Optimal System, Symmetry Reductions and Exact Solutions of the (2 + 1)-Dimensional Seventh-Order Caudrey–Dodd–Gibbon–KP Equation

Abstract

In this paper, the $(2+1)$-dimensional seventh-order Caudrey–Dodd–Gibbon–KP equation is investigated through the Lie group method. The Lie algebra of infinitesimal symmetries, commutative and adjoint tables, and one-dimensional optimal systems is presented. Then, the seventh-order Caudrey–Dodd–Gibbon–KP equation is reduced to nine types of $(1+1)$-dimensional equations with the help of symmetry subalgebras. Finally, the unified algebra method is used to obtain the soliton solutions, trigonometric function solutions, and Jacobi elliptic function solutions of the seventh-order Caudrey–Dodd–Gibbon–KP equation.

1. Introduction

Nonlinear partial differential equations (NPDEs) have a wide range of applications in the natural sciences and engineering. In these fields, it is common to search for exact solutions to NPDEs, as these solutions provide precise descriptions of complex phenomena such as heat conduction and wave propagation. There are currently several powerful methods available for constructing exact solutions of NPDEs. These methods include the Bäcklund transformation method [1], Darboux transformation method [2,3], Lie group method [4,5,6], Hirota bilinear method [7], unified algebraic method [8], and so on.

The Lie group method is well known as a powerful tool for reducing high-dimensional equations into low-dimensional ones, or simplifying equations with a high number of variables into simpler ones. Furthermore, this method enables the discovery of similar solutions through symmetry reduction [9,10,11,12,13,14,15,16,17,18,19,20]. Therefore, it has become a commonly used technique in the field of differential equations research due to its widespread applicability.

The Korteweg–de Vries (KdV) equation, a fundamental equation in the field of integrable systems, is commonly used to describe weakly nonlinear shallow water waves [21],

$$u_t+6u{u}_x+u_{3x}=0,$$

(1)

where $u_{nx}$ represents the nth order partial derivative with respect to x.

As a $(2+1)$-dimensional extension of the KdV equation, the Kadomtsev–Petviashvili (KP) equation can be used to describe the evolution of nonlinear, long waves of small amplitude with slow dependence on the transverse coordinates [22],

$${(u_t+6u{u}_x+u_{3x})}_x+3{\sigma }^2{u}_{2y}=0,$$

(2)

where $u_{2y}$ represents the 2nd order partial derivative with respect to y.

Due to the inability of the KdV Equation (1) to provide the detailed description of many important features observed in laboratory experiments, such as the non-monotonic relationship between solitary wave speed and amplitude, it becomes necessary to consider higher-order forms of the KdV equation such as the fifth-order KdV (fKdV) equation [23],

$$u_t+p_1{u}^2{u}_x+p_2{u}_x{u}_{2x}+p_{3}u{u}_{3x}+u_{5x}=0,$$

(3)

and the seventh-order KdV (sKdV) equation [24],

$$\begin{array}{c}\hfill u_t+h_1{u}^3{u}_x+h_2{u}_x^3+h_{3}u{u}_x{u}_{2x}+h_4{u}^2{u}_{3x}+h_5{u}_{2x}{u}_{3x}+h_6{u}_x{u}_{4x}+h_{7}u{u}_{5x}+u_{7x}=0,\end{array}$$

(4)

where $p_i$$(i=1,\dots ,3)$ and $h_j$$(j=1,2,\dots ,7)$ are arbitrary constants. Due to the arbitrariness of parameters $p_1$, $p_2$ and $p_3$, fKdV Equation (3) often manifests in several distinct representations, such as the Sawada–Kotera (SK) equation, the Caudrey–Dodd–Gibbon (CDG) equation, the Lax equation, the Kaup–Kuperschmidt (KP) equation, and the Ito equation. The fKdV Equation (3) is an essential model for many physical phenomena in fluid dynamics and magneto-acoustic waves. Similarly, the sKdV equation also has various forms including the Sawada–Kotera–Ito equation and the CDG equation, and these higher-order KdV equations play an important role in mathematical physics, engineering and applied sciences.

In this paper, our main focus is on the $(2+1)$-dimensional seventh-order Caudrey–Dodd–Gibbon–KP (sCDG-KP) equation, which is formed by combining the seventh-order CDG equation and the KP equation,

$$\begin{array}{cc}\hfill \Delta =& {(u_t+420u^3{u}_x+210u^2{u}_{3x}+420u{u}_x{u}_{2x}+28u{u}_{5x}+28u_x{u}_{4x}+70u_{2x}{u}_{3x}+u_{7x})}_x\hfill \\ & +\alpha u_{2y}=0,\alpha =\pm 1.\hfill \end{array}$$

(5)

The one and two soliton solutions of sCDG-KP Equation (5) are obtained using the simplified Hirota bilinear method [25].

Because of the complexity of the sCDG-KP Equation (5), there is currently limited research on it. The Lie group method is extremely effective for studying integrable or non-integrable NPDEs. Particularly for high-dimensional NPDEs, the application of the Lie group method can reduce the dimensionality of the given NPDEs, thereby enabling the possibility of obtaining exact solutions to the given NPDEs further. This paper employs the Lie group method and the unified algebraic method to investigate the optimal system of one-dimensional subalgebras, group-invariant solutions, symmetry reduction, and exact solutions of sCDG-KP Equation (5).

The structure of this paper is as follows. Section 2 presents the infinitesimal generators and vector fields that represent the symmetries of sCDG-KP Equation (5). Section 3 derives commutator relations for the infinitesimal generators and uses adjoint relations to construct a one-dimensional optimal system. Section 4 obtains some $(1+1)$-dimensional equations by means of the symmetry reduction. Finally, exact solutions of sCDG-KP Equation (5) are obtained with the help of the unified algebra method.

2. Lie Point Symmetries

The construction of the symmetric algebras of the given NPEDs is a crucial step in the Lie group method. In this section, we first construct the infinitesimals of Equation (5). Based on the Lie group method, the one-parameter Lie group of point transformations in $(x,y,t,u)$ is defined as follows:

$$\begin{array}{cc}\hfill x^{\ast }& =x+ϵ\xi (x,y,t,u)+O\left({ϵ}^2\right),\hfill \\ \hfill y^{\ast }& =y+ϵ\eta (x,y,t,u)+O\left({ϵ}^2\right),\hfill \\ \hfill t^{\ast }& =t+ϵ\tau (x,y,t,u)+O\left({ϵ}^2\right),\hfill \\ \hfill u^{\ast }& =u+ϵ\varphi (x,y,t,u)+O\left({ϵ}^2\right),\hfill \end{array}$$

(6)

where $\xi \equiv \xi (x,y,t,u)$, $\eta \equiv \eta (x,y,t,u)$, $\tau \equiv \tau (x,y,t,u)$, $\varphi \equiv \varphi (x,y,t,u)$ are infinitesimals, and $ϵ$ is a group parameter. The vector field associated with the transformation group is

$$\begin{array}{c}\hfill \mathit{V}=\xi \frac{\partial }{\partial x}+\eta \frac{\partial }{\partial y}+\tau \frac{\partial }{\partial t}+\varphi \frac{\partial }{\partial u}.\end{array}$$

(7)

To obtain the Lie point symmetries for sCDG-KP Equation (5), the associated vector fields must satisfy the following invariant condition:

$$\begin{array}{c}\hfill {\mathit{Pr}}^{\left(\mathbf{8}\right)}{\mathit{V}(\Delta )|}_{\Delta =0}=0,\end{array}$$

(8)

where ${\mathit{Pr}}^{\left(\mathbf{8}\right)}\mathit{V}$ represents the 8th order prolongation of vector field $\mathit{V}$,

$$\begin{array}{cc}\hfill {\mathit{Pr}}^{\left(\mathbf{8}\right)}\mathit{V}=& \mathit{V}+{\varphi }^x\frac{\partial }{\partial u_x}+{\varphi }^y\frac{\partial }{\partial u_y}+{\varphi }^t\frac{\partial }{\partial u_t}+{\varphi }^{2x}\frac{\partial }{\partial u_{2x}}+{\varphi }^{2y}\frac{\partial }{\partial u_{2y}}+{\varphi }^{xt}\frac{\partial }{\partial u_{xt}}+{\varphi }^{3x}\frac{\partial }{\partial u_{3x}}\hfill \\ & +{\varphi }^{4x}\frac{\partial }{\partial u_{4x}}+{\varphi }^{5x}\frac{\partial }{\partial u_{5x}}+{\varphi }^{8x}\frac{\partial }{\partial u_{8x}},\hfill \end{array}$$

(9)

and ${\varphi }^y$, ${\varphi }^t$, ${\varphi }^{xt}$, ${\varphi }^{2y}$, ${\varphi }^{kx}$, $(k=1,2\dots ,8)$ are defined as follows:

$$\begin{array}{cc}\hfill {\varphi }^y& =D_y\varphi -u_x{D}_y\xi -u_y{D}_y\eta -u_t{D}_y\tau ,\hfill \\ \hfill {\varphi }^t& =D_t\varphi -u_x{D}_t\xi -u_y{D}_t\eta -u_t{D}_t\tau ,\hfill \\ \hfill {\varphi }^{xt}& =D_t{\varphi }^x-u_{2x}{D}_t\xi -u_{xy}{D}_t\eta -u_{xt}{D}_t\tau ,\hfill \\ \hfill {\varphi }^{2y}& =D_y{\varphi }^y-u_{xy}{D}_y\xi -u_{2y}{D}_t\eta -u_{yt}{D}_t\tau ,\hfill \\ \hfill {\varphi }^{kx}& =D_x^{kx}(\varphi -\xi u_x-\eta u_y-\tau u_t)+\xi u_{(k+1)x}+\eta u_{kxy}+\tau u_{kxt},\hfill \end{array}$$

(10)

and $D_x$, $D_y$, $D_t$ are the total derivatives of x, y, t. $D_x^{kx}$ represents the k-order total derivative of x. $u_{(k+1)x}=\frac{{\partial }^{k+1}u}{\partial x^{k+1}},u_{kxy}=\frac{{\partial }^{k+1}u}{\partial x^k\partial y},u_{kxt}=\frac{{\partial }^{k+1}u}{\partial x^k\partial t}$.

Thus, invariant Equation (8) can be explicitly written as follows:

$$\begin{array}{cc}\hfill & 420u^3{\varphi }^{2x}+210u^2{\varphi }^{4x}+420{\left(u_x\right)}^2{\varphi }^{2x}+420u_{2x}\varphi +56{\varphi }^x{u}_{5x}+56u_x{\varphi }^{5x}+28u_{6x}\varphi +98u_{4x}{\varphi }^{4x}\hfill \\ & +98u_{2x}{\varphi }^{4x}+140u_{3x}{\varphi }^{3x}+\alpha {\varphi }^{2y}+{\varphi }^{xt}+{\varphi }^{8x}+1260u^2{u}_{2x}\varphi +840u_x{u}_{3x}\varphi +840u{u}_x{\varphi }^{3x}\hfill \\ & +840u{u}_{3x}{\varphi }^x+420u{u}_{4x}\varphi +840u_x{u}_{2x}{\varphi }^x+840u{u}_{2x}{\varphi }^{2x}+2520u{\left(u_x\right)}^2\varphi +2520u^2{u}_x{\varphi }^x=0.\hfill \end{array}$$

(11)

Substituting (10) into (11) and collecting coefficients of u and its various partial derivatives yields a set of determinant equations. The solutions of these determining equations provide the following infinitesimals for Equation (5):

$$\begin{array}{c}\hfill \xi =-\frac{2}{11}{c}_{1}x+c_{2}y+c_3,\eta =-\frac{8}{11}{c}_{1}y-2\alpha c_{2}t+c_4,\tau =-\frac{14}{11}{c}_{1}t+c_5,\varphi =\frac{4}{11}{c}_{1}u,\end{array}$$

(12)

where $c_i$$(i=1,2,\dots ,5)$ are arbitrary constants.

Thus, the Lie algebra of infinitesimal symmetries of Equation (5) is spanned by the five vector fields

$$\begin{array}{c}\hfill V_1=\frac{\partial }{\partial x},V_2=\frac{\partial }{\partial y},V_3=\frac{\partial }{\partial t},V_4=y\frac{\partial }{\partial x}-2\alpha t\frac{\partial }{\partial y},V_5=-\frac{2}{11}(x\frac{\partial }{\partial x}+4y\frac{\partial }{\partial y}+7t\frac{\partial }{\partial t}-2u\frac{\partial }{\partial y}).\end{array}$$

(13)

One of the most significant applications of symmetry theory is the construction of group-invariant solutions. For each subgroup within the symmetry group, there exists a corresponding set of group-invariant solutions. Given that any linear combination of arbitrary infinitesimal quantities remains infinitesimal, this implies that a differential equation possesses infinitely many different symmetry subgroups. Consequently, enumerating all potential group-invariant solutions is virtually an impossible task. This gives rise to the classification problem of group-invariant solutions, also known as the optimal system. Constructing optimal systems of subgroups is equivalent to constructing optimal systems of subalgebras. For one-dimensional subalgebras, we only need to take the most general form of the Lie algebra, and then apply various adjoint transformations on it to simplify the algebraic form as much as possible. Next, we use this method to construct a one-dimensional optimal system of Lie algebra (13).

3. One-Dimensional Optimal System

In this section, we use the Hu–Li–Chen algorithm to construct the optimal system of the sCDG-KP Equation (5) [26,27]. In order to obtain the one-dimensional optimal system of Lie algebra, we first calculate the commutation relations of the Lie algebra (13). Via using these commutation relationships, the infinitesimals shown in (13) can be written as linear combination of $V_i$:

$$V=a_1{V}_1+a_2{V}_2+a_3{V}_3+a_4{V}_4+a_5{V}_5.$$

(14)

Moreover, we derive the adjoint relations and obtain the symmetry subalgebras of the sCDG-KP equation.

3.1. Invariants of Lie Algebra

The Lie bracket for the infinitesimal generators is defined as

$$[V_i,V_j]=V_i{V}_j-V_j{V}_i.$$

(15)

Hence, the commutation relations among the Lie algebra (13) can be derived, as shown in

Table 1. Commutator table.

	${\mathit{V}}_1$	$\mathit{V}2$	${\mathit{V}}_3$	${\mathit{V}}_4$	${\mathit{V}}_5$
$V_1$	0	0	0	0	$-\frac{2}{11}{V}_1$
$V_2$	0	0	0	$V_1$	$-\frac{8}{11}{V}_2$
$V_3$	0	0	0	$-2\alpha V_2$	$-\frac{4}{11}{V}_3$
$V_4$	0	$-V_1$	$2\alpha V_2$	0	$\frac{6}{11}{V}_4$
$V_5$	$\frac{2}{11}{V}_1$	$\frac{8}{11}{V}_2$	$\frac{4}{11}{V}_3$	$-\frac{6}{11}{V}_4$	0

, the entry in row i and the column j representing $[V_i,V_j]$.

Applying the following Lie series formula in conjunction with commutator Table 1, we have

$$\begin{array}{cc}\hfill A{d}_{exp\left(ϵW\right)}\left(V\right)& =V-ϵ[W,V]+\frac{1}{2!}{ϵ}^2[W,[W,V]]-\dots \hfill \\ & =(a_1{V}_1+\dots +a_5{V}_5)-ϵ[b_1{V}_1+\dots +b_5{V}_5,a_1{V}_1+\dots +a_5{V}_5]+O\left({ϵ}^2\right)\hfill \\ \hfill & =(a_1{V}_1+\dots +a_5{V}_5)-ϵ({\Theta }_1{V}_1+\dots +{\Theta }_5{V}_5)+O\left({ϵ}^2\right),\hfill \end{array}$$

(16)

where

$$\begin{array}{cc}\hfill {\Theta }_1& =-\frac{2}{11}{b}_1{a}_5+\frac{2}{11}{b}_5{a}_1+b_2{a}_4-b_4{a}_2,\hfill \\ \hfill {\Theta }_2& =-\frac{8}{11}{b}_2{a}_5+\frac{8}{11}{b}_5{a}_2-2\alpha b_3{a}_4+2\alpha b_4{a}_3,\hfill \\ \hfill {\Theta }_3& =-\frac{4}{11}{b}_3{a}_5+\frac{4}{11}{b}_5{a}_3,{\Theta }_4=\frac{6}{11}{b}_4{a}_5-\frac{6}{11}{b}_5{a}_4,{\Theta }_5=0,\hfill \end{array}$$

and

$$V=\sum _{i=1}^5{a}_i{V}_i,W=\sum _{j=1}^5{b}_j{V}_j,a_i,b_j=\mathrm{constants}.$$

From Equation (16), for any $b_j(j=1,2,\dots ,5)$, the following condition holds:

$$\begin{array}{c}\hfill {\Theta }_1\frac{\partial \psi }{\partial a_1}+{\Theta }_2\frac{\partial \psi }{\partial a_2}+{\Theta }_3\frac{\partial \psi }{\partial a_3}+{\Theta }_4\frac{\partial \psi }{\partial a_4}+{\Theta }_5\frac{\partial \psi }{\partial a_5}=0,\end{array}$$

(17)

where $\psi \equiv \psi (a_1,a_2,a_3,a_4,a_5)$.

Collecting the coefficients of $b_j$ in Equation (17), we obtain

$$\begin{array}{cc}\hfill & \frac{\partial \psi }{\partial a_1}=0,\hfill \\ & a_4\frac{\partial \psi }{\partial a_1}-\frac{8}{11}{a}_5\frac{\partial \psi }{\partial a_2}=0,\hfill \\ & \alpha a_4\frac{\partial \psi }{\partial a_2}+\frac{2}{11}{a}_5\frac{\partial \psi }{\partial a_3}=0,\hfill \\ & a_2\frac{\partial \psi }{\partial a_1}-2\alpha a_3\frac{\partial \psi }{\partial a_2}-\frac{6}{11}{a}_5\frac{\partial \psi }{\partial a_4}=0,\hfill \\ & a_1\frac{\partial \psi }{\partial a_1}+4a_2\frac{\partial \psi }{\partial a_2}+2a_3\frac{\partial \psi }{\partial a_3}-3a_4\frac{\partial \psi }{\partial a_4}=0.\hfill \end{array}$$

(18)

From system (18), it is obvious that $\psi (a_1,a_2,a_3,a_4,a_5)=\psi \left(a_5\right)$.

3.2. Adjoint Matrix

With the help of Formula (16) and Table 1, the adjoint table is given in

Table 2. Adjoint table.

	${\mathit{V}}_1$	$\mathit{V}2$	${\mathit{V}}_3$	${\mathit{V}}_4$	${\mathit{V}}_5$
$V_1$	$V_1$	$V_2$	$V_3$	$V_4$	$\frac{2}{11}ϵV_1+V_5$
$V_2$	$V_1$	$V_2$	$V_3$	$-ϵV_1+V_4$	$\frac{8}{11}ϵV_2+V_5$
$V_3$	$V_1$	$V_2$	$V_3$	$2ϵ\alpha V_2+V_4$	$\frac{4}{11}ϵV_3+V_5$
$V_4$	$V_1$	$ϵV_1+V_2$	$\gamma$	$V_4$	$\frac{6}{11}ϵV_4+V_5$
$V_5$	$e^{-\frac{2}{11}ϵ}{V}_1$	$e^{-\frac{8}{11}ϵ}{V}_2$	$e^{-\frac{4}{11}ϵ}{V}_3$	$e^{\frac{6}{11}ϵ}{V}_4$	0

$\gamma =-\alpha {ϵ}^2{V}_1-2\alpha ϵV_2+V_3$.

.

The adjoint action of $V_i(i=1,2,\dots ,5)$ on V is given by

$$\begin{array}{cc}\hfill A{d}_{exp\left(ϵV_i\right)\left(V\right)}& =(a_1{V}_1+a_2{V}_2+a_3{V}_3+a_4{V}_4+a_5{V}_5)\hfill \\ & -ϵ[V_i,a_1{V}_1+a_2{V}_2+a_3{V}_3+a_4{V}_4+a_5{V}_5]\hfill \\ & -\frac{1}{2!}{ϵ}^2[V_i,[V_i,a_1{V}_1+a_2{V}_2+a_3{V}_3+a_4{V}_4+a_5{V}_5]]-\dots \hfill \\ & =(a_1,a_2,a_3,a_4,a_5)A_i(V_1,V_2,V_3,V_4,V_5),\hfill \end{array}$$

(19)

and therefore we have

$$\begin{array}{c}{A}_1 = \left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ \frac{2}{11}{ϵ}_1& 0& 0& 0& 1\end{array}\right), A_2 = \left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ -{ϵ}_2& 0& 0& 1& 0\\ 0& \frac{8}{11}{ϵ}_2& 0& 0& 1\end{array}\right),\hfill \\ A_3 = \left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 2\alpha {ϵ}_3& 0& 1& 0\\ 0& 0& \frac{4}{11}{ϵ}_3& 0& 1\end{array}\right), A_4 = \left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ {ϵ}_4& 1& 0& 0& 0\\ -\alpha {ϵ}_4^2& -2\alpha {ϵ}_4& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& -\frac{6}{11}{ϵ}_4& 1\end{array}\right),\hfill \\ A_5 = \left(\begin{array}{ccccc}{e}^{-\frac{2}{11}{ϵ}_5}& 0& 0& 0& 0\\ 0& e^{-\frac{8}{11}{ϵ}_5}& 0& 0& 0\\ 0& 0& e^{-\frac{4}{11}{ϵ}_5}& 0& 0\\ 0& 0& 0& e^{\frac{6}{11}{ϵ}_5}& 0\\ 0& 0& 0& 0& 1\end{array}\right).\hfill \end{array}$$

(20)

Thus, the general adjoint transformation matrix A is derived as

$$\begin{array}{cc}\hfill A& =A_1{A}_2{A}_3{A}_4{A}_5\hfill \\ \hfill & =\left(\begin{array}{ccccc}{e}^{-\frac{2}{11}{ϵ}_5}& 0& 0& 0& 0\\ {ϵ}_4{e}^{-\frac{2}{11}{ϵ}_5}& e^{-\frac{8}{11}{ϵ}_5}& 0& 0& 0\\ -\alpha {ϵ}_4^2{e}^{-\frac{2}{11}{ϵ}_5}& -2\alpha {ϵ}_4{e}^{-\frac{8}{11}{ϵ}_5}& e^{-\frac{4}{11}{ϵ}_5}& 0& 0\\ A_{41}& 2\alpha {ϵ}_3{e}^{-\frac{8}{11}{ϵ}_5}& 0& e^{\frac{6}{11}{ϵ}_5}& 0\\ A_{51}& A_{52}& A_{53}& A_{54}& 1\end{array}\right),\hfill \end{array}$$

(21)

where

$$\begin{array}{cc}\hfill A_{41}& =\left(2\alpha {ϵ}_3{ϵ}_4\right)e^{-\frac{2}{11}{ϵ}_5},A_{51}=\frac{2}{11}{e}^{-\frac{2}{11}{ϵ}_5}({ϵ}_1+4{ϵ}_2{ϵ}_4-2\alpha {ϵ}_3{ϵ}_4^2),\hfill \\ \hfill A_{52}& =\frac{8}{11}{e}^{-\frac{8}{11}{ϵ}_5}({ϵ}_2-\alpha {ϵ}_3{ϵ}_4),A_{53}=\frac{4}{11}{ϵ}_3{e}^{-\frac{5}{11}{ϵ}_5},A_{54}=-\frac{6}{11}{ϵ}_4{e}^{\frac{6}{11}{ϵ}_5}.\hfill \end{array}$$

3.3. Classification of Symmetry Algebra

The adjoint transformation equation of sCDG-KP Equation (5) is given by

$$({\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3,{\tilde{a}}_4,{\tilde{a}}_5)=(a_1,a_2,a_3,a_4,a_5)A,$$

(22)

where A is the adjoint transformation matrix (21). Direct calculation yields

$$\begin{array}{cc}\hfill {\tilde{a}}_1& =a_1{e}^{-\frac{2}{11}{ϵ}_5}+a_2{ϵ}_4{e}^{-\frac{2}{11}{ϵ}_5}-a_3\alpha {ϵ}_4^2{e}^{-\frac{2}{11}{ϵ}_5}+a_4(2\alpha {ϵ}_3{ϵ}_4-{ϵ}_2)e^{-\frac{2}{11}{ϵ}_5}+\hfill \\ & \frac{2}{11}{a}_5({ϵ}_1+4{ϵ}_2{ϵ}_4-2\alpha {ϵ}_3{ϵ}_4^2)e^{-\frac{2}{11}{ϵ}_5},\hfill \\ \hfill {\tilde{a}}_2& =a_2{e}^{-\frac{8}{11}{ϵ}_5}-2a_3\alpha {ϵ}_4{e}^{-\frac{8}{11}{ϵ}_5}+2a_4\alpha {ϵ}_3{e}^{-\frac{8}{11}{ϵ}_5}+\frac{8}{11}{a}_5({ϵ}_2-\alpha {ϵ}_3{ϵ}_4)e^{-\frac{8}{11}{ϵ}_5},\hfill \\ \hfill {\tilde{a}}_3& =a_3{e}^{-\frac{4}{11}{ϵ}_5}+\frac{4}{11}{a}_5{ϵ}_3{e}^{-\frac{5}{11}{ϵ}_5},\hfill \\ \hfill {\tilde{a}}_4& =a_4{e}^{\frac{6}{11}{ϵ}_5}-\frac{6}{11}{a}_5{ϵ}_4{e}^{\frac{6}{11}{ϵ}_5},\hfill \\ \hfill {\tilde{a}}_5& =a_5.\hfill \end{array}$$

(23)

According to the Hu–Li–Chen algorithm, there are two cases: $a_5=1$ and $a_5=0$.

Case 1: For $a_5=1$, we select a representative element, $\tilde{V}=V_5$; then, substituting ${\tilde{a}}_i=0$, $i=1,2,3,4$ and ${\tilde{a}}_5=1$ into (23), we have

$$\begin{array}{cc}\hfill {ϵ}_1& =\frac{1331}{32}\alpha a_3{a}_4^2-\frac{121}{16}{a}_2{a}_4-\frac{11}{2}{a}_1,{ϵ}_2=\frac{121}{56}\alpha a_3{a}_4-\frac{11}{8}{a}_2,{ϵ}_3=-\frac{11}{14}{a}_3,{ϵ}_4=\frac{11}{6}{a}_4.\hfill \end{array}$$

Case 2: For $a_5=0$, substituting ${\tilde{a}}_i=0$, $i=1,2,\dots ,5$ into (18), we obtain a new invariant ${\Delta }_1=a_3^3{a}_4^7$. Here are the three cases: ${\Delta }_1=1$, ${\Delta }_1=-1$, and ${\Delta }_1=0$.

Case 2.1: ${\Delta }_1=1.$ Evidently, $a_3>0,a_4>0$ or $a_3<0,a_4<0$.

Case 2.1.1: For $a_3>0,a_4>0$, we let the representative element be $\tilde{V}=V_3+V_4$. Then, by substituting ${\tilde{a}}_3=1,{\tilde{a}}_4=1$ and ${\tilde{a}}_i=0$, $i=1,2,5$ into (23), we obtain

$$\begin{array}{c}\hfill {ϵ}_2=\frac{a_1+\alpha a_3{ϵ}_4^2}{a_4},{ϵ}_3=\frac{-2\alpha a_3{ϵ}_4+a_2}{2\alpha a_4},{ϵ}_5=-\frac{11}{6}ln{a}_4.\end{array}$$

Case 2.1.2: For $a_3<0,a_4<0$, we let the representative element be $\tilde{V}=-V_3-V_4$; by substituting ${\tilde{a}}_3=-1,{\tilde{a}}_4=-1$ and ${\tilde{a}}_i=0$, $i=1,2,5$ into (23), we obtain

$$\begin{array}{c}\hfill {ϵ}_2=\frac{a_1+\alpha a_3{ϵ}_4^2}{a_4},{ϵ}_3=\frac{-2\alpha a_3{ϵ}_4+a_2}{2\alpha a_4},{ϵ}_5=-\frac{11}{6}ln(-\frac{1}{a_4}).\end{array}$$

Essentially, $V_3+V_4$ is equivalent to $-V_3-V_4$.

Case 2.2: ${\Delta }_1=-1$. In this case, $a_3>0,a_4<0$ or $a_3<0,a_3>0$.

Case 2.2.1: For $a_3>0,a_4<0$, we let the representative element be $\tilde{V}=V_3-V_4$; by associating with (23), we obtain

$$\begin{array}{c}\hfill {ϵ}_2=\frac{a_1+\alpha a_3{ϵ}_4^2}{a_4},{ϵ}_3=\frac{2\alpha a_3{ϵ}_4-a_2}{2\alpha a_4},{ϵ}_5=\frac{11}{14}ln{a}_3.\end{array}$$

Case 2.2.2: For $a_3<0,a_4>0$, we select the representative element as $\tilde{V}=-V_3+V_4$.

Substituting ${\tilde{a}}_3=-1,{\tilde{a}}_4=1,{\tilde{a}}_i=0,i=1,2,5$ to (23), we obtain

$$\begin{array}{c}\hfill {ϵ}_2=\frac{a_1+\alpha a_3{ϵ}_4^2}{a_4},{ϵ}_3=\frac{2\alpha a_3{ϵ}_4-a_2}{2\alpha a_4},{ϵ}_5=-\frac{11}{14}ln\left(-\frac{1}{a_3}\right).\end{array}$$

Case 2.3: ${\Delta }_1=0$. In this case, there are three cases: $a_3=0,a_4\ne 0$; $a_3\ne 0,a_4=0$; $a_3=a_4=0$.

Case 2.3.1: $a_3=0,a_4\ne 0$.

For $a_3=0,a_4>0$ and $\tilde{V}=V_4$, by substituting ${\tilde{a}}_4=1$ and ${\tilde{a}}_i=0$, $i=1,2,3,5$ into (23), we have

$$\begin{array}{c}\hfill {ϵ}_2=\frac{a_1}{a_4},{ϵ}_3=-\frac{a_2}{2\alpha a_4},{ϵ}_5=\frac{11}{6}ln{a}_4.\end{array}$$

For $a_3=0,a_4<0$ and $\tilde{V}=-V_4$, by substituting ${\tilde{a}}_4=-1$ and ${\tilde{a}}_i=0$, $i=1,2,3,5$ into (23), we have

$$\begin{array}{c}\hfill {ϵ}_2=\frac{a_1}{a_4},{ϵ}_3=-\frac{a_2}{2\alpha a_4},{ϵ}_5=\frac{11}{6}ln(-\frac{1}{a_4}).\end{array}$$

Case 2.3.2: $a_3\ne 0,a_4=0$.

Substituting $a_4=0,a_5=0$ into (18), we obtain a new invariant

$$\begin{array}{c}\hfill {\Delta }_2=\frac{{(4\alpha a_1{a}_3+a_2^2)}^7}{a_3^8},\end{array}$$

which includes the following three cases: ${\Delta }_2=1$, ${\Delta }_2=-1$, ${\Delta }_2=0$.

When ${\Delta }_2=1,a_3>0$ and $\tilde{V}=V_2+V_3$, by substituting ${\tilde{a}}_2=1,{\tilde{a}}_3=1$ and ${\tilde{a}}_i=0$, $i=1,4,5$ into (23), we obtain

$$\begin{array}{c}\hfill {ϵ}_4=\frac{a_2-a_3^{\frac{4}{7}}}{2\alpha a_3},{ϵ}_5=\frac{11}{14}ln{a}_3.\end{array}$$

When ${\Delta }_2=1,a_3<0$ and $\tilde{V}=V_2-V_3$, by substituting ${\tilde{a}}_2=1,{\tilde{a}}_3=-1$ and ${\tilde{a}}_i=0$, $i=1,4,5$ into (23), we obtain

$$\begin{array}{c}\hfill {ϵ}_4=\frac{a_2-{(-\frac{1}{a_3})}^{\frac{4}{7}}}{2\alpha a_3},{ϵ}_5=-\frac{11}{14}ln(-\frac{1}{a_3}).\end{array}$$

When ${\Delta }_2=-1$ and $\tilde{V}=-\frac{1}{4\alpha }{V}_1+V_3,\tilde{V}=\frac{1}{4\alpha }{V}_1-V_3$, by substituting ${\tilde{a}}_3=1$, ${\tilde{a}}_i=0$, $i=2,4,5$ and ${\tilde{a}}_3=-1$, ${\tilde{a}}_i=0$, $i=2,4,5$ into (23), we obtain

$$\begin{array}{c}\hfill {ϵ}_4=\frac{a_2}{2\alpha a_3},{ϵ}_5=\frac{11}{14}ln{a}_3,a_3>0,\end{array}$$

and

$$\begin{array}{c}\hfill {ϵ}_4=\frac{a_2}{2\alpha a_3},{ϵ}_5=-\frac{11}{14}ln(-\frac{1}{a_3}),a_3<0.\end{array}$$

When ${\Delta }_2=0$, we obtain $\tilde{V}=V_3$, by substituting ${\tilde{a}}_3=1$ and ${\tilde{a}}_i=0$, $i=1,2,4,5$ into (23); the solution is

$$\begin{array}{c}\hfill {ϵ}_4=\frac{a_2}{2\alpha a_3},{ϵ}_5=\frac{11}{14}ln{a}_3,a_3>0.\end{array}$$

Similar to case 2.1.1, when $a_3>0$ and $a_3<0$, the results are essentially the same; thus, we select the result corresponding to $a_3>0$ as the essential element.

Case 2.3.3: $a_3=a_4=0$. We select $\tilde{V}=V_1+V_2$ and $\tilde{V}=V_1-V_2$ as the representative element.

The solutions for (23) are

$$\begin{array}{c}\hfill {ϵ}_4=\frac{a_2^{\frac{1}{4}}-a_1}{a_2},{ϵ}_5=\frac{11}{8}ln{a}_2,a_2>0,\end{array}$$

and

$$\begin{array}{c}\hfill {ϵ}_4=\frac{1}{a_2{a}_2^{-\frac{1}{4}}}-\frac{a_1}{a_2},{ϵ}_5=-\frac{11}{8}ln(-\frac{1}{a_2}),a_2<0.\end{array}$$

Eventually, the symmetry subalgebras of the one-dimensional optimal system for the sCDG-KP Equation (5) are as follows:

$$\begin{array}{cc}\hfill {\tilde{V}}_1& =V_5,{\tilde{V}}_2=V_3+V_4,{\tilde{V}}_3=V_3-V_4,{\tilde{V}}_4=V_4,{\tilde{V}}_5=V_2+V_3,{\tilde{V}}_6=V_2-V_3,\hfill \\ \hfill {\tilde{V}}_7& =\frac{1}{4\alpha }{V}_1-V_3,{\tilde{V}}_8=V_3,{\tilde{V}}_9=V_1+V_2,{\tilde{V}}_{10}=V_1-V_2.\hfill \end{array}$$

(24)

4. Similarity Reductions

In this section, we apply the results of the one-dimensional optimal system discussed earlier to perform various symmetry reductions on sCDG-KP Equation (5). On the basis of (7), its corresponding characteristic equation is

$$\begin{array}{c}\hfill \frac{dx}{\xi }=\frac{dy}{\eta }=\frac{dt}{\tau }=\frac{du}{\varphi }.\end{array}$$

(25)

4.1. Subalgebra ${\tilde{V}}_1=V_5$

Based on vector field $V_5=-\frac{2}{11}(x\frac{\partial }{\partial x}+4y\frac{\partial }{\partial y}+7t\frac{\partial }{\partial t}-2u\frac{\partial }{\partial y})$, Equation (25) becomes

$$\begin{array}{c}\hfill \frac{dx}{x}=\frac{dy}{4y}=\frac{dt}{7t}=\frac{du}{-2u}.\end{array}$$

(26)

Equation (26) provides the invariant functions of Equation (5),

$$\begin{array}{c}\hfill u(x,y,t)=t^{-\frac{2}{7}}F(X,Y)\mathrm{with}X=x{t}^{-\frac{1}{7}},Y=y{t}^{-\frac{4}{7}}.\end{array}$$

(27)

and reduction equation

$$\begin{array}{cc}\hfill & 2940F^3{F}_{2X}+8820F^2{F}_X^2+2940F{F}_{2X}^2+2940F_X^2{F}_{2X}+5880F{F}_X{F}_{3X}+1470F^2{F}_{4X}+\hfill \\ & 686F_{2X}{F}_{4X}-X{F}_{2X}+392F_X{F}_{5X}-4Y{F}_{XY}+490F_{3X}^2+7\alpha F_{YY}+196F{F}_{6X}-3F_X+7F_{8X}=0.\hfill \end{array}$$

(28)

Since the calculation process is exactly the same as that of Case 1, we only list the results hereafter.

4.2. Subalgebra ${\tilde{V}}_2=V_3+V_4$

The vector field is ${\tilde{V}}_2=\frac{\partial }{\partial t}+y\frac{\partial }{\partial x}-2\alpha t\frac{\partial }{\partial y}$ and the characteristic equation is

$$\begin{array}{c}\hfill \frac{dx}{y}=\frac{dy}{-2\alpha t}=\frac{dt}{1}=\frac{du}{0}.\end{array}$$

(29)

Similarity solutions are given by

$$\begin{array}{c}\hfill u(x,y,t)=F(X,Y)\mathrm{with}X=x-\frac{2}{3}\alpha t^3-yt,Y=y+\alpha t^2.\end{array}$$

(30)

The reduction equation is given by

$$\begin{array}{cc}\hfill & 420F^3{F}_{2X}+1260F^2{F}_X^2+840F{F}_X{F}_{3X}+210F^2{F}_{4X}+420F{F}_{2X}^2+420F_X^2{F}_{2X}+\hfill \\ & 70F_{3X}^2+\alpha F_{2Y}+56F_X{F}_{5X}+98F_{2X}{F}_{4X}-Y{F}_{2X}+28F{F}_{6X}+F_{8X}=0.\hfill \end{array}$$

(31)

4.3. Subalgebra ${\tilde{V}}_3=V_3-V_4$

The vector field is ${\tilde{V}}_3=\frac{\partial }{\partial t}-y\frac{\partial }{\partial x}+2\alpha t\frac{\partial }{\partial y}$ and the characteristic equation is

$$\begin{array}{c}\hfill \frac{dx}{-y}=\frac{dy}{2\alpha t}=\frac{dt}{1}=\frac{du}{0}.\end{array}$$

(32)

Similarity solutions are given by

$$\begin{array}{c}\hfill u(x,y,t)=F(X,Y)\mathrm{with}X=x-\frac{2}{3}\alpha t^3+yt,Y=y+\alpha t^2.\end{array}$$

(33)

The reduction equation is given by

$$\begin{array}{cc}\hfill & 420F^3{F}_{2X}+1260F^2{F}_X^2+840F{F}_X{F}_{3X}+210F^2{F}_{4X}+420F{F}_{2X}^2+420F_X^2{F}_{2X}+\hfill \\ & 70F_{3X}^2+\alpha F_{2Y}+56F_X{F}_{5X}+98F_{2X}{F}_{4X}+Y{F}_{2X}+28F{F}_{6X}+F_{8X}=0.\hfill \end{array}$$

(34)

4.4. Subalgebra ${\tilde{V}}_4=V_4$

The vector field is ${\tilde{V}}_4=y\frac{\partial }{\partial x}-2\alpha t\frac{\partial }{\partial y}$ and the characteristic equation is

$$\begin{array}{c}\hfill \frac{dx}{y}=\frac{dy}{-2\alpha t}=\frac{dt}{0}=\frac{du}{0}.\end{array}$$

(35)

Similarity solutions are given by

$$\begin{array}{c}\hfill u(x,y,t)=F(X,T)\mathrm{with}X=x+\frac{y^2}{4\alpha t},T=t.\end{array}$$

(36)

The reduction equation is given by

$$\begin{array}{cc}\hfill & 2T{F}_{8X}+56TF{F}_{6X}+112T{F}_X{F}_{5X}+420T(F^2+\frac{7}{15}{F}_{2X})F_{4X}+140T{F}_{3X}^2+\hfill \\ & 1680TF{F}_X{F}_{3X}+840TF{F}_{2X}^2+840T(F^3+F_X^2)F_{2X}+2520T{F}^2{F}_X^2+2T{F}_{TX}+F_X=0.\hfill \end{array}$$

(37)

4.5. Subalgebra ${\tilde{V}}_5=V_2+V_3$

The vector field is ${\tilde{V}}_5=\frac{\partial }{\partial y}+\frac{\partial }{\partial t}$ and the characteristic equation is

$$\begin{array}{c}\hfill \frac{dx}{0}=\frac{dy}{1}=\frac{dt}{1}=\frac{du}{0}.\end{array}$$

(38)

Similarity solutions are given by

$$\begin{array}{c}\hfill u(x,y,t)=F(X,T)\mathrm{with}X=x,T=y-t.\end{array}$$

(39)

The reduction equation is given by

$$\begin{array}{cc}\hfill & F_{8X}+28F{F}_{6X}+56F_X{F}_{5X}+(210F^2+98F_{2X})F_{4X}+70F_{3X}^2+840F{F}_X{F}_{3X}+420F{F}_{2X}^2\hfill \\ & +(420F^3+420F_X^2)F_{2X}+1260F^2{F}_X^2+1260F^2{F}_X^2+\alpha F_{2T}-F_{XT}=0.\hfill \end{array}$$

(40)

4.6. Subalgebra ${\tilde{V}}_6=V_2-V_3$

The vector field is ${\tilde{V}}_6=\frac{\partial }{\partial y}-\frac{\partial }{\partial t}$ and the characteristic equation is

$$\begin{array}{c}\hfill \frac{dx}{0}=\frac{dy}{1}=\frac{dt}{-1}=\frac{du}{0}.\end{array}$$

(41)

Similarity solutions are given by

$$\begin{array}{c}\hfill u(x,y,t)=F(X,T)\mathrm{with}X=x,T=t+y.\end{array}$$

(42)

The reduction equation is given by

$$\begin{array}{cc}\hfill & F_{8X}+28F{F}_{6X}+56F_X{F}_{5X}+(210F^2+98F_{2X})F_{4X}+70F_{3X}^2+840F{F}_X{F}_{3X}+\hfill \\ & (420F^3+420F_X^2)F_{2X}+420F{F}_{2X}^2+1260F^2{F}_X^2+\alpha F_{2T}+F_{XT}=0.\hfill \end{array}$$

(43)

4.7. Subalgebra ${\tilde{V}}_7=\frac{1}{4\alpha }{V}_1-V_3$

The vector field is ${\tilde{V}}_7=\frac{1}{4\alpha }\frac{\partial }{\partial x}-\frac{\partial }{\partial t}$ and the characteristic equation is

$$\begin{array}{c}\hfill \frac{dx}{\frac{1}{4\alpha }}=\frac{dy}{0}=\frac{dt}{-1}=\frac{du}{0}.\end{array}$$

(44)

Similarity solutions are given by

$$\begin{array}{c}\hfill u(x,y,t)=F(X,T)\mathrm{with}X=x+\frac{t}{4\alpha },T=y.\end{array}$$

(45)

The reduction equation is given by

$$\begin{array}{cc}\hfill & 4\alpha F_{8X}+112\alpha F{F}_{6X}+224\alpha F_X{F}_{5X}+280\alpha F_{3X}^2+840\alpha (F^2+\frac{7}{15}{F}_{2X})F_{4X}+3360\alpha F{F}_X{F}_{3X}\hfill \\ & +(1680\alpha F^3+1680\alpha F_X^2+1)F_{2X}+1680\alpha F{F}_{2X}^2+5040\alpha F^2{F}_X^2+4{\alpha }^2{F}_{2T}=0.\hfill \end{array}$$

(46)

4.8. Subalgebra ${\tilde{V}}_8=V_3$

The vector field is ${\tilde{V}}_8=\frac{\partial }{\partial t}$ and the characteristic equation is

$$\begin{array}{c}\hfill \frac{dx}{0}=\frac{dy}{0}=\frac{dt}{1}=\frac{du}{0}.\end{array}$$

(47)

Similarity solutions are given by

$$\begin{array}{c}\hfill u(x,y,t)=F(X,Y)\mathrm{with}X=x,Y=y.\end{array}$$

(48)

The reduction equation is given by

$$\begin{array}{cc}\hfill & F_{8X}+28F{F}_{6X}+56F_X{F}_{5X}+(210F^2+98F_{2X})F_{4X}+70F_{3X}^2+840F{F}_X{F}_{3X}+\hfill \\ & 420F{F}_{2X}^2+(420F^3+420F_X^2)F_{2X}+1260F^2{F}_X^2+\alpha F_{2Y}=0.\hfill \end{array}$$

(49)

4.9. Subalgebra ${\tilde{V}}_9=V_1+V_2$

The vector field is ${\tilde{V}}_9=\frac{\partial }{\partial x}+\frac{\partial }{\partial y}$ and the characteristic equation is

$$\begin{array}{c}\hfill \frac{dx}{1}=\frac{dy}{1}=\frac{dt}{0}=\frac{du}{0}.\end{array}$$

(50)

Similarity solutions are given by

$$\begin{array}{c}\hfill u(x,y,t)=F(X,T)\mathrm{with}X=x-y,T=t.\end{array}$$

(51)

The reduction equation is given by

$$\begin{array}{cc}\hfill & F_{8X}+28F{F}_{6X}+56F_X{F}_{5X}+(210F^2+98F_{2X})F_{4X}+70F_{3X}^2+840F{F}_X{F}_{3X}+\hfill \\ & 420F{F}_{2X}^2+(420F^3+420F_X^2+\alpha )F_{2X}+1260F^2{F}_X^2+F_{TX}=0.\hfill \end{array}$$

(52)

4.10. Subalgebra ${\tilde{V}}_{10}=V_1-V_2$

The vector field is ${\tilde{V}}_{10}=\frac{\partial }{\partial x}-\frac{\partial }{\partial y}$ and the characteristic equation is

$$\begin{array}{c}\hfill \frac{dx}{1}=\frac{dy}{-1}=\frac{dt}{0}=\frac{du}{0}.\end{array}$$

(53)

Similarity solutions are given by

$$\begin{array}{c}\hfill u(x,y,t)=F(X,T)\mathrm{with}X=x+y,T=t.\end{array}$$

(54)

The reduction equation is given by

$$\begin{array}{cc}\hfill & F_{8X}+28F{F}_{6X}+56F_X{F}_{5X}+(210F^2+98F_{2X})F_{4X}+70F_{3X}^2+840F{F}_X{F}_{3X}+\hfill \\ & 420F{F}_{2X}^2+(420F^3+420F_X^2+\alpha )F_{2X}+1260F^2{F}_X^2+F_{TX}=0.\hfill \end{array}$$

(55)

One can observe that the $(1+1)$-dimensional equation resulting from the reduction using ${\tilde{V}}_9$ and ${\tilde{V}}_{10}$ is identical.

5. Exact Solution

The unified algebraic method, which is one of the effective approaches for solving NPDEs, is employed in this section to derive solutions for sCDG-KP Equation (5).

For sCDG-KP Equation (5), we consider its solution to be of the following form:

$$\begin{array}{cc}\hfill u& =a_0+a_1\varphi +a_2{\varphi }^2,\hfill \\ \hfill {\varphi }^{\prime }& =ϵ\sqrt{c_0+c_1\varphi +c_2{\varphi }^2+c_3{\varphi }^3+c_4{\varphi }^4},\hfill \end{array}$$

(56)

where $\varphi \equiv \varphi \left(\xi \right),\xi =kx+\eta y+\tau t$. By substituting (56) into Equation (5), we obtain the single soliton solution and the trigonometric solution

$$\begin{array}{cc}\hfill u_1& =a_0-a_2\frac{c_2}{c_4}{\mathrm{sech}}^2\left(\sqrt{c_2}\xi \right),c_2>0,c_4<0,\hfill \end{array}$$

(57a)

$$\begin{array}{cc}\hfill u_2& =a_0-a_2\frac{c_2}{c_4}{\sec }^2\left(\sqrt{-c_2}\xi \right),c_2<0,c_4>0,\hfill \end{array}$$

(57b)

as well as three Jacobi periodic solutions

$$\begin{array}{cc}\hfill u_3& =a_0-a_2\frac{c_2{m}^2}{c_4(2m^2-1)}{\mathrm{cn}}^2\left(\sqrt{\frac{c_2}{2m^2-1}}\xi \right),c_2>0,\hfill \end{array}$$

(58a)

$$\begin{array}{cc}\hfill u_4& =a_0-a_2\frac{c_2{m}^2}{c_4(m^2+1)}{\mathrm{sn}}^2\left(\sqrt{-\frac{c_2}{m^2+1}}\xi \right),c_2<0,\hfill \end{array}$$

(58b)

$$\begin{array}{cc}\hfill u_5& =a_0-a_2\frac{c_2}{c_4(2-m^2)}{\mathrm{dn}}^2\left(\sqrt{\frac{c_2}{2-m^2}}\xi \right),c_2>0,\hfill \end{array}$$

(58c)

where

$$\begin{array}{cc}\hfill a_0& =-\frac{5{\left[370881({\eta }^2\alpha +\tau k)k\right]}^{\frac{1}{3}}}{1218k},c_2=\frac{{\left[370881({\eta }^2\alpha +\tau k)k\right]}^{\frac{1}{3}}}{116k^3},c_4=-\frac{a_2}{2k^2}.\hfill \end{array}$$

The physical characteristics of the exact solutions of sCDG-KP Equation (5) can be seen more clearly via Figure 1, Figure 2 and Figure 3. As is well known, soliton solutions are a type of localized and stable solution with broad applications in fields such as fiber optics, material physics, and nuclear physics. The Jacobi elliptic function solutions can degenerate into soliton solutions as $m\to 1$.

6. Conclusions

The exact solutions of high-dimensional NPDEs have always been a hot topic in the field of mathematical physics. In this paper, the Lie group method is employed to derive four infinitesimals along with their corresponding symmetry algebras for sCDG-KP Equation (5). With the help of the one-dimensional optimal system and similarity reduction, sCDG-KP Equation (5) is reduced to nine classes of $(1+1)$-dimensional NPDEs. Furthermore, by applying the unified algebraic method, various types of exact solutions are obtained, including soliton solutions, trigonometric function solutions, and Jacobi elliptic function solutions for sCDG-KP Equation (5).