (3) is given by

$$\begin{array}{cc}\hfill X^{\left(k\right)}=& {\xi }_i(x,u){\displaystyle \frac{\partial }{\partial x_i}}+{\eta }^{\mu }(x,u){\displaystyle \frac{\partial }{\partial u^{\mu }}}+{\eta }_i^{\left(1\right)\mu }\left(x,u,u_{\left(1\right)}\right){\displaystyle \frac{\partial }{\partial u_i^{\mu }}}+\cdots \hfill \\ \hfill & +{\eta }_{i_1{i}_2\cdots i_k}^{\left(k\right)\mu }\left(x,u,u_{\left(1\right)},\dots ,u_{\left(k\right)}\right){\displaystyle \frac{\partial }{\partial u_{i{i}_2\cdots i_k}^{\mu }}},k\ge 1\hfill \end{array}$$

(4)

with the extended infinitesimals ${\eta }_{i{i}_2\cdots i_k}^{\left(k\right)}$ given by

$${\eta }_i^{\left(1\right)\mu }=D_i{\eta }^{\mu }-\left(D_i{\xi }_j\right)u_j^{\mu },$$

and

$${\eta }_{i{i}_2\cdots i_k}^{\left(k\right)\mu }=D_{i_k}{\eta }_{i{i}_2\cdots i_{k-1}}^{(k-1)\mu }-\left(D_{i_k}{\xi }_j\right)u_{i{i}_i\cdots i_{k-1}j}^{\mu },$$

$i_{\mathsf{\ell }}=1,2,\dots ,n$ for $\mathsf{\ell }=1,2,\dots ,k$ with $k\ge 2$, where $D_i$ is the total derivative operator with respect to $x^i$ defined by

$$D_i={\displaystyle \frac{\partial }{\partial x^i}}+u_i^{\mu }{\displaystyle \frac{\partial }{\partial u^{\mu }}}+u_{i{i}_1}^{\mu }{\displaystyle \frac{\partial }{\partial u_{i_1}^{\mu }}}+u_{i{i}_1{i}_2}^{\mu }{\displaystyle \frac{\partial }{\partial u_{i_1{i}_2}^{\mu }}}+\cdots ,i=1,2,\dots ,n.$$

(5)

In (5), $u_i^{\mu }$ represents the first derivative of $u^{\mu }$ with respect to $x^i$, and $u_{ij}^{\mu }$ denotes the second derivative of $u^{\mu }$ with respect to $x^i$ and $x^j$, and so on for higher-order derivatives.

We now present some well-known definitions and results (see, e.g., [13,15,20,24]).

• Conservation Law (from [25]): A conservation law for the PDE system (2) is represented as an expression in divergence form:

  $$D_i\left(C^i\right)=0,$$

  (6)

  which holds for all solutions of the system (2). The functions $C^i=C^i\left(x,u,u_{\left(1\right)},\dots ,u_{\left(r\right)}\right)$, for $i=1,\dots ,n$, are referred to as the fluxes of the conservation law.
• Multiplier (from [26]): A multiplier for the PDE system (2) is a set of non-singular functions:

  $$\mathsf{\Lambda }=\left\{{\mathsf{\Lambda }}_1\left(x,u,u_{\left(1\right)},\dots ,u_{\left(s\right)}\right),\dots ,{\mathsf{\Lambda }}_m\left(x,u,u_{\left(1\right)},\dots ,u_{\left(s\right)}\right)\right\},$$

  (7)

  defined on the solution space that satisfies the condition

  $${\mathsf{\Lambda }}_{\mu }{F}^{\mu }=D_i{C}^i,$$

  (8)

  for some flux functions $C^i$ and arbitrary functions $u\left(x\right)$.
• A set of multipliers (7) generates a local conservation law for the PDE system (2) if and only if the following set of identities holds:

  $${\displaystyle \frac{\delta }{\delta u^{\mu }}}\left({\mathsf{\Lambda }}_{\beta }\left(x,u,u_{\left(1\right)},\dots ,u_{\left(s\right)}\right)F^{\beta }\left(x,u,u_{\left(1\right)},\dots ,u_{\left(r\right)}\right)\right)=0,\mu =1,2,\dots ,N,$$

  (9)

  for arbitrary functions $u=u\left(x\right)$. Here, the Euler operator ${\displaystyle \frac{\delta }{\delta u^{\mu }}}$, for each $\mu$, is defined as

  $${\displaystyle \frac{\delta }{\delta u^{\mu }}}={\displaystyle \frac{\partial }{\partial u^{\mu }}}+\sum _{s=1}^{\infty }{(-1)}^s{D}_{i_1}\cdots D_{i_s}{\displaystyle \frac{\partial }{\partial u_{i_1\cdots i_s}^{\mu }}}.$$

  (10)
  The system of equations in (9) is an over-determined linear PDE system, and its solutions are the desired multipliers.
• Lie–Bäcklund Symmetry Generator: A Lie–Bäcklund symmetry generator X, given by (4), is said to be associated with a conserved vector C for the system (2) if X and C satisfy the condition:

  $$X\left(C^i\right)+C^i{D}_j{\xi }^j-C^j{D}_j{\xi }^i=0,i=1,\dots ,n.$$

  (11)

Theorem 1. 

Suppose $D_i{C}^i=0$ is a conservation law of the PDE system (2). Then under a similarity transformation of a symmetry X of the form (3) for the PDE, there exist functions ${\tilde{C}}^i$ such that X is still symmetry for the PDE ${\tilde{D}}_i{\tilde{C}}^i=0$, where ${\tilde{C}}^i$ is given by

$$\left(\begin{array}{c}{\tilde{C}}^1\\ {\tilde{C}}^2\\ ·\\ ·\\ ·\\ {\tilde{C}}^n\end{array}\right)=J{\left(A^{-1}\right)}^T\left(\begin{array}{c}{C}^1\\ C^2\\ ·\\ ·\\ ·\\ C^n\end{array}\right),$$

(12)

where

$$A=\left(\begin{array}{cccc}{\tilde{D}}_1{x}_1& {\tilde{D}}_1{x}_2& \dots & {\tilde{D}}_1{x}_n\\ {\tilde{D}}_2{x}_1& {\tilde{D}}_2{x}_2& \dots & {\tilde{D}}_2{x}_n\\ ⋮& ⋮& ⋮& ⋮\\ {\tilde{D}}_n{x}_1& {\tilde{D}}_n{x}_2& \dots & {\tilde{D}}_n{x}_n\end{array}\right),A^{-1}=\left(\begin{array}{cccc}{D}_1{\tilde{x}}_1& D_1{\tilde{x}}_2& \dots & D_1{\tilde{x}}_n\\ D_2{\tilde{x}}_1& D_2{\tilde{x}}_2& \dots & D_2{\tilde{x}}_n\\ ⋮& ⋮& ⋮& ⋮\\ D_n{\tilde{x}}_1& D_n{\tilde{x}}_2& \dots & D_n{\tilde{x}}_n\end{array}\right),$$

and $J=\det \left(A\right).$

Colollary 1. 

(The necessary and sufficient condition for reduced conserved form [13]). The conserved form $D_i{C}^i=$ 0 of the PDE system (2) can be reduced under a similarity transformation of a symmetry X to a reduced conserved form ${\tilde{D}}_i{\tilde{C}}^i=0$ if and only if X is associated with the conservation law C.

Colollary 2. 

(see [13]). A nonlinear system of qth-order PDEs with n independent and m dependent variables which admits a nontrivial conserved form that has at least one associated symmetry in every reduction from the n reductions (the first step of double reduction) can be reduced to a $(q-1)$ th-order nonlinear system of ODEs.

Colollary 3. 

(The Inherited Symmetries [13]). Any symmetry Y for the conserved form $D_i{C}^i=0$ of PDE system (2) can be transformed under the similarity transformation of a symmetry X for the PDE to the symmetry $\tilde{Y}$ for the PDE ${\tilde{D}}_i{\tilde{C}}^i=0$.

3. Lie Point Symmetries and Conservation Laws of the BK System (1)

Many software packages are available for computing Lie symmetries of differential equations [27,28,29,30,31,32,33]. Using the mathematical software Maple, the following Lie point symmetries of the BK system (1) were obtained:

$$\begin{array}{cc}\hfill X_1& ={\displaystyle \frac{\partial }{\partial t}}\hfill \\ \hfill X_2& =2t{\displaystyle \frac{\partial }{\partial t}}+x{\displaystyle \frac{\partial }{\partial x}}-u{\displaystyle \frac{\partial }{\partial u}}-2v{\displaystyle \frac{\partial }{\partial v}}\hfill \\ \hfill X_3& ={\displaystyle \frac{\partial }{\partial x}}\hfill \\ \hfill X_4& =t{\displaystyle \frac{\partial }{\partial x}}-{\displaystyle \frac{\partial }{\partial u}}.\hfill \end{array}$$

(13)

While a more general framework exists on relationships between symmetries and conservation laws for non-Lagrangian equations (see, e.g., [20,34,35]), in this article, we adopt the multiplier method. Specifically, we investigate the conservation laws of the system (1) arising from first-order multipliers:

$$\begin{array}{cc}\hfill & {\mathsf{\Lambda }}_1={\mathsf{\Lambda }}_1\left(t,x,u,v,u_t,u_x,v_t,v_x\right),\hfill \\ \hfill & {\mathsf{\Lambda }}_2={\mathsf{\Lambda }}_2\left(t,x,u,v,u_t,u_x,v_t,v_x\right).\hfill \end{array}$$

(14)

For the two independent variables $(t,x)$ and two dependent variables $(u,v)$ in the BK system (1), Equation (8) takes the following form:

$${\mathsf{\Lambda }}_1\left(u_t+{\displaystyle \frac{1}{2}}{u}_{xx}-v_x-u{u}_x\right)+{\mathsf{\Lambda }}_2\left(v_t-{\displaystyle \frac{1}{2}}{v}_{xx}-{\left(uv\right)}_x\right)=D_t{C}^t+D_x{C}^x,$$

(15)

where $D_t$ and $D_x$ are the total derivative operators defined in (5).

The determining equations for the multipliers ${\mathsf{\Lambda }}_1$ and ${\mathsf{\Lambda }}_2$ are then given by

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\delta }{\delta u}}\left[{\mathsf{\Lambda }}_1\left(u_t+{\displaystyle \frac{1}{2}}{u}_{xx}-v_x-u{u}_x\right)+{\mathsf{\Lambda }}_2\left(v_t-{\displaystyle \frac{1}{2}}{v}_{xx}-{\left(uv\right)}_x\right)\right]=0,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\delta }{\delta v}}\left[{\mathsf{\Lambda }}_1\left(u_t+{\displaystyle \frac{1}{2}}{u}_{xx}-v_x-u{u}_x\right)+{\mathsf{\Lambda }}_2\left(v_t-{\displaystyle \frac{1}{2}}{v}_{xx}-{\left(uv\right)}_x\right)\right]=0,\hfill \end{array}$$

(17)

where the Euler operators ${\displaystyle \frac{\delta }{\delta u}}$ and ${\displaystyle \frac{\delta }{\delta v}}$ are given by (10). Solving the determining Equations (16) and (17), we obtain

$$\begin{array}{cc}\hfill {\mathsf{\Lambda }}_1& ={\displaystyle \frac{1}{2}}{c}_1{u}^{2}v+{\displaystyle \frac{c_{1}u{v}_x}{6}}-{\displaystyle \frac{c_1{u}_{x}v}{3}}+{\displaystyle \frac{c_1{v}^2}{2}}+{\displaystyle \frac{c_1{v}_t}{3}}+c_{2}uv+{\displaystyle \frac{c_2{v}_x}{2}}+c_{3}tv+c_{4}v+c_6,\hfill \\ \hfill {\mathsf{\Lambda }}_2& ={\displaystyle \frac{c_1{u}^3}{6}}-{\displaystyle \frac{c_{1}u{u}_x}{6}}+c_{1}uv-{\displaystyle \frac{c_1{u}_t}{3}}+{\displaystyle \frac{c_1{v}_x}{3}}+{\displaystyle \frac{c_2{u}^2}{2}}-{\displaystyle \frac{c_2{u}_x}{2}}+c_{2}v+c_{3}tu+c_{3}x\hfill \\ \hfill & +c_{4}u+c_5,\hfill \end{array}$$

(18)

where $c_i,i=1,\dots ,6$ are arbitrary constants.

The solution set (18) of the determining Equations (16) and (17) represents a six-parameter family of multipliers. As is customary, we assign a value of one to each parameter individually while setting the others to zero. This yields six distinct sets of multipliers, namely,

$$\begin{array}{ccc}\hfill {\mathsf{\Lambda }}^1& :& \left\{\begin{array}{cc}\hfill {\mathsf{\Lambda }}_1^1& ={\displaystyle \frac{u^{2}v}{2}}+{\displaystyle \frac{u{v}_x}{6}}-{\displaystyle \frac{u_{x}v}{3}}+{\displaystyle \frac{v^2}{2}}+{\displaystyle \frac{v_t}{3}},\hfill \\ \hfill {\mathsf{\Lambda }}_2^1& ={\displaystyle \frac{u^3}{6}}-{\displaystyle \frac{u{u}_x}{6}}+uv-{\displaystyle \frac{u_t}{3}}+{\displaystyle \frac{v_x}{3}},\hfill \end{array}\right.\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill {\mathsf{\Lambda }}^2& :& \left\{\begin{array}{cc}\hfill {\mathsf{\Lambda }}_1^2& =uv+{\displaystyle \frac{v_x}{2}},\hfill \\ \hfill {\mathsf{\Lambda }}_2^2& ={\displaystyle \frac{u^2}{2}}-{\displaystyle \frac{u_x}{2}}+v,\hfill \end{array}\right.\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill {\mathsf{\Lambda }}^3& :& \left\{\begin{array}{cc}\hfill {\mathsf{\Lambda }}_1^3& =tv,\hfill \\ \hfill {\mathsf{\Lambda }}_2^3& =tu+x,\hfill \end{array}\right.\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill {\mathsf{\Lambda }}^4& :& \left\{\begin{array}{cc}\hfill {\mathsf{\Lambda }}_1^4& =v,\hfill \\ \hfill {\mathsf{\Lambda }}_2^4& =u,\hfill \end{array}\right.\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\mathsf{\Lambda }}^5& :& \left\{\begin{array}{cc}\hfill {\mathsf{\Lambda }}_1^5& =0,\hfill \\ \hfill {\mathsf{\Lambda }}_2^5& =1,\hfill \end{array}\right.\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\mathsf{\Lambda }}^6& :& \left\{\begin{array}{cc}\hfill {\mathsf{\Lambda }}_1^6& =1,\hfill \\ \hfill {\mathsf{\Lambda }}_2^6& =0.\hfill \end{array}\right.\hfill \end{array}$$

(24)

The conserved vectors of (1) corresponding to the multipliers (19)–(24) are

$$\begin{array}{ccc}\hfill C_1& :& \left\{\begin{array}{cc}\hfill C_1^t& =-{\displaystyle \frac{1}{2}}t{v}^2{v}_x+{\displaystyle \frac{u^{3}v}{6}}+{\displaystyle \frac{u^2{v}_x}{4}}+{\displaystyle \frac{u{v}^2}{2}}-{\displaystyle \frac{u_x{v}_x}{6}},\hfill \\ \hfill C_1^x& ={\displaystyle \frac{1}{2}}t{v}^2{v}_t-{\displaystyle \frac{u^{4}v}{6}}-{\displaystyle \frac{u^3{v}_x}{12}}+{\displaystyle \frac{1}{4}}{u}^2{u}_{x}v-{\displaystyle \frac{3u^2{v}^2}{4}}-{\displaystyle \frac{u^2{v}_t}{4}}+{\displaystyle \frac{u{u}_x{v}_x}{12}}\hfill \\ & -{\displaystyle \frac{uv{v}_x}{2}}+{\displaystyle \frac{u_t{v}_x}{6}}-{\displaystyle \frac{{u_x}^{2}v}{12}}+{\displaystyle \frac{u_x{v}^2}{4}}+{\displaystyle \frac{u_x{v}_t}{6}}-{\displaystyle \frac{{v_x}^2}{12}},\hfill \end{array}\right.\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill C_2& :& \left\{\begin{array}{cc}\hfill C_2^t& ={\displaystyle \frac{u^{2}v}{2}}+{\displaystyle \frac{u{v}_x}{2}}+{\displaystyle \frac{v^2}{2}},\hfill \\ \hfill C_2^x& =-{\displaystyle \frac{u^{3}v}{2}}-{\displaystyle \frac{u^2{v}_x}{4}}+{\displaystyle \frac{u{u}_{x}v}{2}}-u{v}^2-{\displaystyle \frac{u{v}_t}{2}}+{\displaystyle \frac{u_x{v}_x}{4}}-{\displaystyle \frac{v{v}_x}{2}},\hfill \end{array}\right.\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill C_3& :& \left\{\begin{array}{cc}\hfill C_3^t& =-{\displaystyle \frac{1}{2}}{t}^{2}v{v}_x+tuv+{\displaystyle \frac{t{v}_x}{2}}+vx,\hfill \\ \hfill C_3^x& ={\displaystyle \frac{1}{2}}{t}^{2}v{v}_t-t{u}^{2}v-{\displaystyle \frac{tu{v}_x}{2}}+{\displaystyle \frac{t{u}_{x}v}{2}}-{\displaystyle \frac{t{v}_t}{2}}-uvx-{\displaystyle \frac{v_{x}x}{2}},\hfill \end{array}\right.\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill C_4& :& \left\{\begin{array}{cc}\hfill C_4^t& =uv-tv{v}_x,\hfill \\ \hfill C_4^x& =tv{v}_t-u^{2}v-{\displaystyle \frac{u{v}_x}{2}}+{\displaystyle \frac{u_{x}v}{2}},\hfill \end{array}\right.\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill C_5& :& \left\{\begin{array}{cc}\hfill C_5^t& =v,\hfill \\ \hfill C_5^x& =-uv-{\displaystyle \frac{v_x}{2}},\hfill \end{array}\right.\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill C_6& :& \left\{\begin{array}{cc}\hfill C_6^t& =u-t{v}_x,\hfill \\ \hfill C_6^x& =t{v}_t-{\displaystyle \frac{u^2}{2}}+{\displaystyle \frac{u_x}{2}}.\hfill \end{array}\right.\hfill \end{array}$$

(30)

The associations between the symmetries (13) and the conservation laws (25)–(30) are established using (11) and are presented in

Table 1. Association of symmetries and conservation laws.

	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$	${\mathit{C}}_5$	${\mathit{C}}_6$
$X_1$		✓			✓
$X_2$			✓			✓
$X_3$	✓	✓		✓	✓	✓
$X_4$			✓		✓

. In this table, a tick below a conservation law $C_i$ and to the right of a symmetry $X_j$ indicates an association between the symmetry and the conservation law.

4. Double Reduction of the BK System (1)

Suppose a symmetry

$$X={\xi }^1{\displaystyle \frac{\partial }{\partial t}}+{\xi }^2{\displaystyle \frac{\partial }{\partial x}}+{\eta }^1{\displaystyle \frac{\partial }{\partial u}}+{\eta }^2{\displaystyle \frac{\partial }{\partial v}}$$

(31)

of the BK system (1) is associated with the conservation law of the form

$$C=\left(C^t,C^x\right)$$

(32)

according to (11). The operator X has the form $\partial /\partial s$ under canonical variables

$$r=r(t,x),s=s(t,x),w\left(r\right)=u,k\left(r\right)=v,$$

(33)

obtained from the characteristic equations

$${\displaystyle \frac{dt}{{\xi }^1}}={\displaystyle \frac{dx}{{\xi }^2}}={\displaystyle \frac{du}{{\eta }^1}}={\displaystyle \frac{dv}{{\eta }^2}}.$$

(34)

Suppose the inverse canonical coordinates of (33) are given by

$$t=t(r,s),s=s(r,s),u=w,v=k.$$

(35)

According to Theorem 1, the conservation law (32) is transformed under the canonical variable (33) into $\tilde{C}=\left(C^r,C^s\right)$ as follows:

$$\left(\begin{array}{c}{C}^r\\ C^s\end{array}\right)=J{\left(A^{-1}\right)}^T\left(\begin{array}{c}{C}^t\\ C^x\end{array}\right),$$

(36)

where

$$A=\left(\begin{array}{cc}{D}_{r}t& D_{r}x\\ D_{s}t& D_{s}x\end{array}\right),A^{-1}=\left(\begin{array}{cc}{D}_{t}r\hfill & D_{t}s\hfill \\ D_{x}r\hfill & D_{x}s\hfill \end{array}\right),$$

(37)

and

$$J=\det \left(A\right).$$

(38)

The reduced conservation law is then

$$D_r\left(C^r\right)=0.$$

(39)

4.1. Case 1: Reduction of (1) Using $X_1$

The symmetry $X_1$ is associated with the conservation laws $C_2$ and $C_5$, and it has a canonical form ${\tilde{X}}_1=\partial /\partial s$ when

$${\displaystyle \frac{dt}{1}}={\displaystyle \frac{dx}{0}}={\displaystyle \frac{du}{0}}={\displaystyle \frac{dv}{0}}={\displaystyle \frac{dr}{0}}={\displaystyle \frac{ds}{1}}={\displaystyle \frac{dw}{0}}={\displaystyle \frac{dq}{0}}.$$

(40)

The characteristic Equation (40) yields the following canonical variables:

$$r=x,s=t,w\left(r\right)=u,q\left(r\right)=v.$$

(41)

The inverse canonical coordinates are given by

$$t=s,x=r,u=w,v=q.$$

(42)

The first-order partial derivatives $u_x$, $v_x$, and $v_t$ in terms of the canonical coordinates (41) are given by

$$u_x=w_r,v_t=0,v_x=q_r.$$

(43)

According to Theorem 1, a conservation law $C=\left(C^t,C^x\right)$ is transformed under the canonical variable (41) into

$$\left(\begin{array}{c}{C}^r\\ C^s\end{array}\right)=J{\left(A^{-1}\right)}^T\left(\begin{array}{c}{C}^t\\ C^x\end{array}\right),$$

(44)

where

$$A=\left(\begin{array}{cc}{D}_{r}t\hfill & D_{r}x\hfill \\ D_{s}t\hfill & D_{s}x\hfill \end{array}\right)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),$$

$${\left(A^{-1}\right)}^T=\left(\begin{array}{cc}{D}_{t}r& D_{x}r\\ D_{t}s& D_{x}s\end{array}\right)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),$$

and

$$J=\det \left(A\right)=-1.$$

Therefore, under the canonical variable (41), the conserved vectors $C_2$ and $C_5$ become

$$\begin{array}{ccc}\hfill \widetilde{C_2}& :& \left\{\begin{array}{cc}\hfill C_2^r& ={\displaystyle \frac{1}{4}}(2qw+q_r)\left(2q+w^2-w_r\right),\hfill \\ \hfill C_2^s& =-{\displaystyle \frac{1}{2}}\left(q^2+q{w}^2+w{q}_r\right),\hfill \end{array}\right.\hfill \end{array}$$

(45)

$$\begin{array}{ccc}\hfill \widetilde{C_5}& :& \left\{\begin{array}{cc}\hfill C_5^r& =qw+{\displaystyle \frac{q_r}{2}},\hfill \\ \hfill C_5^s& =-q,\hfill \end{array}\right.\hfill \end{array}$$

(46)

and the reduced conservation laws $D_r{C}_2^r=0$ and $D_r{C}_5^r=0$ lead to

$$\begin{array}{cc}\hfill (2qw+q_r)\left(2q+w^2-w_r\right)& =d_1,\hfill \\ \hfill 2qw+q_r& =d_2,\hfill \end{array}$$

(47)

respectively, where $d_1$ and $d_2$ are arbitrary constants.

If we set $d_1=0$ in Equation (47) and solve for q in terms $w,$ we obtain

$$q={\displaystyle \frac{1}{2}}\left(w_r-w^2\right).$$

(48)

Substituting (48) in the second equation of (47), we obtain the second-order ordinary differential equation (ODE)

$$w_{rr}=2\left(d_2+w^3\right).$$

(49)

Using the fact that in the case $d_2=0,$ Equation (49) admits the two symmetries

$$Z_1={\partial }_r\mathrm{and}{Z}_2=r{\partial }_r-w{\partial }_w,$$

(50)

we use the method of differential invariants [36] to reduce (49) in the case $d_2=0$ to the first-order ODE

$$w_r=w^2,$$

(51)

the solution of which is

$$w^2={\displaystyle \frac{1}{{(\kappa +r)}^2}},$$

(52)

or

$$w=\pm {\displaystyle \frac{1}{\kappa +r}},$$

(53)

where $\kappa$ is an arbitrary constant. The following solution of the BK system (1) follows from (53), (48) and (41):

$$\begin{array}{ccc}\hfill u(t,x)& =& \pm {\displaystyle \frac{1}{\kappa +x}},\hfill \end{array}$$

(54)

$$\begin{array}{ccc}\hfill v(t,x)& =& {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial u(t,x)}{\partial x}}-u{(t,x)}^2\right).\hfill \end{array}$$

(55)

4.2. Case 2: Reduction of (1) Using $X_2$

In this case, the symmetry $X_2$ is associated with the conservation laws $C_3$ and $C_6$. Canonical variables resulting from $X_2$ are

$$r={\displaystyle \frac{x}{\sqrt{t}}},s={\displaystyle \frac{lnt}{2}},w\left(r\right)=u\sqrt{t},q\left(r\right)=tv,$$

(56)

for which the inverse canonical coordinates are

$$t=e^{2s},x=r{e}^s,u=-e^{-s}w,v=q{e}^{-2s}.$$

(57)

The first-order partial derivatives $u_x$, $v_x$, and $v_t$ in terms of the canonical variables are

$$u_x=-e^{-2s}{w}_r,v_t=-e^{-4s}(2q+q_{r}r)/2,v_x=q_r{e}^{-3s}.$$

(58)

Utilizing Theorem 1, the conserved vectors $C_3$ and $C_6$ transform under the canonical variables (56) as follows:

$$\begin{array}{ccc}\hfill \widetilde{C_3}& :& \left\{\begin{array}{cc}\hfill C_3^r& =q^2+q{r}^2-3qrw+2q{w}^2+q{w}_r-q+q_{r}r-q_{r}w,\hfill \\ \hfill C_3^s& ={\displaystyle \frac{q{q}_r}{2}}-qr+qw-{\displaystyle \frac{q_r}{2}}.\hfill \end{array}\right.\hfill \end{array}$$

(59)

$$\begin{array}{ccc}\hfill \widetilde{C_6}& :& \left\{\begin{array}{cc}\hfill C_6^r& =2q-rw+w^2+w_r,\hfill \\ \hfill C_6^s& =q_r+w.\hfill \end{array}\right.\hfill \end{array}$$

(60)

Therefore, from the reduced conservation laws $D_r{C}_3^r=0$ and $D_r{C}_6^r=0$, we obtain the following equations for w and q:

$$\begin{array}{cc}\hfill q{q}_r-2qr+2qw-q_r& =d_1,\hfill \\ \hfill 2q-rw+w^2+w_r& =d_2,\hfill \end{array}$$

(61)

where $d_1$ and $d_2$ are arbitrary constants. It follows from (61) that w is a solution of the second-order ODE

$$\begin{array}{cc}\hfill & d_{2}w-d_{2}r-{\displaystyle \frac{1}{4}}{d}_2{w}_{rr}-{\displaystyle \frac{1}{2}}{d}_{2}w{w}_r+{\displaystyle \frac{1}{4}}{d}_{2}r{w}_r+{\displaystyle \frac{1}{4}}{w}^2{w}_{rr}-{\displaystyle \frac{1}{4}}rw{w}_{rr}+{\displaystyle \frac{w_{rr}}{2}}\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}{r}^{2}w{w}_r+{\displaystyle \frac{1}{2}}{w}^3{w}_r-{\displaystyle \frac{3}{4}}r{w}^2{w}_r+{\displaystyle \frac{1}{2}}w{w}_r^2-{\displaystyle \frac{1}{4}}r{w}_r^2+{\displaystyle \frac{1}{2}}r{w}_r\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}{w}_r{w}_{rr}-r^{2}w-w^3+2r{w}^2=d_1,\hfill \end{array}$$

(62)

and q is given by

$$q\left(r\right)={\displaystyle \frac{1}{2}}\left(d_2+rw-w^2-w_r\right).$$

(63)

Equation (62) is a highly nonlinear ODE and does not admit any Lie point symmetries. An analytic solution via symmetry methods is unlikely, and only numerical approaches may be viable.

We conclude that a solution of the BK system (1) arising from $X_2$ via $C_3$ and $C_6$ is

$$\begin{array}{ccc}\hfill u(t,x)& =& {\displaystyle \frac{1}{\sqrt{t}}}w\left(x/\sqrt{t}\right),\hfill \end{array}$$

(64)

$$\begin{array}{ccc}\hfill v(t,x)& =& {\displaystyle \frac{1}{t}}q\left(x/\sqrt{t}\right),\hfill \end{array}$$

(65)

where w is a solution of (62) and q is qiven by (63).

4.3. Case 3: Reduction of (1) Using $X_3$

The symmetry $X_3$ is associated with all the conservation laws of the BK system (1) except for $C_3$. The operator $X_3$ yields the following canonical variables:

$$r=t,s=x,w\left(r\right)=u,q\left(r\right)=v,$$

(66)

for which the inverse canonical coordinates are

$$t=r,x=s,u=w,v=q.$$

(67)

The first-order partial derivatives $u_x$, $u_t$, $v_x$, and $v_t$ in terms of these canonical variables are

$$u_t=w_r,u_x=0,v_t=q_r,v_x=0.$$

(68)

Applying Theorem 1 with these canonical variables (66), the conserved vectors $C_1$, $C_2$, $C_4$, $C_5$, and $C_6$ reduce as follows:

$$\begin{array}{ccc}\hfill \widetilde{C_1}& :& \left\{\begin{array}{cc}\hfill C_1^r& ={\displaystyle \frac{q^{2}w}{2}}+{\displaystyle \frac{q{w}^3}{6}},\hfill \\ \hfill C_1^s& ={\displaystyle \frac{1}{2}}{q}^2{q}_{r}r-{\displaystyle \frac{3q^2{w}^2}{4}}-{\displaystyle \frac{q{w}^4}{6}}-{\displaystyle \frac{q_r{w}^2}{4}}.\hfill \end{array}\right.\hfill \end{array}$$

(69)

$$\begin{array}{ccc}\hfill \widetilde{C_2}& :& \left\{\begin{array}{cc}\hfill C_2^r& ={\displaystyle \frac{q^2}{2}}+{\displaystyle \frac{q{w}^2}{2}},\hfill \\ \hfill C_2^s& =-q^{2}w-{\displaystyle \frac{q{w}^3}{2}}-{\displaystyle \frac{q_{r}w}{2}},\hfill \end{array}\right.\hfill \end{array}$$

(70)

$$\begin{array}{ccc}\hfill \widetilde{C_4}& :& \left\{\begin{array}{cc}\hfill C_4^r& =qw,\hfill \\ \hfill C_4^s& =q{q}_{r}r-q{w}^2,\hfill \end{array}\right.\hfill \end{array}$$

(71)

$$\begin{array}{ccc}\hfill \widetilde{C_5}& :& \left\{\begin{array}{cc}\hfill C_5^r& =q,\hfill \\ \hfill C_5^s& =-qw,\hfill \end{array}\right.\hfill \end{array}$$

(72)

$$\begin{array}{ccc}\hfill \widetilde{C_6}& :& \left\{\begin{array}{cc}\hfill C_6^r& =w,\hfill \\ \hfill C_6^s& =q_{r}r-{\displaystyle \frac{w^2}{2}},\hfill \end{array}\right.\hfill \end{array}$$

(73)

From the reduced conservation laws $D_r{C}_i^r=0,$$i=1,2,4,5,6$, we obtain the following equations for w and q:

$$\begin{array}{cc}\hfill 3q^{2}w+q{w}^3& =d_1,\hfill \\ \hfill q^2+q{w}^2& =d_2,\hfill \\ \hfill qw& =d_4,\hfill \\ \hfill q& =d_5,\hfill \\ \hfill w& =d_6.\hfill \end{array}$$

(74)

where $d_1$, $d_2$, $d_4$, $d_5$, and $d_6$ are arbitrary constants. The only solution of the BK system (1) in this case is the trivial solution in which both u and v are constant.

4.4. Case 4: Reduction of (1) Using $X_4$

The symmetry $X_4$ is associated with the conservation laws $C_3$ and $C_5$. The operator $X_4$ leads to the canonical variables

$$r=t,s={\displaystyle \frac{x}{t}},w\left(r\right)=u+{\displaystyle \frac{x}{t}},q\left(r\right)=v,$$

(75)

and the corresponding inverse canonical coordinates

$$t=r,x=rs,u=w-s,v=q.$$

(76)

The first-order partial derivatives $u_x$, $v_x$, and $v_t$ in terms of these canonical variables are

$$u_x=-{\displaystyle \frac{1}{r}},v_t=q_r,v_x=0.$$

(77)

Applying Theorem 1 with the canonical variables (75), the conserved vectors $C_3$ and $C_5$ reduce as follows:

$$\begin{array}{ccc}\hfill \widetilde{C_3}& :& \left\{\begin{array}{cc}\hfill C_3^r& =q{r}^{2}w,\hfill \\ \hfill C_3^s& ={\displaystyle \frac{1}{2}}q{q}_r{r}^2-qr{w}^2-{\displaystyle \frac{q}{2}}-{\displaystyle \frac{q_{r}r}{2}},\hfill \end{array}\right.\hfill \end{array}$$

(78)

$$\begin{array}{ccc}\hfill \widetilde{C_5}& :& \left\{\begin{array}{cc}\hfill C_5^r& =2qr,\hfill \\ \hfill C_5^s& =-qw.\hfill \end{array}\right.\hfill \end{array}$$

(79)

From the reduced conservation laws $D_r{C}_3^r=0$ and $D_r{C}_5^r=0$, we obtain the following equations for w and q:

$$\begin{array}{ccc}\hfill q{r}^{2}w& =& d_1,\hfill \end{array}$$

(80)

$$\begin{array}{ccc}\hfill qr& =& d_2,\hfill \end{array}$$

(81)

where $d_1$ and $d_2$ are arbitrary constants.

Solving the system of Equations (80) and (81) simultaneously for w and q and using (75) yields the solution

$$\begin{array}{ccc}\hfill u(t,x)& =& {\displaystyle \frac{d_1}{d_{2}t}}-{\displaystyle \frac{x}{t}},\hfill \end{array}$$

(82)

$$\begin{array}{ccc}\hfill v(t,x)& =& {\displaystyle \frac{d_2}{t}},\hfill \end{array}$$

(83)

for the BK system (1).

5. Concluding Remarks

In this study, we successfully applied the generalized double reduction theory to the (1 + 1)-dimensional Broer–Kaup (BK) system, which is a PDE model that describes long wave propagation in shallow water. We computed four Lie point symmetries and six conservation laws using the multiplier method based on low-order multipliers. Each symmetry was found to be associated with at least two conservation laws. By using these symmetries and conservation laws, we performed double reductions, simplifying the original system to first-order ODEs or algebraic equations, ultimately obtaining exact solutions.

These findings deepen the understanding of the BK system and its relevance to shallow water wave dynamics. Moreover, the symmetries, conservation laws, and exact solutions presented here lay a foundation for further research on the BK system. The approach of combining symmetry analysis with conservation laws proves to be a powerful tool for solving complex PDE systems, and the methods demonstrated here can be extended to other nonlinear PDE models.