Exploring Wave Interactions and Conserved Quantities of KdV–Caudrey–Dodd–Gibbon Equation Using Lie Theory

Abstract

This study introduces the KdV–Caudrey–Dodd–Gibbon (KdV-CDGE) equation to describe long water waves, acoustic waves, plasma waves, and nonlinear optics. Employing a generalized new auxiliary equation scheme, we derive exact analytical wave solutions, revealing rational, exponential, trigonometric, and hyperbolic trigonometric structures. The model also produces periodic, dark, bright, singular, and other soliton wave profiles. We compute classical and translational symmetries to develop abelian algebra, and visualize our results using selected parameters.

1. Introduction

Nonlinear evolution equations (NLEEs), which people have been desperately pursuing for a range of advanced applications, particularly the exact solution problem, remain the focus of both mathematicians and physicists [1,2]. Accordingly, investigations of the wave patterns for NLEEs have adequate significance in looking through nonlinear normal occasions [3]. NLEEs have numerous applications, including plasma physical science, computational physical science, optical fibers, water wave mechanics, the control hypothesis, meteorology, the electromagnetic hypothesis, mechanics, biogenetics, and more. Because of their repetitive appearance in different applications in physical science, science, design, the control hypothesis, money, and elements, the wave patterns to NLPDEs have drawn the consideration of many investigations. The analytical results of NLPDEs assume a significant part in the investigation of nonlinear actual peculiarities [4].

It is essential to recall that analytical solutions to nonlinear PDEs (partial differential equations) are effective tools for describing measurable and unmeasurable features existing in nonlinear phenomena across different scientific fields. These include plasma physics, solid-state physics, optical fibers, and solitary wave theory, among many others. There are various methods to find the analytical solution of NLPDEs, such as the exponential-function technique [5], the HAM [6], the Hirota approach [7,8], the F-expansion approach [9], the Exp-function technique [10], the sine-Gordon approach [11], the modified Kudryashov technique [12], the first integral scheme [13], the power series [14], the unified scheme [15,16], the multi-symplectic RK approach [17], the generalized unified approach [18,19], the khater approach [20], and some other approaches [21,22,23,24,25,26,27,28,29,30,31].

The Lie symmetry method, which is being highlighted by experts in this field, is a fundamental method that can be used to discover exact solutions of NLPDEs. The success of the Newton–Raphson method is mainly attributed to its simplicity, concision, and effectiveness; this method is a landmark in solving nonlinear differential equations. It is necessary to emphasize that the Lie point symmetry method refers to one of the most powerful and efficient sets of tools to classify the group invariant solutions of differential equations [32,33,34,35]. It provides a lot of help in generating analytical solutions to nonlinear differential equations utilizing similarity variables, which helps you to reduce complex partial differential equations into ordinary differential equations to solve. In addition, optimality is an essential second aspect of Lie groups theory, where both partial differential equations are converted to either linear or nonlinear ordinary differential equations.

In this study, we tackle a nonlinear KdV–Caudrey–Dodd–Gibbon Equation [36,37] of the following form:

$${\mathcal{Q}}_{\tau }+k{\left({\mathcal{Q}}_{\chi \chi }+{\displaystyle \frac{1}{5}}\alpha {\mathcal{Q}}^2\right)}_{\chi }+p{\left({\displaystyle \frac{1}{15}}\alpha {\mathcal{Q}}^3+\alpha \mathcal{Q}{\mathcal{Q}}_{\chi \chi }+{\mathcal{Q}}_{\chi \chi \chi \chi }\right)}_{\chi }=0.$$

(1)

The KdV-CDG equation plays an important role in laser optics, plasma physics, other nonlinear sciences, and ocean dynamics [38,39]. In this research, the used boundary conditions are ${\mathcal{Q}}^{\prime }\to 0$ and ${\mathcal{Q}}^{″}\to 0$, and $k,p,$ and $\alpha$ are arbitrary positive quantities. When $p=0$ in Equation (1), it turns into the Korteweg–de Vries (KdV) Equation [40]. When $k=0$, it turns into the Caudrey–Dodd–Gibbon (CDG) Equation [41]. From the literature, some work has been carried out on this model. The adequate soliton solutions of Equation (1) were computed in [36]. In [37,42], soliton solutions of Equation (1) were computed with the extended hyperbolic function method.

GNAEM is used to construct the new wave patterns for the KdV-CDG equation. This method was effectively applied to our proposed equation, yielding more generalized and intriguing results that are quite valuable in nonlinear research, particularly applied mathematics. GNAEM is a more powerful tool than other tools present in the literature that are used for the construction of wave solutions. GNAEM gives us interesting wave solutions in the form of trigonometric and hyperbolic trigonometric function forms.

This research follows this structure: In Section 2, GNAEM and the multiplier effect will be discussed. Section 3 represents a Lie theory application to Equation (1). Section 4 presents the generalized system, a scheme for the reduction of similarity, and an application method. Section 5 contains a graphical solution behavior. Lastly, Section 6 reveals the research results.

2. Preliminaries

The Idea of the Proposed Method

In this portion, the general pattern of GNAEM is represented. Consider the general form of PDE [43],

$$\mathcal{P}(\mathcal{Q},{\mathcal{Q}}_{\chi },{\mathcal{Q}}_{\chi \chi },\dots )=0,$$

(2)

where $\mathcal{Q}$ is a dependent variable.

Let us assume the following:

$$\mathcal{Q}(\chi ,\tau )=\mathcal{U}\left(\mathbb{G}\right),\mathbb{G}=\chi -\mathbb{S}\tau ,$$

(3)

where $\mathbb{S}$ is arbitrary constant. Taking the following transformation, Equation (3) converts Equation (2) to an ODE:

$$\mathcal{P}(\mathcal{U}\left(\mathbb{G}\right),{\mathcal{U}}^{\prime }\left(\mathbb{G}\right),{\mathcal{U}}^{″}\left(\mathbb{G}\right),\dots )=0.$$

(4)

Suppose Equation (4) has solutions as follows:

$$\mathcal{U}\left(\mathbb{G}\right)={\mathcal{K}}_0+\sum _{j=1}^N{\mathcal{K}}_j{\mathbb{H}}^j\left(\mathbb{G}\right),{\mathcal{K}}_N\ne 0,$$

(5)

where ${\mathcal{K}}_0,$ and ${\mathcal{K}}_j$ are constants. We combine the best order nonlinear and linear terms in the balancing approach to derive the value of N using Equation (4):

$${\mathbb{H}}^{\prime }\left(\mathbb{G}\right)=\sqrt{{\displaystyle \sum _{i_2=1}^3}{\mathcal{A}}_{i_2}^{★}{\mathbb{H}}^{i_2+1}\left(\mathbb{G}\right)},{\mathcal{A}}_{i_2}^{★}\in \mathbb{R}.$$

(6)

The following results of Equation (6) hold:

• Type 1: When ${\mathcal{A}}_1^{★}>0$,

  $${\mathbb{H}}_1\left(\mathbb{G}\right)={\displaystyle \frac{-{\mathcal{A}}_1^{★}{\mathcal{A}}_2^{★}{\mathrm{sech}}^2\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{{\mathcal{A}}_2^{★}}^2-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}{\left(1\pm tanh\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)\right)}^2}},$$

  (7)

  $${\mathbb{H}}_2\left(\mathbb{G}\right)={\displaystyle \frac{{\mathcal{A}}_1^{★}{\mathcal{A}}_2^{★}{\mathrm{csch}}^2\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{{\mathcal{A}}_2^{★}}^2-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}{\left(1\pm coth\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)\right)}^2}}.$$

  (8)
• Type 2: When ${\mathcal{A}}_1^{★}>0$, $\Delta >0$,

  $${\mathbb{H}}_3\left(\mathbb{G}\right)={\displaystyle \frac{2{\mathcal{A}}_1^{★}{\mathrm{sech}}^2\left(\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}\right)}{\pm \sqrt{\Delta }-{\mathcal{A}}_2^{★}\mathrm{sech}\left(\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}\right)}},$$

  (9)

  $${\mathbb{H}}_4\left(\mathbb{G}\right)={\displaystyle \frac{2{\mathcal{A}}_1^{★}{\mathrm{csch}}^2\left(\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}\right)}{\pm \sqrt{-\Delta }-{\mathcal{A}}_2^{★}\mathrm{csch}\left(\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}\right)}}.$$

  (10)
• Type 3: When ${\mathcal{A}}_1^{★}>0$, ${\mathcal{A}}_3^{★}>0$,

  $${\mathbb{H}}_5\left(\mathbb{G}\right)={\displaystyle \frac{-{\mathcal{A}}_1^{★}{\mathrm{sech}}^2\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{\mathcal{A}}_2^{★}\pm 2\sqrt{{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}tanh\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}},$$

  (11)

  $${\mathbb{H}}_6\left(\mathbb{G}\right)={\displaystyle \frac{{\mathcal{A}}_1^{★}{\mathrm{csch}}^2\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{\mathcal{A}}_2^{★}\pm 2\sqrt{{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}coth\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}}.$$

  (12)
• Type 4: When ${\mathcal{A}}_1^{★}>0$, $\Delta =0$,

  $${\mathbb{H}}_7\left(\mathbb{G}\right)=-{\displaystyle \frac{{\mathcal{A}}_1^{★}}{{\mathcal{A}}_2^{★}}}\left(1\pm tanh\left({\displaystyle \frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}}\mathbb{G}\right)\right),$$

  (13)

  $${\mathbb{H}}_8\left(\mathbb{G}\right)=-{\displaystyle \frac{{\mathcal{A}}_1^{★}}{{\mathcal{A}}_2^{★}}}\left(1\pm coth\left({\displaystyle \frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}}\mathbb{G}\right)\right).$$

  (14)
• Type 5: When ${\mathcal{A}}_1^{★}<0$, $\Delta >0$,

  $${\mathbb{H}}_9\left(\mathbb{G}\right)={\displaystyle \frac{2{\mathcal{A}}_1^{★}{sec}^2\left(\sqrt{-{\mathcal{A}}_1^{★}}\mathbb{G}\right)}{\pm \sqrt{\Delta }-{\mathcal{A}}_2^{★}sec\left(\sqrt{-{\mathcal{A}}_1^{★}}\mathbb{G}\right)}},$$

  (15)

  $${\mathbb{H}}_{10}\left(\mathbb{G}\right)={\displaystyle \frac{2{\mathcal{A}}_1^{★}{csc}^2\left(\sqrt{-{\mathcal{A}}_1^{★}}\mathbb{G}\right)}{\pm \sqrt{\Delta }-{\mathcal{A}}_2^{★}csc\left(\sqrt{-{\mathcal{A}}_1^{★}}\mathbb{G}\right)}}.$$

  (16)
• Type 6: When ${\mathcal{A}}_1^{★}<0$, ${\mathcal{A}}_3^{★}>0$,

  $${\mathbb{H}}_{11}\left(\mathbb{G}\right)={\displaystyle \frac{-{\mathcal{A}}_1^{★}{sec}^2\left(\frac{\sqrt{-{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{\mathcal{A}}_2^{★}\pm 2\sqrt{-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}tan\left(\frac{\sqrt{-{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}},$$

  (17)

  $${\mathbb{H}}_{12}\left(\mathbb{G}\right)={\displaystyle \frac{-{\mathcal{A}}_1^{★}{csc}^2\left(\frac{\sqrt{-{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{\mathcal{A}}_2^{★}\pm 2\sqrt{-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}cot\left(\frac{\sqrt{-{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}}.$$

  (18)
• Type 7: When ${\mathcal{A}}_1^{★}>0$,

  $${\mathbb{H}}_{13}\left(\mathbb{G}\right)={\displaystyle \frac{4{\mathcal{A}}_1^{★}{e}^{\pm \sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}}}{{(e^{\pm \sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}}-{\mathcal{A}}_2^{★})}^2-4{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}}.$$

  (19)
• Type 8: When ${\mathcal{A}}_1^{★}>0$, ${\mathcal{A}}_2^{★}=0$,

  $${\mathbb{H}}_{14}\left(\mathbb{G}\right)={\displaystyle \frac{\pm 4{\mathcal{A}}_1^{★}{e}^{\pm \sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}}}{1-4{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}{e}^{\pm 2\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}}}}.$$

  (20)
• Type 9: When ${\mathcal{A}}_1^{★}=0$,

  $${\mathbb{H}}_{15}\left(\mathbb{G}\right)={\displaystyle \frac{\pm {\mathcal{A}}_1^{★}{\mathcal{A}}_2^{★}}{{{\mathcal{A}}_2^{★}}^2{\mathbb{G}}^2-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}}.$$

  (21)
• Type 10: When ${\mathcal{A}}_1^{★}=0$, ${\mathcal{A}}_2^{★}=0$,

  $${\mathbb{H}}_{16}\left(\mathbb{G}\right)=\pm {\displaystyle \frac{1}{\sqrt{{\mathcal{A}}_3^{★}}\mathbb{G}}},$$

  (22)

  where $\Delta ={{\mathcal{A}}_2^{★}}^2-4{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}$. The above results can be arranged as hyperbolic trigonometric, trigonometric, and rational. Putting Equations (5) and (6) into Equation (4), we then obtain the system of the equation. After solving the system by maple, we obtain the values of the unknowns.

3. Symmetry Analysis of Equation (1)

The Lie method is used to investigate the stated Equation (1). Assume the one-parameter Lie group of infinitesimal transformations [44,45,46,47]:

$$\begin{array}{cc}\hfill {\tau }^{★}=& \tau +\epsilon {\xi }^1(\tau ,\chi ,\mathcal{Q})+O\left({\epsilon }^2\right),\\ \hfill {\chi }^{★}=& \chi +\epsilon {\xi }^2(\tau ,\chi ,\mathcal{Q})+O\left({\epsilon }^2\right),\\ \hfill \overline{\mathcal{Q}}=& \mathcal{Q}+\epsilon \Phi (\tau ,\chi ,\mathcal{Q})+O\left({\epsilon }^2\right),\end{array}$$

where $\epsilon \ll 1$ is a group parameter. The corresponding Lie algebra of infnitesimal symmetries is generated utilizing vector fields:

$$\mathcal{B}={\xi }^1{\displaystyle \frac{\partial }{\partial \tau }}+{\xi }^2{\displaystyle \frac{\partial }{\partial \chi }}+\Phi {\displaystyle \frac{\partial }{\partial \mathcal{Q}}}.$$

(23)

Equation (23) produces a symmetry of Equation (1), and $\mathcal{B}$ passes the Lie group specifications:

$$P{r}^{\left(5\right)}\mathcal{B}\left({\mathcal{Q}}_{\tau }+k{\left({\mathcal{Q}}_{\chi \chi }+{\displaystyle \frac{1}{5}}\alpha {\mathcal{Q}}^2\right)}_{\chi }+p{\left({\displaystyle \frac{1}{15}}\alpha {\mathcal{Q}}^3+\alpha \mathcal{Q}{\mathcal{Q}}_{\chi \chi }+{\mathcal{Q}}_{\chi \chi \chi \chi }\right)}_{\chi }\right){|}_{\Xi =0}=0,$$

(24)

where

$$\Xi ={\mathcal{Q}}_{\tau }+k{\left({\mathcal{Q}}_{\chi \chi }+{\displaystyle \frac{1}{5}}\alpha {\mathcal{Q}}^2\right)}_{\chi }+p{\left({\displaystyle \frac{1}{15}}\alpha {\mathcal{Q}}^3+\alpha \mathcal{Q}{\mathcal{Q}}_{\chi \chi }+{\mathcal{Q}}_{\chi \chi \chi \chi }\right)}_{\chi },$$

$$\begin{array}{cc}\hfill P{r}^{\left(5\right)}\mathcal{B}& =\mathcal{B}+{\Phi }^{\tau }{\displaystyle \frac{\partial }{\partial {\mathcal{Q}}_{\tau }}}+{\Phi }^{\chi }{\displaystyle \frac{\partial }{\partial {\mathcal{Q}}_{\chi }}}+{\Phi }^{\chi \chi }{\displaystyle \frac{\partial }{\partial {\mathcal{Q}}_{\chi \chi }}}+{\Phi }^{\chi \chi \chi }{\displaystyle \frac{\partial }{\partial {\mathcal{Q}}_{\chi \chi \chi }}}+{\Phi }^{\chi \chi \chi \chi \chi }{\displaystyle \frac{\partial }{\partial {\mathcal{Q}}_{\chi \chi \chi \chi \chi }}}.\hfill \end{array}$$

We have

$$\left\{\begin{array}{c}{\Phi }^{\tau }={\mathcal{D}}_{\tau }(\Phi )-{\mathcal{Q}}_{\tau }{\mathcal{D}}_{\tau }\left({\xi }^1\right)-{\mathcal{Q}}_{\chi }{\mathcal{D}}_{\tau }\left({\xi }^2\right),\hfill \\ {\Phi }^{\chi }={\mathcal{D}}_{\chi }(\Phi )-{\mathcal{Q}}_{\tau }{\mathcal{D}}_{\chi }\left({\xi }^1\right)-{\mathcal{Q}}_{\chi }{\mathcal{D}}_{\chi }\left({\xi }^2\right),\hfill \\ {\Phi }^{\chi \chi }={\mathcal{D}}_{\chi }\left({\Phi }^{\chi }\right)-{\mathcal{Q}}_{\chi \tau }{\mathcal{D}}_{\chi }\left({\xi }^1\right)-{\mathcal{Q}}_{\chi \chi }{\mathcal{D}}_{\chi }\left({\xi }^2\right),\hfill \\ {\Phi }^{\chi \chi \chi }={\mathcal{D}}_{\chi }\left({\Phi }^{\chi \chi }\right)-{\mathcal{Q}}_{\chi \chi \tau }{\mathcal{D}}_{\chi }\left({\xi }^1\right)-{\mathcal{Q}}_{\chi \chi \chi }{\mathcal{D}}_{\chi }\left({\xi }^2\right),\hfill \\ {\Phi }^{\chi \chi \chi \chi \chi }={\mathcal{D}}_{\chi }\left({\Phi }^{\chi \chi \chi \chi }\right)-{\mathcal{Q}}_{\chi \chi \chi \chi \tau }{\mathcal{D}}_{\chi }\left({\xi }^1\right)-{\mathcal{Q}}_{\chi \chi \chi \chi \chi }{\mathcal{D}}_{\chi }\left({\xi }^2\right),\hfill \end{array}\right.$$

(25)

where ${\mathcal{D}}_i$ can be written as follows:

$${\mathcal{D}}_i={\displaystyle \frac{\partial }{\partial x^i}}+{\mathcal{Q}}_i{\displaystyle \frac{\partial }{\partial \mathcal{Q}}}+{\mathcal{Q}}_{ij}{\displaystyle \frac{\partial }{\partial {\mathcal{Q}}_j}}+\dots ,1\le i,j\le 2.$$

By applying the Lie symmetry invariance condition to Equation (1), we have an over-determined system of PDEs; then, solving this system, we have the following infinitesimals:

$${\xi }^1=C_1,{\xi }^2=C_2,\Phi =0.$$

(26)

We obtain the following two-dimensional Lie algebra:

$${\mathcal{B}}_1={\displaystyle \frac{\partial }{\partial \tau }},{\mathcal{B}}_2={\displaystyle \frac{\partial }{\partial \chi }}.$$

(27)

We observe that

$$[{\mathcal{B}}_i,{\mathcal{B}}_j]=0,\mathrm{where}1\le i,j\le 2·$$

4. Traveling Waves of Equation (1) by Abelian Algebra

According to Equation (27), the vector field $\mathcal{B}=\{{\mathcal{B}}_1,{\mathcal{B}}_2\}$ develops an abelian algebra. We can present the most effective system for (23) as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\pounds }_1=& <{\mathcal{B}}_1>,\hfill \\ \hfill {\pounds }_2=& <{\mathcal{B}}_1+\mathbb{S}{\mathcal{B}}_2>.\hfill \end{array}\end{array}$$

(28)

We have the following similarity variables:

$$\begin{array}{c}\hfill \mathcal{Q}(\tau ,\chi )=\mathcal{U}\left(\mathbb{G}\right),\mathrm{where}\mathbb{G}=\chi -\mathbb{S}\tau ,\end{array}$$

(29)

using Equations (1) and (29), and we obtain the following ODE:

$$-\mathbb{S}{\mathcal{U}}^{\prime }+k{\left({\mathcal{U}}^{″}+{\displaystyle \frac{1}{5}}\alpha {\mathcal{U}}^2\right)}^{\prime }+p{\left({\displaystyle \frac{1}{15}}\alpha {\mathcal{U}}^3+\alpha \mathcal{U}{\mathcal{U}}^{″}+{\mathcal{U}}^{⁗}\right)}^{\prime }=0.$$

(30)

To find soliton solutions, we integrate Equation (30) while keeping the integration constant to have the value zero:

$$-\mathbb{S}\mathcal{U}+k\left({\mathcal{U}}^{″}+{\displaystyle \frac{1}{5}}\alpha {\mathcal{U}}^2\right)+p\left({\displaystyle \frac{1}{15}}\alpha {\mathcal{U}}^3+\alpha \mathcal{U}{\mathcal{U}}^{″}+{\mathcal{U}}^{⁗}\right)=0.$$

(31)

Balancing ${\mathcal{U}}^{⁗}$ with ${\mathcal{U}}^3$, we obtain $N=2$. Therefore, the solution according to Equation (1) can be formulated as

$$\mathcal{U}\left(\mathbb{G}\right)={\mathcal{K}}_0+{\mathcal{K}}_1\mathbb{H}\left(\mathbb{G}\right)+{\mathcal{K}}_2{\mathbb{H}}^2\left(\mathbb{G}\right).$$

(32)

Inserting Equation (32) into Equation (31) and letting the coefficient of ${\mathcal{U}}^j\left(\mathbb{G}\right)$, $(0\le j\le 7)$ approach zero results in the following:

$$\left\{\begin{array}{cc}\hfill \mathbf{Set}\mathbf{1}.& \alpha =1,k=\pm \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-5p{\mathcal{A}}_1^{★},{\mathcal{K}}_2=-30{\mathcal{A}}_3^{★},{\mathcal{K}}_1=-15{\mathcal{A}}_2^{★},\hfill \\ \hfill & {\mathcal{K}}_0=\left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★},\mathbb{S}=\left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right)p{\left({\mathcal{A}}_1^{★}\right)}^2+8p{\left({\mathcal{A}}_1^{★}\right)}^2.\hfill \\ \hfill \mathbf{Set}\mathbf{2}.& \alpha =1,k=-5p{\mathcal{A}}_1^{★},\mathbb{S}=-4p{\left({\mathcal{A}}_1^{★}\right)}^2,{\mathcal{K}}_0=0,{\mathcal{K}}_1=-15{\mathcal{A}}_2^{★},{\mathcal{K}}_2=-30{\mathcal{A}}_3^{★}.\hfill \end{array}\right.$$

(33)

Using Equation (33) and the solution sets defined in Section 2, Equation (1) yields the following descriptions for its solutions:

• For Family 1: When ${\mathcal{A}}_1^{★}>0$,

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_1(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{{\displaystyle \frac{-{\mathcal{A}}_1^{★}{\mathcal{A}}_2^{★}{\mathrm{sech}}^2\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{{\mathcal{A}}_2^{★}}^2-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}{\left(1\pm tanh\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)\right)}^2}}\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{-{\mathcal{A}}_1^{★}{\mathcal{A}}_2^{★}{\mathrm{sech}}^2\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{{\mathcal{A}}_2^{★}}^2-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}{\left(1\pm tanh\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)\right)}^2}}\right\}}^2,\hfill \end{array}$$

  (34)

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_2(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{{\displaystyle \frac{{\mathcal{A}}_1^{★}{\mathcal{A}}_2^{★}{\mathrm{csch}}^2\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{{\mathcal{A}}_2^{★}}^2-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}{\left(1\pm coth\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)\right)}^2}}\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{{\mathcal{A}}_1^{★}{\mathcal{A}}_2^{★}{\mathrm{csch}}^2\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{{\mathcal{A}}_2^{★}}^2-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}{\left(1\pm coth\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)\right)}^2}}\right\}}^2.\hfill \end{array}$$

  (35)
• For Family 2: When ${\mathcal{A}}_1^{★}>0$, $\Delta >0$,

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_3(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{{\displaystyle \frac{2{\mathcal{A}}_1^{★}{\mathrm{sech}}^2\left(\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}\right)}{\pm \sqrt{\Delta }-{\mathcal{A}}_2^{★}\mathrm{sech}\left(\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}\right)}}\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{2{\mathcal{A}}_1^{★}{\mathrm{sech}}^2\left(\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}\right)}{\pm \sqrt{\Delta }-{\mathcal{A}}_2^{★}\mathrm{sech}\left(\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}\right)}}\right\}}^2,\hfill \end{array}$$

  (36)

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_4(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{{\displaystyle \frac{2{\mathcal{A}}_1^{★}{\mathrm{csch}}^2\left(\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}\right)}{\pm \sqrt{-\Delta }-{\mathcal{A}}_2^{★}\mathrm{csch}\left(\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}\right)}}\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{2{\mathcal{A}}_1^{★}{\mathrm{csch}}^2\left(\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}\right)}{\pm \sqrt{-\Delta }-{\mathcal{A}}_2^{★}\mathrm{csch}\left(\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}\right)}}\right\}}^2.\hfill \end{array}$$

  (37)
• For Family 3: When ${\mathcal{A}}_1^{★}>0$, ${\mathcal{A}}_3^{★}>0$,

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_5(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{{\displaystyle \frac{-{\mathcal{A}}_1^{★}{\mathrm{sech}}^2\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{\mathcal{A}}_2^{★}\pm 2\sqrt{{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}tanh\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}}\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{-{\mathcal{A}}_1^{★}{\mathrm{sech}}^2\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{\mathcal{A}}_2^{★}\pm 2\sqrt{{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}tanh\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}}\right\}}^2,\hfill \end{array}$$

  (38)

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_6(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{{\displaystyle \frac{{\mathcal{A}}_1^{★}{\mathrm{csch}}^2\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{\mathcal{A}}_2^{★}\pm 2\sqrt{{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}coth\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}}\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{{\mathcal{A}}_1^{★}{\mathrm{csch}}^2\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{\mathcal{A}}_2^{★}\pm 2\sqrt{{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}coth\left(\frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}}\right\}}^2.\hfill \end{array}$$

  (39)
• For Family 4: When ${\mathcal{A}}_1^{★}>0$, $\Delta =0$,

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_7(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{-{\displaystyle \frac{{\mathcal{A}}_1^{★}}{{\mathcal{A}}_2^{★}}}\left(1\pm tanh\left({\displaystyle \frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}}\mathbb{G}\right)\right)\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{-{\displaystyle \frac{{\mathcal{A}}_1^{★}}{{\mathcal{A}}_2^{★}}}\left(1\pm tanh\left({\displaystyle \frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}}\mathbb{G}\right)\right)\right\}}^2,\hfill \end{array}$$

  (40)

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_8(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{-{\displaystyle \frac{{\mathcal{A}}_1^{★}}{{\mathcal{A}}_2^{★}}}\left(1\pm coth\left({\displaystyle \frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}}\mathbb{G}\right)\right)\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{-{\displaystyle \frac{{\mathcal{A}}_1^{★}}{{\mathcal{A}}_2^{★}}}\left(1\pm coth\left({\displaystyle \frac{\sqrt{{\mathcal{A}}_1^{★}}}{2}}\mathbb{G}\right)\right)\right\}}^2.\hfill \end{array}$$

  (41)
• For Family 5: When ${\mathcal{A}}_1^{★}<0$, $\Delta >0$,

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_9(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{{\displaystyle \frac{2{\mathcal{A}}_1^{★}{sec}^2\left(\sqrt{-{\mathcal{A}}_1^{★}}\mathbb{G}\right)}{\pm \sqrt{\Delta }-{\mathcal{A}}_2^{★}sec\left(\sqrt{-{\mathcal{A}}_1^{★}}\mathbb{G}\right)}}\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{2{\mathcal{A}}_1^{★}{sec}^2\left(\sqrt{-{\mathcal{A}}_1^{★}}\mathbb{G}\right)}{\pm \sqrt{\Delta }-{\mathcal{A}}_2^{★}sec\left(\sqrt{-{\mathcal{A}}_1^{★}}\mathbb{G}\right)}}\right\}}^2,\hfill \end{array}$$

  (42)

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_{10}(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{{\displaystyle \frac{2{\mathcal{A}}_1^{★}{csc}^2\left(\sqrt{-{\mathcal{A}}_1^{★}}\mathbb{G}\right)}{\pm \sqrt{\Delta }-{\mathcal{A}}_2^{★}csc\left(\sqrt{-{\mathcal{A}}_1^{★}}\mathbb{G}\right)}}\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{2{\mathcal{A}}_1^{★}{csc}^2\left(\sqrt{-{\mathcal{A}}_1^{★}}\mathbb{G}\right)}{\pm \sqrt{\Delta }-{\mathcal{A}}_2^{★}csc\left(\sqrt{-{\mathcal{A}}_1^{★}}\mathbb{G}\right)}}\right\}}^2.\hfill \end{array}$$

  (43)
• For Family 6: When ${\mathcal{A}}_1^{★}<0$, ${\mathcal{A}}_3^{★}>0$,

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_{11}(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{{\displaystyle \frac{-{\mathcal{A}}_1^{★}{sec}^2\left(\frac{\sqrt{-{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{\mathcal{A}}_2^{★}\pm 2\sqrt{-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}tan\left(\frac{\sqrt{-{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}}\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{-{\mathcal{A}}_1^{★}{sec}^2\left(\frac{\sqrt{-{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{\mathcal{A}}_2^{★}\pm 2\sqrt{-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}tan\left(\frac{\sqrt{-{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}}\right\}}^2,\hfill \end{array}$$

  (44)

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_{12}(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{{\displaystyle \frac{-{\mathcal{A}}_1^{★}{csc}^2\left(\frac{\sqrt{-{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{\mathcal{A}}_2^{★}\pm 2\sqrt{-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}cot\left(\frac{\sqrt{-{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}}\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{-{\mathcal{A}}_1^{★}{csc}^2\left(\frac{\sqrt{-{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}{{\mathcal{A}}_2^{★}\pm 2\sqrt{-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}cot\left(\frac{\sqrt{-{\mathcal{A}}_1^{★}}}{2}\mathbb{G}\right)}}\right\}}^2.\hfill \end{array}$$

  (45)
• For Family 7: When ${\mathcal{A}}_1^{★}>0$,

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_{13}(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-15{\mathcal{A}}_2^{★}\left\{{\displaystyle \frac{4{\mathcal{A}}_1^{★}{e}^{\pm \sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}}}{{(e^{\pm \sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}}-{\mathcal{A}}_2^{★})}^2-4{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}}\right\}\hfill \\ & -30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{4{\mathcal{A}}_1^{★}{e}^{\pm \sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}}}{{(e^{\pm \sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}}-{\mathcal{A}}_2^{★})}^2-4{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}}\right\}}^2.\hfill \end{array}$$

  (46)
• For Family 8: When ${\mathcal{A}}_1^{★}>0$ and ${\mathcal{A}}_2^{★}=0,$

  $$\begin{array}{cc}\hfill {\mathcal{Q}}_{14}(\tau ,\chi )=& \left(-{\displaystyle \frac{15}{2}}\pm {\displaystyle \frac{i}{2}}\sqrt{15}\right){\mathcal{A}}_1^{★}-30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{\pm 4{\mathcal{A}}_1^{★}{e}^{\pm \sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}}}{1-4{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}{e}^{\pm 2\sqrt{{\mathcal{A}}_1^{★}}\mathbb{G}}}}\right\}}^2.\hfill \end{array}$$

  (47)
• For Family 9: When ${\mathcal{A}}_1^{★}=0$,

  $${\mathcal{Q}}_{15}(\tau ,\chi )=-15{\mathcal{A}}_2^{★}\left\{{\displaystyle \frac{\pm {\mathcal{A}}_1^{★}{\mathcal{A}}_2^{★}}{{{\mathcal{A}}_2^{★}}^2{\mathbb{G}}^2-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}}\right\}-30{\mathcal{A}}_3^{★}{\left\{{\displaystyle \frac{\pm {\mathcal{A}}_1^{★}{\mathcal{A}}_2^{★}}{{{\mathcal{A}}_2^{★}}^2{\mathbb{G}}^2-{\mathcal{A}}_1^{★}{\mathcal{A}}_3^{★}}}\right\}}^2.$$

  (48)
• For Family 10: When ${\mathcal{A}}_1^{★}=0$, ${\mathcal{A}}_2^{★}=0$,

  $${\mathcal{Q}}_{16}(\tau ,\chi )=-30{\mathcal{A}}_3^{★}{\left\{\pm {\displaystyle \frac{1}{\sqrt{{\mathcal{A}}_3^{★}}\mathbb{G}}}\right\}}^2.$$

  (49)

5. Graphical Representations

The section for graphical representation shows how our results would be useful for emphasizing the relevance of a nonlinear wave equation. To achieve this, we exemplify the graphical representation of the solutions by assigning reasonable values to the parameters. It could be used to model the behavior of shallow water waves since the solutions derived are obtained. These models can be applied for forecasting the patterns of waves and their interactions, which can be essential for coast constructions, tsunami warnings, and marine transportation. These derived solutions can be used back in plasma physics for the description of the wave behavior in plasma like in fusion reactors or in space plasmas, which have importance in energy and space science.

• Figure 1 demonstrates the 3D and 2D versions of ${\mathcal{Q}}_1(\chi ,\tau )$ for ${\mathcal{A}}_1^{★}=0$, ${\mathcal{A}}_2^{★}=1$, ${\mathcal{A}}_3^{★}=1.5$, $\alpha =1$, $\mathbb{S}=1$, and $p=1$.
• The plotted curves of ${\mathcal{Q}}_3(\chi ,\tau )$ can be seen for ${\mathcal{A}}_1^{★}=1$, ${\mathcal{A}}_2^{★}=3$, ${\mathcal{A}}_3^{★}=1$, $\alpha =1$, $\mathbb{S}=2$, and $p=1$. Their corresponding values are shown in Figure 2.
• The graphics of ${\mathcal{Q}}_7(\chi ,\tau )$ for ${\mathcal{A}}_1^{★}=1$, ${\mathcal{A}}_2^{★}=1$, ${\mathcal{A}}_3^{★}=0$, $\alpha =1$ $\mathbb{S}=1$, and $p=1$ are shown in Figure 3. It shows the effect of the velocity of the soliton.
• Furthermore, Figure 4 shows the 3D and 2D graph of ${\mathcal{Q}}_7(\chi ,\tau )$ for ${\mathcal{A}}_1^{★}=1$, ${\mathcal{A}}_2^{★}=1$, ${\mathcal{A}}_3^{★}=0$, $\alpha =1$, $\mathbb{S}=-1$, and $p=1$.

6. Conclusions

The present work presents a thorough examination of the KdV-CDG equation, emphasizing its numerous applications in nonlinear sciences. Through the application of the Lie group method, we successfully modeled and analyzed the equation, transforming nonlinear PDEs into manageable ODEs using similarity reduction. The implementation of GNAEM demonstrated its effectiveness in generating various wave patterns, underscoring its utility in solving nonlinear PDEs. Our graphical analysis clarified the physical consequences of our findings. By combining these results, we obtained a better understanding of the dynamics of gravity–capillary waves, shallow water waves, and magneto–sound wave interactions in plasma. Most importantly, our discovery of novel results presents new avenues for future research, marking a significant advancement in the field.