Qualitative Study of Solitary Wave Profiles in a Dissipative Nonlinear Model

Abstract

The convective Cahn–Hilliard–Oono equation is analyzed under the conditions ${\mu }_1\ge 0$ and ${\mu }_3+{\mu }_4\le 0$. The Lie invariance criteria are examined through symmetry generators, leading to the identification of Lie algebra, where translation symmetries exist in both space and time variables. By employing Lie group methods, the equation is transformed into a system of highly nonlinear ordinary differential equations using appropriate similarity transformations. The extended direct algebraic method are utilized to derive various soliton solutions, including kink, anti-kink, singular soliton, bright, dark, periodic, mixed periodic, mixed trigonometric, trigonometric, peakon soliton, anti-peaked with decay, shock, mixed shock-singular, mixed singular, complex solitary shock, singular, and shock wave solutions. The characteristics of selected solutions are illustrated in 3D, 2D, and contour plots for specific wave number effects. Additionally, the model’s stability is examined. These results contribute to advancing research by deepening the understanding of nonlinear wave structures and broadening the scope of knowledge in the field.

1. Introduction

Partial differential equations (PDEs) operate as fundamental mathematical tools throughout various scientific fields including technological applications. PDEs function as crucial base models for complex applications like fluid dynamics and quantum mechanics and heat transfer [1], enabling their extensive use across domains that include engineering, and medicine [2] and chemotactic interactions [3], as well as communication networks [4] and many more connected fields. Science and technology move forward through predictive capabilities and new discoveries, which require fundamental knowledge of PDE theory and solution techniques. Man-made and scientific advancements rely heavily on PDE expertise for their successful development. Nonlinear PDEs yield accurate representations of real-world phenomena because they represent natural systems better than linear variants despite their nonlinear characteristics, which improve modeling accuracy. The complexity of these equations demands alternative solution methods when attempting to acquire both approximate and numerical results. Despite these challenges, techniques such as the modified generalized Riccati equation mapping method [5], the modified simple equation method [6], the generalized Arnous method [7], the modified Kudryashov method [8], the sine-Gordon equation method [9], and Lie symmetry [10] have successfully provided analytical solutions for specific PDEs. We examine the NPDE [11]

$${\vartheta }_{\tau }+{\vartheta }_{\eta \eta \eta \eta }+{\mu }_2{\vartheta }_{\eta \eta }+{\mu }_1\vartheta +{\mu }_3{\left({\vartheta }^2\right)}_{\eta }+{\mu }_4{\left({\vartheta }^3\right)}_{\eta \eta }=0.$$

(1)

Here, ${\mu }_1,{\mu }_2,{\mu }_3$, and ${\mu }_4$ represent the linear growth, diffusion, quadratic convection, and nonlinear diffusion coefficients.

• We assume that ${\mu }_1\ge 0,{\mu }_3+{\mu }_4\le 0$; Equation (1) can be referred to as the convective Cahn–Hilliard–Oono equation [11].
• If ${\mu }_1={\mu }_3=0$, it reduces to a form of the classical Cahn–Hilliard equation.
• When ${\mu }_1=0,{\mu }_3\le 0$, the equation is known as the convective Cahn–Hilliard equation [12].
• Lastly, if ${\mu }_3=0,{\mu }_1>0$ and a dissipative term is incorporated into Equation (1), this leads to its designation as the Cahn–Hilliard–Oono equation [13].

In this work, we employ Lie symmetry reduction to derive various analytical solutions, beyond traveling-wave solutions, of the convective Cahn–Hilliard–Oono equation and also conduct a stability analysis of the equation. In [14], a modified Cahn–Hilliard equation describes the interface behavior between fluid layers. A more comprehensive discussion of the physical aspects associated with the Cahn–Hilliard equation is available in [12,13,14]. These observations partly justify the proposed name for Equation (1). By incorporating an additional term into Equation (1), one can derive the Kuramoto–Sivashinsky equation [15] under specific conditions.

The Lie group analysis method [10] is crucial for obtaining exact and explicit solutions to both ordinary and partial differential equations. Its core concept involves considering tangent structural equations under one or more parameterized Lie groups of point transformations. By utilizing symmetry reductions, one can derive exact solutions, including similarity solutions and more general group-invariant solutions [16,17]. Extensive research and numerous survey articles have focused on symmetry analysis for integer-order differential equations [18,19]. Nadjafikhah et al. [20] investigated the two-dimensional Ricci flow equation using the Lie group analysis method. Beenish et al. [21] studied the generalized integrable Kadomtsev–Petviashvili equation through symmetry analysis. Hussain et al. [22] explored multidimensional nonlinear wave equations using Lie group analysis. The core understanding of nonlinear wave propagation through soliton theory shows solitons as particular wave configurations that maintain both shape and speed when they encounter other waveforms [23]. The equilibrium between dispersion effects and nonlinearity enables solitons to resist dispersion-related degradation of their form and velocity.

Soliton theory stands as an essential element of present-day scientific investigation because it appears throughout various fields of study [24]. Fiber optic communications stands as one of the primary uses of solitons [25]. The propagation properties of solitons represent special waveform configurations that preserve their form together with their speed while passing through other waveforms but their behavior is explained by soliton theory. Soliton characteristics emerge from subtle interactions between dispersion and nonlinearity forces that keep them intact even though ordinary waves typically spread across time. Since this phenomena is present in many different fields, soliton theory is an essential part of modern scientific research [26]. In optical fiber communications, solitons are used in one of the most well-known applications: allowing data to be transferred over long distances without signal deterioration. The phenomenon of solitons appears in fluid dynamic systems to depict stable water waves that operate in canal formations as well as marine currents and atmospheric wave structures. Plasma physicists deploy solitons as models to examine magnetized plasma nonlinear wave interactions, thus obtaining essential knowledge about space and astrophysical phenomena [27]. Bose–Einstein condensates require soliton behavior for understanding the quantum fluid mechanics that functions at extreme low temperatures. The mechanisms of biological energy transport through proteins as well as nerve signal transmission can be explained through the concept of solitons in living organisms.

In 1834, John Scott Russell observed the first solitary wave that sustained itself independently during experiments in a canal, thus starting the study of solitons [27]. The formulation of a solid mathematical framework for soliton theory completed its development through the inverse scattering transform method during the 1960s. This crucial advancement led to the discovery of soliton solutions inside the Korteweg–de Vries equation that describes shallow-water wave behavior. Soliton theory has expanded widely since its discovery to encompass various scientific applications, which have delivered notable breakthroughs to nonlinear optics as well as condensed matter physics and financial modeling. Scientists continue studying solitons in integrable and non-integrable systems because this research leads to fresh discoveries in applied mathematics and physics. The multi-disciplinary nature of soliton theory makes it essential to studying nonlinear dynamics throughout various scientific fields. Nowadays, symmetry analysis and soliton solutions are prominent and widely studied topics in nonlinear science. Ejaz Hussain et al. [28] investigated soliton solutions of the Sharma–Tasso–Olver–Burgers equation using the $\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)$ method. San et al. [29] explored soliton solutions of the fractional magneto-electro-elastic system via the direct algebraic method. Yasin et al. [30] studied soliton solutions of the Ivancevic option pricing model using the unified method. Samreen et al [31] studied soliton solutions of the stochastic potential Korteweg–de Vries equation using the new extended direct algebraic method.

The paper is organized as follows: Section 2 examines Lie point symmetry, while Section 3 covers group-invariant solutions. Section 4 outlines the methodology, and Section 5 presents analytical solutions via the extended direct algebraic method. Section 6 visualizes the results, followed by a stability analysis in Section 8. Section 7 discusses a comparative evaluation. Finally, Section 9 concludes with key findings.

2. Investigating Lie Point Symmetries of Equation (1)

This section presents a brief review of infinite symmetries under a one-parameter group of transformations. The one-parameter transformations for Equation (1) are considered as [31]

$$\begin{array}{cc}\hfill {\eta }^{*}=& \eta +\Omega {\Lambda }_1(\eta ,\tau ,\vartheta )+\mathcal{O}\left({\Omega }^2\right),\hfill \\ \hfill {\tau }^{*}=& \tau +\Omega {\Lambda }_2(\eta ,\tau ,\vartheta )+\mathcal{O}\left({\Omega }^2\right),\hfill \\ \hfill {\vartheta }^{*}=& \vartheta +\Omega ϵ(\eta ,\tau ,\vartheta )+\mathcal{O}\left({\Omega }^2\right).\hfill \end{array}$$

(2)

Here, $\Omega$ represents the group parameter, while ${\Lambda }_1$, ${\Lambda }_1$, and $ϵ$ are the infinitesimals dependent on both the independent variables $\eta ,\tau$ and the dependent variable $\vartheta$, which need to be determined. Accordingly, the corresponding vector field for the convective Cahn–Hilliard–Oono equation (1) is given by

$$\mathcal{W}={\Lambda }_1(\eta ,\tau ,\vartheta ){\displaystyle \frac{\partial }{\partial \eta }}+{\Lambda }_2(\eta ,\tau ,\vartheta ){\displaystyle \frac{\partial }{\partial \tau }}+ϵ(\eta ,\tau ,\vartheta ){\displaystyle \frac{\partial }{\partial \vartheta }}.$$

(3)

Here, ${\Lambda }_1$, ${\Lambda }_2$, and $ϵ$ represent the infinitesimals associated with $\eta$, $\tau$, and $\vartheta$, which need to be determined. The fourth prolongation of $\mathcal{W}$ is expressed as

$$\begin{array}{cc}\hfill \mathcal{P}{r}^{\left[4\right]}\mathcal{W}=& \mathcal{W}+{ϵ}^{\tau }{\displaystyle \frac{\partial }{\partial {\vartheta }_{\tau }}}+{ϵ}^{\eta }{\displaystyle \frac{\partial }{\partial {\vartheta }_{\eta }}}+{ϵ}^{\eta \eta }{\displaystyle \frac{\partial }{\partial {\vartheta }_{\eta \eta }}}+{ϵ}^{\eta \eta \eta \eta }{\displaystyle \frac{\partial }{\partial {\vartheta }_{\eta \eta \eta \eta }}}.\hfill \end{array}$$

(4)

Utilizing the fourth prolongation formula under the invariance condition $\mathcal{P}{r}^{\left[4\right]}\mathcal{W}=0$ applied to Equation (1), the corresponding invariant surface condition is obtained as

$$\begin{array}{cc}\hfill & {ϵ}^{\tau }+{ϵ}^{\eta \eta \eta \eta }+{\mu }_2{ϵ}^{\eta \eta }+{\mu }_1ϵ+2{\mu }_3\vartheta {ϵ}^{\eta }+2{\mu }_3ϵ{\vartheta }^{\eta }+24{\mu }_4\vartheta ϵ{\vartheta }_{\eta }+12{\mu }_4{\vartheta }^2{ϵ}^{\eta }+6{\mu }_4\vartheta ϵ{\vartheta }^{\eta \eta }\hfill \\ & +3{\mu }_4{\vartheta }^2{ϵ}^{\eta \eta }=0,\hfill \end{array}$$

(5)

with the corresponding coefficient given by [21]

$$\left\{\begin{array}{c}{ϵ}^{\tau }={\mathfrak{D}}_{\tau }\left(ϵ\right)-{\vartheta }_{\eta }{\mathfrak{D}}_{\tau }\left({\Lambda }_1\right)-{\vartheta }_{\tau }{\mathfrak{D}}_{\tau }\left({\Lambda }_2\right),\hfill \\ {ϵ}^{\eta }={\mathfrak{D}}_{\eta }\left(ϵ\right)-{\vartheta }_{\eta }{\mathfrak{D}}_{\eta }\left({\Lambda }_1\right)-{\vartheta }_{\tau }{\mathfrak{D}}_{\eta }\left({\Lambda }_2\right),\hfill \\ {ϵ}^{\eta \eta }={\mathfrak{D}}_{\eta }\left({ϵ}^{\eta }\right)-{\vartheta }_{\eta \eta }{\mathfrak{D}}_{\eta }\left({\Lambda }_1\right)-{\vartheta }_{\tau \eta }{\mathfrak{D}}_{\eta }\left({\Lambda }_2\right),\hfill \\ {ϵ}^{\eta \eta \eta }={\mathfrak{D}}_{\eta }\left({ϵ}^{\eta \eta }\right)-{\vartheta }_{\eta \eta \eta }{\mathfrak{D}}_{\eta }\left({\Lambda }_1\right)-{\vartheta }_{\tau \eta \eta }{\mathfrak{D}}_{\eta }\left({\Lambda }_2\right),\hfill \\ {ϵ}^{\eta \eta \eta \eta }={\mathfrak{D}}_{\eta }\left({ϵ}^{\eta \eta \eta }\right)-{\vartheta }_{\eta \eta \eta }{\mathfrak{D}}_{\eta }\left({\Lambda }_1\right)-{\vartheta }_{\tau \eta \eta \eta }{\mathfrak{D}}_{\eta }\left({\Lambda }_2\right).\hfill \end{array}\right.$$

(6)

Employing the $\mathcal{P}{r}^{\left[4\right]}\mathcal{K}$ on Equation (1):

$$\mathcal{P}{r}^{\left[4\right]}\mathcal{W}({\vartheta }_{\tau }+{\vartheta }_{\eta \eta \eta \eta }+{\mu }_2{\vartheta }_{\eta \eta }+{\mu }_1\vartheta +{\mu }_3{\left({\vartheta }^2\right)}_{\eta }+{\mu }_4{\left({\vartheta }^3\right)}_{\eta \eta }){|}_{Equation\left(1\right)=0}=0.$$

(7)

The total derivative operators ${\mathfrak{D}}_{\eta }$ and ${\mathfrak{D}}_{\tau }$ are defined as [22]

$$\begin{array}{cc}\hfill & {\mathfrak{D}}_{\eta }={\displaystyle \frac{\partial }{\partial \eta }}+{\vartheta }_{\eta }{\displaystyle \frac{\partial }{\partial \vartheta }}+{\vartheta }_{\eta \tau }{\displaystyle \frac{\partial }{\partial {\vartheta }_{\tau }}}+{\vartheta }_{\eta \eta }{\displaystyle \frac{\partial }{\partial {\vartheta }_{\eta }}}+{\vartheta }_{\eta \eta \eta }{\displaystyle \frac{\partial }{\partial {\vartheta }_{\eta \eta }}}+{\vartheta }_{\eta \eta \eta \eta }{\displaystyle \frac{\partial }{\partial {\vartheta }_{\eta \eta \eta }}},\hfill \\ \hfill & {\mathfrak{D}}_{\tau }={\displaystyle \frac{\partial }{\partial \tau }}+{\vartheta }_{\eta }{\displaystyle \frac{\partial }{\partial \vartheta }}+{\vartheta }_{\eta \tau }{\displaystyle \frac{\partial }{\partial {\vartheta }_{\eta }}}+{\vartheta }_{\tau \tau }{\displaystyle \frac{\partial }{\partial {\vartheta }_{\tau }}}+{\vartheta }_{\tau \tau \tau }{\displaystyle \frac{\partial }{\partial {\vartheta }_{\tau \tau }}}+{\vartheta }_{\tau \tau \tau \tau }{\displaystyle \frac{\partial }{\partial {\vartheta }_{\tau \tau \tau }}}·\hfill \end{array}$$

(8)

By substituting Equation (6) into (5) and setting the coefficients of the partial derivatives to zero, we derive the system of determining equations:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial {\Lambda }_2}{\partial \eta }}=0,{\displaystyle \frac{\partial {\Lambda }_1}{\partial \eta }}=0,{\displaystyle \frac{\partial {\Lambda }_2}{\partial \tau }}=0,{\displaystyle \frac{\partial ϵ}{\partial \vartheta }}=0,\hfill \\ {\displaystyle \frac{\partial {\Lambda }_1}{\partial \tau }}=0,{\displaystyle \frac{\partial {\Lambda }_2}{\partial \vartheta }}=0,{\displaystyle \frac{\partial {\Lambda }_1}{\partial \vartheta }}=0·\hfill \end{array}\right.$$

(9)

As a result, solving Equation (9) yields the following infinitesimals:

$${\Lambda }_1={\mathfrak{a}}_1,{\Lambda }_2={\mathfrak{a}}_2,ϵ=0·$$

(10)

Hence, the Lie algebra of symmetries for Equation (1) is spanned by the following vector fields:

$${\mathcal{W}}_1={\displaystyle \frac{\partial }{\partial \tau }},{\mathcal{W}}_2={\displaystyle \frac{\partial }{\partial \eta }}·$$

(11)

3. Group-Invariant Solutions of Equation (1)

This section focuses on constructing group-invariant solutions for the convective Cahn–Hilliard–Oono equation.

Group 1: For the generator ${\mathcal{W}}_1={\displaystyle \frac{\partial }{\partial \tau }}$, the associated characteristic equation is

$${\displaystyle \frac{d\eta }{0}}={\displaystyle \frac{d\tau }{1}}={\displaystyle \frac{d\vartheta }{0}}$$

(12)

based on the similarity transformation $\vartheta (\eta ,\tau )=\zeta \left(\gamma \right)$ with $\gamma =\eta$, which is employed to derive the reduced form of Equation (1):

$${\displaystyle \frac{d^4\zeta }{d{\gamma }^4}}+{\mu }_2\left({\displaystyle \frac{d^2\zeta }{d{\gamma }^2}}\right)+{\mu }_1\zeta +2{\mu }_3\zeta \left({\displaystyle \frac{d\zeta }{d\gamma }}\right)+12{\mu }_4{\zeta }^2\left({\displaystyle \frac{d\zeta }{d\gamma }}\right)+3{\mu }_4{\zeta }^2\left({\displaystyle \frac{d^2\zeta }{d{\gamma }^2}}\right)=0.$$

(13)

Group 2: Considering the generator ${\mathcal{W}}_2={\displaystyle \frac{\partial }{\partial \eta }}$, the corresponding characteristic equation for ${\mathcal{W}}_2$ is

$${\displaystyle \frac{d\eta }{1}}={\displaystyle \frac{d\tau }{0}}={\displaystyle \frac{d\vartheta }{0}}$$

(14)

which leads to the similarity transformation $\vartheta (\eta ,\tau )=\zeta \left(\gamma \right)$ with $\gamma =\tau$. This transformation simplifies Equation (1) into its reduced form:

$${\displaystyle \frac{d\zeta }{d\gamma }}+{\mu }_1\zeta =0.$$

(15)

The solution of Equation (16) is given by

$$\zeta =f_1{e}^{-{\mu }_1\gamma }.$$

(16)

After applying the inverse transformation, the soliton solution of Equation (1) is obtained as

$$\vartheta (\eta ,\tau )=f_1{e}^{-{\mu }_1\tau }.$$

(17)

Group 3: When considering the generator ${\mathcal{W}}_1+\sigma {\mathcal{W}}_2={\displaystyle \frac{\partial }{\partial \tau }}+\sigma {\displaystyle \frac{\partial }{\partial \eta }}$, the characteristic equation implies

$${\displaystyle \frac{d\eta }{\sigma }}={\displaystyle \frac{d\tau }{0}}={\displaystyle \frac{d\vartheta }{0}}$$

(18)

with the transformation $\vartheta (\eta ,\tau )=\zeta \left(\gamma \right)$ with $\gamma =\eta -\sigma \tau$ facilitating the reduction of Equation (1) into a simpler form:

$$-\sigma \left({\displaystyle \frac{d\zeta }{d\gamma }}\right)+{\displaystyle \frac{d^4\zeta }{d{\gamma }^4}}+{\mu }_2\left({\displaystyle \frac{d^2\zeta }{d{\gamma }^2}}\right)+{\mu }_1\zeta +2{\mu }_3\zeta \left({\displaystyle \frac{d\zeta }{d\gamma }}\right)+12{\mu }_4{\zeta }^2\left({\displaystyle \frac{d\zeta }{d\gamma }}\right)+3{\mu }_4{\zeta }^2\left({\displaystyle \frac{d^2\zeta }{d{\gamma }^2}}\right)=0·$$

(19)

4. Investigation of the Proposed Strategies

The procedure consists of the following key steps:

Initial stage: Considering an NLPDE for $\vartheta$, we represent it in the following form:

$${\mathfrak{J}}_1(\vartheta ,{\vartheta }_{\eta },{\vartheta }_{\tau },{\vartheta }_{\eta \eta },{\vartheta }_{\tau \tau },{\vartheta }_{\tau \eta },{\vartheta }_{\eta \eta \eta },{\vartheta }_{\tau \tau \tau },....)=0·$$

(20)

By applying the transformations $\vartheta =\zeta \left(\gamma \right)$ and $\gamma =\eta -\sigma \tau$, the NLPDE is transformed into NODE in the following form:

$${\mathfrak{J}}_2(\zeta ,{\zeta }^{\prime },{\zeta }^{\prime \prime },{\zeta }^{\prime \prime \prime },{\zeta }^{\prime \prime \prime \prime }....)=0·$$

(21)

where $\sigma$ represents the wave velocity.

Second stage: Presentation of the initial solution for Equation (21):

$$\zeta \left(\gamma \right)=\sum _{h=0}^{\mathfrak{G}}{g}_h{\mathcal{Q}}^h\left(\gamma \right),g_{\mathfrak{G}}\ne 0·$$

(22)

Here, $g_h$ represents arbitrary parameters to be determined, and $\mathfrak{G}$ yields balanced numbers by using higher-order linear derivatives and the highest-order nonlinear term.

Review of the Extended Direct Algebraic Method Strategy

Within this method, the function ${\mathcal{Q}}^{\prime }\left(\gamma \right)$ adheres to the following auxiliary equation:

$${\mathcal{Q}}^{\prime }\left(\gamma \right)=ln(\Xi )(\Psi +\Phi \mathcal{Q}\left(\gamma \right)+\Theta {\mathcal{Q}}^2\left(\gamma \right)),$$

(23)

where $\Xi \ne 0,1$ and $\Psi$, $\Phi$, and $\Theta$ are constants. Defining $\rho ={\Phi }^2-4\Theta \Psi$, the solutions of Equation (23) can then be expressed as follows.

Family 1: If $\rho <0$ and $\Theta \ne 0$, the corresponding trigonometric solutions are derived as follows:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_1\left(\gamma \right)=-{\displaystyle \frac{\Phi }{2\Theta }}+{\displaystyle \frac{\sqrt{-\rho }}{2\Theta }}{tan}_{\Xi }\left({\displaystyle \frac{\sqrt{-\rho }}{2}}\gamma \right)·\hfill \\ \hfill & {\mathcal{Q}}_2\left(\gamma \right)=-{\displaystyle \frac{\Phi }{2\Theta }}-{\displaystyle \frac{\sqrt{-\rho }}{2\Theta }}{cot}_{\Xi }\left({\displaystyle \frac{\sqrt{-\rho }}{2}}\gamma \right)·\hfill \\ \hfill & {\mathcal{Q}}_3\left(\gamma \right)=-{\displaystyle \frac{\Phi }{2\Theta }}+{\displaystyle \frac{\sqrt{-\rho }}{2\Theta }}({tan}_{\Xi }\left(\sqrt{-\rho }\gamma \right)\pm \sqrt{r_1{r}_2}{sec}_{\Xi }\left(\sqrt{-\rho }\gamma \right))·\hfill \\ \hfill & {\mathcal{Q}}_4\left(\gamma \right)=-{\displaystyle \frac{\Phi }{2\Theta }}-{\displaystyle \frac{\sqrt{-\rho }}{2\Theta }}({cot}_{\Xi }\left(\sqrt{-\rho }\gamma \right)\pm \sqrt{r_1{r}_2}{csc}_{\Xi }\left(\sqrt{-\rho }\gamma \right))·\hfill \\ \hfill & {\mathcal{Q}}_5\left(\gamma \right)=-{\displaystyle \frac{\Phi }{2\Theta }}+{\displaystyle \frac{\sqrt{-\rho }}{4\Theta }}({tan}_{\Xi }\left({\displaystyle \frac{\sqrt{-\rho }}{4}}\gamma \right)-{cot}_{\Xi }\left({\displaystyle \frac{\sqrt{-\rho }}{4}}\gamma \right)).\hfill \end{array}$$

(24)

Family 2: If $\rho >0$ and $\Theta \ne 0$, the corresponding hyper trigonometric solutions are derived as follows:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_6\left(\gamma \right)=-{\displaystyle \frac{\Phi }{2\Theta }}-{\displaystyle \frac{\sqrt{\rho }}{2\Theta }}{tanh}_{\Xi }\left({\displaystyle \frac{\sqrt{\rho }}{2}}\gamma \right)·\hfill \\ \hfill & {\mathcal{Q}}_7\left(\gamma \right)=-{\displaystyle \frac{\Phi }{2\Theta }}-{\displaystyle \frac{\sqrt{\rho }}{2\Theta }}{coth}_{\Xi }\left({\displaystyle \frac{\sqrt{\rho }}{2}}\gamma \right)·\hfill \\ \hfill & {\mathcal{Q}}_8\left(\gamma \right)=-{\displaystyle \frac{\Phi }{2\Theta }}-{\displaystyle \frac{\sqrt{\rho }}{2\Theta }}({tanh}_{\Xi }\left(\sqrt{\rho }\gamma \right)\pm \iota \sqrt{r_1{r}_2}sec{h}_{\Xi }\left(\sqrt{\rho }\gamma \right))·\hfill \\ \hfill & {\mathcal{Q}}_9\left(\gamma \right)=-{\displaystyle \frac{\Phi }{2\Theta }}-{\displaystyle \frac{\sqrt{\rho }}{2\Theta }}({coth}_{\Xi }\left(\sqrt{\rho }\gamma \right)\pm \sqrt{r_1{r}_2}csc{h}_{\Xi }\left(\sqrt{\rho }\gamma \right))·\hfill \\ \hfill & {\mathcal{Q}}_{10}\left(\gamma \right)=-{\displaystyle \frac{\Phi }{2\Theta }}-{\displaystyle \frac{\sqrt{\rho }}{4\Theta }}({tanh}_{\Xi }\left({\displaystyle \frac{\sqrt{\rho }}{4}}\gamma \right)+{coth}_{\Xi }\left({\displaystyle \frac{\sqrt{\rho }}{4}}\gamma \right))·\hfill \end{array}$$

(25)

Family 3: If $\Theta \Psi >0$ and $\Phi =0$, the corresponding trigonometric solutions are derived as follows:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{11}\left(\gamma \right)=\sqrt{{\displaystyle \frac{\Psi }{\Theta }}}{tan}_{\Xi }\left(\sqrt{\Theta \Psi }\gamma \right)·\hfill \\ \hfill & {\mathcal{Q}}_{12}\left(\gamma \right)=-\sqrt{{\displaystyle \frac{\Psi }{\Theta }}}{cot}_{\Xi }\left(\sqrt{\Theta \Psi }\gamma \right)·\hfill \\ \hfill & {\mathcal{Q}}_{13}\left(\gamma \right)=\sqrt{{\displaystyle \frac{\Psi }{\Theta }}}({tan}_{\Xi }\left(2\sqrt{\Theta \Psi }\gamma \right)\pm \sqrt{r_1{r}_2}{sec}_{\Xi }\left(2\sqrt{\Theta \Psi }\gamma \right))·\hfill \\ \hfill & {\mathcal{Q}}_{14}\left(\gamma \right)=\sqrt{{\displaystyle \frac{\Psi }{\Theta }}}(-{cot}_{\Xi }\left(2\sqrt{\Theta \Psi }\gamma \right)\pm \sqrt{r_1{r}_2}{csc}_{\Xi }\left(2\sqrt{\Theta \Psi }\gamma \right))·\hfill \\ \hfill & {\mathcal{Q}}_{15}\left(\gamma \right)={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\Psi }{\Theta }}}({tan}_{\Xi }\left({\displaystyle \frac{\sqrt{\Theta \Psi }}{2}}\gamma \right)-{cot}_{\Xi }\left({\displaystyle \frac{\sqrt{\Theta \Psi }}{2}}\gamma \right))·\hfill \end{array}$$

(26)

Family 4: If $\Theta \Psi <0$ and $\Phi =0$, the corresponding hyper trigonometric solutions are derived as follows:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{16}\left(\gamma \right)=-\sqrt{-{\displaystyle \frac{\Psi }{\Theta }}}{tanh}_{\Xi }\left(\sqrt{-\Theta \Psi }\gamma \right)·\hfill \\ \hfill & {\mathcal{Q}}_{17}\left(\gamma \right)=-\sqrt{-{\displaystyle \frac{\Psi }{\Theta }}}{coth}_{\Xi }\left(\sqrt{-\Theta \Psi }\gamma \right)·\hfill \\ \hfill & {\mathcal{Q}}_{18}\left(\gamma \right)=-\sqrt{-{\displaystyle \frac{\Psi }{\Theta }}}({tanh}_{\Xi }\left(2\sqrt{-\Theta \Psi }\gamma \right)\pm \iota \sqrt{r_1{r}_2}sec{h}_{\Xi }\left(2\sqrt{-\Theta \Psi }\gamma \right))·\hfill \\ \hfill & {\mathcal{Q}}_{19}\left(\gamma \right)=-\sqrt{-{\displaystyle \frac{\Psi }{\Theta }}}({coth}_{\Xi }\left(2\sqrt{-\Theta \Psi }\gamma \right)\pm \sqrt{r_1{r}_2}csc{h}_{\Xi }\left(2\sqrt{-\Theta \Psi }\gamma \right))·\hfill \\ \hfill & {\mathcal{Q}}_{20}\left(\gamma \right)=-{\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{\Psi }{\Theta }}}({tanh}_{\Xi }\left({\displaystyle \frac{\sqrt{-\Theta \Psi }}{2}}\gamma \right)+{coth}_{\Xi }\left({\displaystyle \frac{\sqrt{-\Theta \Psi }}{2}}\gamma \right))·\hfill \end{array}$$

(27)

Family 5: If $\Phi =0$ and $\Theta =\Psi$, the corresponding trigonometric solutions are derived as follows:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{21}\left(\gamma \right)={tan}_{\Xi }(\Psi \gamma )·\hfill \\ \hfill & {\mathcal{Q}}_{22}\left(\gamma \right)=-{cot}_{\Xi }(\Psi \gamma )·\hfill \\ \hfill & {\mathcal{Q}}_{23}\left(\gamma \right)={tan}_{\Xi }(2\Psi \gamma )\pm \sqrt{r_1{r}_2}{sec}_{\Xi }(2\Psi \gamma )·\hfill \\ \hfill & {\mathcal{Q}}_{24}\left(\gamma \right)=-{cot}_{\Xi }(2\Psi \gamma )\pm \sqrt{r_1{r}_2}{csc}_{\Xi }(2\Psi \gamma )·\hfill \\ \hfill & {\mathcal{Q}}_{25}\left(\gamma \right)={\displaystyle \frac{1}{2}}({tan}_{\Xi }\left({\displaystyle \frac{\Psi }{2}}\gamma \right)-{cot}_{\Xi }\left({\displaystyle \frac{\Psi }{2}}\gamma \right))·\hfill \end{array}$$

(28)

Family 6: If $\Phi =0$ and $\Theta =-\Psi$, the corresponding hyper trigonometric solutions are derived as follows:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{26}\left(\gamma \right)=-{tanh}_{\Xi }(\Psi \gamma )·\hfill \\ \hfill & {\mathcal{Q}}_{27}\left(\gamma \right)=-{coth}_{\Xi }(\Psi \gamma )·\hfill \\ \hfill & {\mathcal{Q}}_{28}\left(\gamma \right)=-{tanh}_{\Xi }(2\Psi \gamma )\pm \iota \sqrt{r_1{r}_2}sec{h}_{\Xi }(2\Psi \gamma )·\hfill \\ \hfill & {\mathcal{Q}}_{29}\left(\gamma \right)=-{coth}_{\Xi }(2\Psi \gamma )\pm \sqrt{r_1{r}_2}csc{h}_{\Xi }(2\Psi \gamma )·\hfill \\ \hfill & {\mathcal{Q}}_{30}\left(\gamma \right)=-{\displaystyle \frac{1}{2}}({tanh}_{\Xi }\left({\displaystyle \frac{\Psi }{2}}\gamma \right)+{coth}_{\Xi }\left({\displaystyle \frac{\Psi }{2}}\gamma \right))·\hfill \end{array}$$

(29)

Family 7: If ${\Phi }^2=4\Theta \Psi$, then

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{31}\left(\gamma \right)={\displaystyle \frac{-2\Psi (\Phi \gamma ln(\Xi )+2)}{{\Phi }^2\gamma ln(\Xi )}}·\hfill \end{array}$$

(30)

Family 8: If $\Phi ={\lambda }_0$, $\Psi =r_3{\lambda }_0(r_3\ne 0)$, and $\Theta =0$, then

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{32}\left(\gamma \right)={\Xi }^{\lambda \gamma }-p·\hfill \end{array}$$

(31)

Family 9: If $\Phi =\Theta =0$, then

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{33}\left(\gamma \right)=\Psi \gamma ln(\Xi ).·\hfill \end{array}$$

(32)

Family 10: If $\Phi =\Psi =0$, then

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{34}\left(\gamma \right)={\displaystyle \frac{-1}{\Theta \gamma ln(\Xi )}}.·\hfill \end{array}$$

(33)

Family 11: If $\Psi =0$ and $\Phi \ne 0$, then

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{35}\left(\gamma \right)=-{\displaystyle \frac{r\Phi }{\Theta ({cosh}_{\Xi }(\Phi \gamma )-{sinh}_{\Xi }(\Phi \gamma )+r_1)}}·\hfill \\ \hfill & {\mathcal{Q}}_{36}\left(\gamma \right)=-{\displaystyle \frac{\Phi ({sinh}_{\Xi }(\Phi \gamma )+{cosh}_{\Xi }(\Phi \gamma ))}{\Theta ({sinh}_{\Xi }(\Phi \gamma )+{cosh}_{\Xi }(\Phi \gamma )+r_2)}}·\hfill \end{array}$$

(34)

Family 12: If $\Phi ={\lambda }_0$, $\Theta =r_3{\lambda }_0(r_3\ne 0)$, and $\Psi =0$, then

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{37}\left(\gamma \right)={\displaystyle \frac{r_1{\Xi }^{\lambda \gamma }}{r_2-r_3{r}_1{\Xi }^{\lambda \gamma }}}·\hfill \end{array}$$

(35)

The hyperbolic and trigonometric functions are introduced as follows:

$$\begin{array}{cc}\hfill & {sinh}_{\Xi }\left(\gamma \right)={\displaystyle \frac{r_1{\Xi }^{\gamma }-r_2{\Xi }^{-\gamma }}{2}},{cosh}_{\Xi }\left(\gamma \right)={\displaystyle \frac{r_1{\Xi }^{\gamma }+r_2{\Xi }^{-\gamma }}{2}}·\hfill \\ \hfill & {tanh}_{\Xi }\left(\gamma \right)={\displaystyle \frac{r_1{\Xi }^{\gamma }-r_2{\Xi }^{-\gamma }}{r_1{\Xi }^{\gamma }+r_2{\Xi }^{-\gamma }}},{coth}_{\Xi }\left(\gamma \right)={\displaystyle \frac{r_1{\Xi }^{\gamma }+r_2{\Xi }^{-\gamma }}{r_1{\Xi }^{\gamma }-r_2{\Xi }^{-\gamma }}}·\hfill \\ \hfill & csc{h}_{\Xi }\left(\gamma \right)={\displaystyle \frac{2}{r_1{\Xi }^{\gamma }-r_2{\Xi }^{-\gamma }}},sec{h}_{\Xi }\left(\gamma \right)={\displaystyle \frac{2}{r_1{\Xi }^{\gamma }+r_2{\Xi }^{-\gamma }}}·\hfill \\ \hfill & {sin}_{\Xi }\left(\gamma \right)={\displaystyle \frac{r_1{\Xi }^{\iota \gamma }-r_2{\Xi }^{-\iota \gamma }}{2\iota }},{cos}_{\Xi }\left(\gamma \right)={\displaystyle \frac{r_1{\Xi }^{\iota \gamma }+r_2{\Xi }^{-\iota \gamma }}{2}}·\hfill \\ \hfill & {tan}_{\Xi }\left(\gamma \right)=-\iota {\displaystyle \frac{r_1{\Xi }^{\iota \gamma }-r_2{\Xi }^{-\iota \gamma }}{r_1{\Xi }^{\iota \gamma }+r_2{\Xi }^{-\iota \gamma }}},{cot}_{\Xi }\left(\gamma \right)=\iota {\displaystyle \frac{r_1{\Xi }^{\iota \gamma }+r_2{\Xi }^{-\iota \gamma }}{}}·\hfill \\ \hfill & cs{c}_{\Xi }\left(\gamma \right)={\displaystyle \frac{2\iota }{r_1{\Xi }^{\iota \gamma }-r_2{\Xi }^{-\iota \gamma }}},se{c}_{\Xi }\left(\gamma \right)={\displaystyle \frac{2}{r_1{\Xi }^{\iota \gamma }+r_2{\Xi }^{-\iota \gamma }}}·\hfill \end{array}$$

(36)

5. Explicit Analytical Solution of Equation (1)

In this section, we explore the explicit analytical solution of Equation (1) using extended direct algebraic method. Determine the degree of the considered outcome by balancing the highest nonlinear and dispersive terms in Equation (19), yielding the balance number $\mathfrak{G}=1$. The initial solution in this case is given by

$$\zeta \left(\gamma \right)=g_0+g_1\mathcal{Q}\left(\gamma \right)·$$

(37)

By incorporating Equations (23) and (37) into Equation (19) and solving for $g_0,g_1,{\mu }_1,{\mu }_2,{\mu }_3$, and ${\mu }_4$, we obtain the following solution:

$$\left\{\begin{array}{c}{g}_0={\displaystyle \frac{\Phi ln(\Xi )\left({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{4{\mu }_3}},\hfill \\ g_1={\displaystyle \frac{\Theta ln(\Xi )\left({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{2{\mu }_3}},\hfill \\ {\mu }_4={\displaystyle \frac{-8{\mu }_3^2}{{({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2)}^2}},{\mu }_1={\mu }_1,{\mu }_2={\mu }_2·\hfill \end{array}\right.$$

(38)

Substituting the above values into Equation (37), the solutions of Equation (1) are obtained as follows:

Family 1: If $\rho <0$, ${\mu }_3\ne 0$, and $\Theta \ne 0$, the corresponding trigonometric solutions are derived as follows:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_1(\eta ,\tau )={\displaystyle \frac{ln(\Xi )\left({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{2{\mu }_3}}\left({\displaystyle \frac{\sqrt{-\rho }}{2}}{tan}_{\Xi }\left({\displaystyle \frac{\sqrt{-\rho }}{2}}(\eta -\sigma \tau )\right)\right)·\hfill \\ \hfill & {\mathcal{Q}}_2(\eta ,\tau )={\displaystyle \frac{ln(\Xi )\left({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{2{\mu }_3}}\left(-{\displaystyle \frac{\sqrt{-\rho }}{2}}{cot}_{\Xi }\left({\displaystyle \frac{\sqrt{-\rho }}{2}}(\eta -\sigma \tau )\right)\right)·\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill {\mathcal{Q}}_3(\eta ,\tau )& ={\displaystyle \frac{ln(\Xi )\left({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{2{\mu }_3}}\left({\displaystyle \frac{\sqrt{-\rho }}{2}}\right({tan}_{\Xi }\left(\sqrt{-\rho }(\eta -\sigma \tau )\right)\hfill \\ & \pm \sqrt{r_1{r}_2}{sec}_{\Xi }\left(\sqrt{-\rho }(\eta -\sigma \tau )\right)\left)\right)·\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill {\mathcal{Q}}_4(\eta ,\tau )& ={\displaystyle \frac{ln(\Xi )\left({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{2{\mu }_3}}\left({\displaystyle \frac{\sqrt{-\rho }}{2}}\right({cot}_{\Xi }\left(\sqrt{-\rho }(\eta -\sigma \tau )\right)\hfill \\ & \pm \sqrt{r_1{r}_2}{csc}_{\Xi }\left(\sqrt{-\rho }(\eta -\sigma \tau )\right)\left)\right)·\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill {\mathcal{Q}}_5(\eta ,\tau )& ={\displaystyle \frac{ln(\Xi )\left({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{2{\mu }_3}}\left({\displaystyle \frac{\sqrt{-\rho }}{4}}\right({tan}_{\Xi }\left({\displaystyle \frac{\sqrt{-\rho }}{4}}(\eta -\sigma \tau )\right)\hfill \\ & -{cot}_{\Xi }\left({\displaystyle \frac{\sqrt{-\rho }}{4}}(\eta -\sigma \tau )\right)\left)\right).\hfill \end{array}$$

(42)

Family 2: If $\rho >0$, ${\mu }_3\ne 0$, and $\Theta \ne 0$, the corresponding hyper trigonometric solutions are derived as follows:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_6(\eta ,\tau )={\displaystyle \frac{ln(\Xi )\left({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{2{\mu }_3}}\left(-{\displaystyle \frac{\sqrt{\rho }}{2}}{tanh}_{\Xi }\left({\displaystyle \frac{\sqrt{\rho }}{2}}(\eta -\sigma \tau )\right)\right)·\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_7(\eta ,\tau )={\displaystyle \frac{ln(\Xi )\left({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{2{\mu }_3}}\left(-{\displaystyle \frac{\sqrt{\rho }}{2}}{coth}_{\Xi }\left({\displaystyle \frac{\sqrt{\rho }}{2}}(\eta -\sigma \tau )\right)\right)·\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill {\mathcal{Q}}_8(\eta ,\tau )& ={\displaystyle \frac{ln(\Xi )\left({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{2{\mu }_3}}\left(\right(-{\displaystyle \frac{\sqrt{\rho }}{2}}({tanh}_{\Xi }\left({\displaystyle \frac{\sqrt{\rho }}{2}}(\eta -\sigma \tau )\right)\hfill \\ & \pm \iota \sqrt{r_1{r}_2}sec{h}_{\Xi }\left({\displaystyle \frac{\sqrt{\rho }}{2}}(\eta -\sigma \tau )\right)\left)\right))·\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\mathcal{Q}}_9(\eta ,\tau )& ={\displaystyle \frac{ln(\Xi )\left({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{2{\mu }_3}}(-{\displaystyle \frac{\sqrt{\rho }}{2}}{coth}_{\Xi }\left({\displaystyle \frac{\sqrt{\rho }}{2}}(\eta -\sigma \tau )\right)\hfill \\ & \pm \sqrt{r_1{r}_2}csc{h}_{\Xi }\left({\displaystyle \frac{\sqrt{\rho }}{2}}(\eta -\sigma \tau )\right))·\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\mathcal{Q}}_{10}(\eta ,\tau )& ={\displaystyle \frac{ln(\Xi )\left({[ln(\Xi )]}^2{\Phi }^2-4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{2{\mu }_3}}(-{\displaystyle \frac{\sqrt{\rho }}{4}}({tanh}_{\Xi }\left({\displaystyle \frac{\sqrt{\rho }}{4}}(\eta -\sigma \tau )\right)\hfill \\ & +{coth}_{\Xi }\left({\displaystyle \frac{\sqrt{\rho }}{4}}(\eta -\sigma \tau )\right)\left)\right)·\hfill \end{array}$$

(47)

Family 3: If $\Theta \Psi >0$, ${\mu }_3\ne 0$, and $\Phi =0$, the corresponding trigonometric solutions are derived as follows:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{11}(\eta ,\tau )={\displaystyle \frac{-{\mu }_{2}ln(\Xi )}{{\mu }_3}}\left(\sqrt{\Psi \Theta }{tan}_{\Xi }\left(\sqrt{\Theta \Psi }(\eta -\sigma \tau )\right)\right)·\hfill \\ \hfill & {\mathcal{Q}}_{12}(\eta ,\tau )={\displaystyle \frac{-{\mu }_{2}ln(\Xi )}{{\mu }_3}}\left(\sqrt{\Psi \Theta }{cot}_{\Xi }\left(\sqrt{\Theta \Psi }(\eta -\sigma \tau )\right)\right)·\hfill \\ \hfill & {\mathcal{Q}}_{13}(\eta ,\tau )={\displaystyle \frac{-{\mu }_{2}ln(\Xi )}{{\mu }_3}}\left(\sqrt{\Theta \Psi }({tan}_{\Xi }\left(2\sqrt{\Theta \Psi }(\eta -\sigma \tau )\right)\pm \sqrt{r_1{r}_2}{sec}_{\Xi }\left(2\sqrt{\Theta \Psi }(\eta -\sigma \tau )\right))\right)·\hfill \\ \hfill & {\mathcal{Q}}_{14}(\eta ,\tau )={\displaystyle \frac{-{\mu }_{2}ln(\Xi )}{{\mu }_3}}(\sqrt{\Theta \Psi }\left(-{cot}_{\Xi }\left(2\sqrt{\Theta \Psi }(\eta -\sigma \tau )\right)\pm \sqrt{r_1{r}_2}{csc}_{\Xi }\left(2\sqrt{\Theta \Psi }(\eta -\sigma \tau ))\right)\right)·\hfill \\ \hfill & {\mathcal{Q}}_{15}(\eta ,\tau )={\displaystyle \frac{-{\mu }_{2}ln(\Xi )}{{\mu }_3}}\left({\displaystyle \frac{\sqrt{\Theta \Psi }}{2}}\left({tan}_{\Xi }\left({\displaystyle \frac{\sqrt{\Theta \Psi }}{2}}(\eta -\sigma \tau )\right)-{cot}_{\Xi }\left({\displaystyle \frac{\sqrt{\Theta \Psi }}{2}}(\eta -\sigma \tau )\right)\right)\right)·\hfill \end{array}$$

(48)

Family 4: If $\Theta \Psi <0$, ${\mu }_3\ne 0$, and $\Phi =0$, the corresponding hyper trigonometric solutions are derived as follows:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{16}(\eta ,\tau )={\displaystyle \frac{\sqrt{-\Psi }{\mu }_{2}ln(\Xi )}{{\mu }_3}}\left({tanh}_{\Xi }\left(\sqrt{-\Theta \Psi }(\eta -\sigma \tau )\right)\right)·\hfill \\ \hfill & {\mathcal{Q}}_{17}(\eta ,\tau )={\displaystyle \frac{\sqrt{-\Psi }{\mu }_{2}ln(\Xi )}{{\mu }_3}}\left({coth}_{\Xi }\left(\sqrt{-\Theta \Psi }(\eta -\sigma \tau )\right)\right)·\hfill \\ \hfill & {\mathcal{Q}}_{18}(\eta ,\tau )={\displaystyle \frac{\sqrt{-\Psi }{\mu }_{2}ln(\Xi )}{{\mu }_3}}\left({tanh}_{\Xi }\left(2\sqrt{-\Theta \Psi }(\eta -\sigma \tau )\right)\pm \iota \sqrt{r_1{r}_2}sec{h}_{\Xi }\left(2\sqrt{-\Theta \Psi }(\eta -\sigma \tau )\right)\right)·\hfill \\ \hfill & {\mathcal{Q}}_{19}(\eta ,\tau )={\displaystyle \frac{\sqrt{-\Psi }{\mu }_{2}ln(\Xi )}{{\mu }_3}}\left({coth}_{\Xi }\left(2\sqrt{-\Theta \Psi }(\eta -\sigma \tau )\right)\pm \sqrt{r_1{r}_2}csc{h}_{\Xi }\left(2\sqrt{-\Theta \Psi }(\eta -\sigma \tau )\right)\right)·\hfill \\ \hfill & {\mathcal{Q}}_{20}(\eta ,\tau )={\displaystyle \frac{\sqrt{-\Psi }{\mu }_{2}ln(\Xi )}{2{\mu }_3}}\left({tanh}_{\Xi }\left(\frac{\sqrt{-\Theta \Psi }}{2}(\eta -\sigma \tau )\right)+{coth}_{\Xi }\left(\frac{\sqrt{-\Theta \Psi }}{2}(\eta -\sigma \tau )\right)\right).\hfill \end{array}$$

(49)

Family 5: If $\Phi =0$, ${\mu }_3\ne 0$, and $\Theta =\Psi$, the corresponding trigonometric solutions are derived as follows:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{21}(\eta ,\tau )={\displaystyle \frac{-{\mu }_2\Psi {tan}_{\Xi }(\Psi (\eta -\sigma \tau ))}{{\mu }_3}}·\hfill \\ \hfill & {\mathcal{Q}}_{22}(\eta ,\tau )={\displaystyle \frac{-ln(\Xi ){\mu }_2\Psi {cot}_{\Xi }(\Psi (\eta -\sigma \tau ))}{{\mu }_3}}·\hfill \\ \hfill & {\mathcal{Q}}_{23}(\eta ,\tau )={\displaystyle \frac{-\Psi ln(\Xi ){\mu }_2\left({tan}_{\Xi }(2\Psi (\eta -\sigma \tau ))\pm \sqrt{r_1{r}_2}{sec}_{\Xi }(2\Psi (\eta -\sigma \tau ))\right)}{{\mu }_3}}·\hfill \\ \hfill & {\mathcal{Q}}_{24}(\eta ,\tau )={\displaystyle \frac{-2ln(\Xi ){\mu }_2\Psi \left(-{cot}_{\Xi }(2\Psi (\eta -\sigma \tau ))\pm \sqrt{r_1{r}_2}{csc}_{\Xi }(2\Psi (\eta -\sigma \tau ))\right)}{2{\mu }_3}}·\hfill \\ \hfill & {\mathcal{Q}}_{25}(\eta ,\tau )={\displaystyle \frac{-2ln(\Xi ){\mu }_2\Psi \left(\frac{1}{2}({tan}_{\Xi }\left(\frac{\Psi }{2}(\eta -\sigma \tau )\right)-{cot}_{\Xi }\left(\frac{\Psi }{2}(\eta -\sigma \tau )\right))\right)}{{\mu }_3}}·\hfill \end{array}$$

(50)

Family 6: If $\Phi =0$, ${\mu }_3\ne 0$, and $\Theta =-\Psi$, the corresponding hyper trigonometric solutions are derived as follows:

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{26}(\eta ,\tau )={\displaystyle \frac{-ln(\Xi ){\mu }_2\Psi {tanh}_{\Xi }(\Psi (\eta -\sigma \tau ))}{{\mu }_3}}·\hfill \\ \hfill & {\mathcal{Q}}_{27}(\eta ,\tau )={\displaystyle \frac{-ln(\Xi ){\mu }_2\Psi {coth}_{\Xi }(\Psi (\eta -\sigma \tau ))}{{\mu }_3}}·\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{28}(\eta ,\tau )={\displaystyle \frac{\Psi ln(\Xi ){\mu }_2(-{tanh}_{\Xi }(2\Psi (\eta -\sigma \tau ))\pm \iota \sqrt{r_1{r}_2}sec{h}_{\Xi }(2\Psi (\eta -\sigma \tau )))}{{\mu }_3}}·\hfill \\ \hfill & {\mathcal{Q}}_{29}(\eta ,\tau )={\displaystyle \frac{\Psi ln(\Xi ){\mu }_2(-{coth}_{\Xi }(2\Psi (\eta -\sigma \tau ))\pm \iota \sqrt{r_1{r}_2}csc{h}_{\Xi }(2\Psi (\eta -\sigma \tau )))}{{\mu }_3}}·\hfill \\ \hfill & {\mathcal{Q}}_{30}(\eta ,\tau )={\displaystyle \frac{-\Psi ln(\Xi ){\mu }_2(-{tanh}_{\Xi }(2\Psi (\eta -\sigma \tau ))\pm \iota \sqrt{r_1{r}_2}{coth}_{\Xi }(2\Psi (\eta -\sigma \tau )))}{2{\mu }_3}}·\hfill \end{array}$$

(52)

Family 7: If ${\Phi }^2=4\Theta \Psi$, then

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{31}(\eta ,\tau )={\displaystyle \frac{ln(\Xi )\left({[ln(\Xi )]}^{2}4\Theta \Psi -4{[ln(\Xi )]}^2\Theta \Psi -2{\mu }_2\right)}{2{\mu }_3}}\left({\displaystyle \frac{\Phi }{2}}+\Theta \left({\displaystyle \frac{-\Psi (\Phi (\eta -\sigma \tau )ln(\Xi )+1)}{2\Theta \Psi (\eta -\sigma \tau )ln(\Xi )}}\right)\right)·\hfill \end{array}$$

(53)

Family 8: If $\Phi ={\lambda }_0$, $\Psi =r_3{\lambda }_0(r_3\ne 0)$, and $\Theta =0$, then

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{32}(\eta ,\tau )={\displaystyle \frac{ln(\Xi )\Phi ({[ln(\Xi )]}^2{\lambda }_0^2-2{\mu }_2)}{4{\mu }_3}}·\hfill \end{array}$$

(54)

Family 9: If $\Phi =\Psi =0$, then

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{34}(\eta ,\tau )={\displaystyle \frac{{\mu }_2}{{\mu }_3(\eta -\sigma \tau )}}·\hfill \end{array}$$

(55)

Family 10: If $\Psi =0$ and $\Phi \ne 0$, then

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{35}(\eta ,\tau )={\displaystyle \frac{ln(\Xi )r_3{\mu }_2\Phi }{{\mu }_3({cosh}_{\Xi }(\Phi (\eta -\sigma \tau ))-{sinh}_{\Xi }(\Phi (\eta -\sigma \tau ))+r_1)}}·\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{36}(\eta ,\tau )={\displaystyle \frac{-ln(\Xi ){\mu }_2}{{\mu }_3}}\left({\displaystyle \frac{\Phi }{2}}-{\displaystyle \frac{\Phi ({sinh}_{\Xi }(\Phi (\eta -\sigma \tau ))+{cosh}_{\Xi }(\Phi (\eta -\sigma \tau )))}{({sinh}_{\Xi }(\Phi (\eta -\sigma \tau ))+{cosh}_{\Xi }(\Phi (\eta -\sigma \tau ))+r_2)}}\right)·\hfill \end{array}$$

(57)

Family 11: If $\Phi ={\lambda }_0$, $\Theta =r_3{\lambda }_0(r_3\ne 0)$, and $\Psi =0$, then

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_{37}(\eta ,\tau )={\displaystyle \frac{ln(\Xi )({[ln(\Xi )]}^2{\lambda }_0^2-2{\mu }_2)}{2{\mu }_3}}\left({\displaystyle \frac{{\lambda }_0}{2}}+{\displaystyle \frac{r_1{r}_3\lambda {\Xi }^{\lambda (\eta -\sigma \tau )}}{r_2-r_3{r}_1{\Xi }^{\lambda (\eta -\sigma \tau )}}}\right)·\hfill \end{array}$$

(58)

6. Graphical Interpretations and Analysis

This section presents a graphical representation of the results obtained through the extended direct algebraic method by selecting distinct parameter values. Various wave structures, including Kink waves, periodic waves, anti-kink waves, complex solitary singular waves, singular periodic waves, and shock waves, are effectively demonstrated [32,33].

Figure 1: The solution ${\vartheta }_6(\eta ,\tau )$ with parameters $\Psi =0.88,\Theta =3.56,\Xi =e,{\mu }_2=1.25,{\mu }_3=0.89,\rho =8.2736$ exhibits a kink soliton. The simulation is performed on the domain $\eta ,\tau \in [-10,10]$, with wavenumber $k=0.79$. Kink solitons are important in condensed matter physics, where they describe domain walls and phase-transition fronts in nonlinear lattices and field theories. Figure 2 The solution ${\vartheta }_8(\eta ,\tau )$ for $\Psi =0.78,\Theta =4.23,\Xi =e,{\mu }_2=0.895,{\mu }_3=1.2,\rho =14.6169$ yields a bright soliton. Computations use the domain $\eta ,\tau \in [-10,10]$, with wavenumber $k=1.27$. Bright solitons play a key role in optical fiber communications, as they enable the transmission of stable light pulses over long distances without distortion [34,35].

Figure 3: The solution ${\vartheta }_{28}(\eta ,\tau )$ with $\Psi =1.56,\Theta =-1.56,\Phi =0,\Xi =e,{\mu }_2=1.09,{\mu }_3=0.89,r_1=0.45,r_2=0.25,\rho =8.2736$ exhibits kink-type soliton behavior. The domain is $\eta ,\tau \in [-10,10]$, with wavenumber $k=1.19$. Such kink-type structures are widely applied in plasma physics and fluid dynamics, where they model nonlinear wave fronts and shock-like solitary excitations [36,37]. Figure 4: The solution ${\vartheta }_{36}(\eta ,\tau )$ for $\Psi =1.56,\Theta =-1.56,\Phi =0,\Xi =e,{\mu }_2=1.09,{\mu }_3=0.89,r_1=0.45,r_2=0.25,\rho =8.2736$ produces a periodic soliton train.The results are shown on $\eta ,\tau \in [-10,10]$, with wavenumber $k=3.89$. Periodic soliton trains are significant in nonlinear optics and photonics, where they support soliton lattices used for frequency conversion, switching, and robust energy transport in metamaterials [38,39]

7. Comparative Evaluation

Table 1. Result comparison analysis.

Reference	Work
[11]	A generalized Cahn–Hilliard equation with convection and dissipation is studied under Neumann boundary conditions. Local bifurcations, stability, and inhomogeneous equilibria are analyzed using dynamical systems methods. The results are compared with previous work on periodic boundary conditions.
[13]	This article reviews the Cahn–Hilliard equation and its variants, highlighting their applications in areas such as biology and image inpainting.
[14]	The free energy of a nonuniform isotropic system depends on composition gradients and a parameter related to temperature. This formulation explains interfacial properties, predicting that interface thickness grows with temperature and diverges at the critical point, consistent with experiments.
Current study	The convective Cahn–Hilliard–Oono equation is analyzed under specified conditions, and its Lie symmetries are explored through symmetry generators. Using Lie group methods, the equation is reduced to nonlinear ODEs and solved via theextended direct algebraic method. A wide range of soliton solutions is obtained, including periodic, anti-kink, bright, dark, shock, kink, and mixed types, with their features illustrated through 2D, 3D, and contour plots. Stability analysis further demonstrates the model’s ability to capture diverse nonlinear wave structures.

compares prior studies and highlights this work’s contributions.

8. Examining the Stability of Equation (1)

Consider a perturbed solution of the form [40]:

$$\vartheta (\eta ,\tau )={\mu }_0+{\Delta }_0\varsigma (\eta ,\tau ).$$

(59)

It is clear that any constant ${\mu }_0$ can act as a stable solution for Equation (1). By substituting Equation (59) into Equation (1), we obtain the following linearization form:

$${\Delta }_0{\varsigma }_{\tau }+{\Delta }_0{\varsigma }_{\eta \eta \eta \eta }+{\mu }_2{\Delta }_0{\varsigma }_{\eta \eta }+{\mu }_1{\Delta }_0\varsigma +2{\mu }_3\Delta {\mu }_0{\varsigma }_{\eta }+12{\mu }_4{\Delta }_0{\mu }_0^2{\varsigma }_{\eta \eta }+3{\mu }_4{\Delta }_0{\mu }_0^2{\varsigma }_{\eta \eta }=0.$$

(60)

Take into account Equation (60), which admits a solution of the form

$$\varsigma (\eta ,\tau )=e^{i({\rm Y}\eta +\alpha \tau )}.$$

(61)

By substituting Equation (61) into Equation (60) and solving for ${\Delta }_0$, we derive $\alpha$, where $\alpha$ represents the normalized wave number:

$$\alpha =-2{\mu }_3{\mu }_0{\rm Y}+i({\mu }_1+{{\rm Y}}^4-\mu {{\rm Y}}^2)·$$

(62)

The behavior of $\alpha$, which controls the growth or decay of perturbations over time, can be used to analytically analyze the stability of the perturbed solution. The growth or decay of perturbations is determined by the real component of $\alpha$. The solution’s oscillatory nature is determined by the imaginary portion.

8.1. Stability Conditions

• If $Re\left(\alpha \right)<0$, i.e., $-2{\mu }_3{\mu }_0{\rm Y}<0$, then perturbations diminish over time, leading to stability. This holds when ${\mu }_3{\mu }_0>0$, as shown in Figure 5a.
• If $Re\left(\alpha \right)>0$, perturbations amplify, leading to instability, as shown in Figure 5a. This occurs when ${\mu }_3{\mu }_0<0$.
• If $Re\left(\alpha \right)=0$, the solution neither grows nor decays, implying marginal stability, as shown in Figure 5a.

8.2. Physical Interpretation

The coefficient ${\mu }_3$, which modifies the impact of perturbations on the wave propagation, determines the system’s stability. In the event that ${\mu }_3$ and ${\mu }_0$ have opposing signs, perturbations will increase and may result in turbulent or chaotic behavior. Wave dispersion is influenced by the term ${{\rm Y}}^4-{\mu }_2{{\rm Y}}^2$ in the imaginary component, which alters the oscillation frequency, as shown in Figure 5b. Complex eigenvalues imply that disturbances will oscillate as they either decay or become amplified over time.

9. Conclusions

This study explores a nonlinear evolutionary partial differential equation, formulated as a generalized version of the convective Cahn–Hilliard–Oono equation, incorporating additional terms to account for convection and dissipation effects. The Lie algebra of the proposed model is constructed, leading to the development of an ideal system of subalgebras. Translational symmetry is established, reinforcing the equation’s structural consistency. To analyze the model, which provides different classes of distinct solutions, we employ the extended direct algebraic method. This approach simplifies problem solving, offering efficient and straightforward solutions applicable across various scientific and mathematical domains. The obtained solitary wave solutions, including hyperbolic and trigonometric forms, are entirely novel and have not been previously published, marking a significant advancement in the field. Their physical relevance is demonstrated through an analysis of dispersion effects on pattern formation in liquid droplets, considering molecular motion and particle displacement due to evaporation. Graphical representations, including 3D, 2D, and contour plots, are utilized to illustrate the solutions and their implications, as shown in Figure 1, Figure 2, Figure 3 and Figure 4. Analytical methods provide exact, closed-form solutions that enhance the understanding of nonlinear phenomena while being computationally efficient compared to numerical simulations. Furthermore, the study conducts a stability analysis, revealing that the model remains stable when ${\mu }_3{\mu }_0>0$ and becomes unstable when ${\mu }_3{\mu }_0<0$, as shown in Figure 5. The interplay between real and imaginary components determines whether the system exhibits steady decay, oscillatory growth, or chaotic transitions. These findings contribute to advancing theoretical research and broadening the understanding of nonlinear dynamic systems.

Future directions: In the future, this work can be extended in several directions. One potential avenue is the construction of lump, breather, and multi-soliton solutions. In addition, various numerical methods such as the Variational Iteration Method, Adomian Decomposition Method, and perturbation techniques may be applied to study this problem more effectively. Another promising direction is the investigation of the system’s dynamical behavior through chaos, bifurcation, and sensitivity analyses. Furthermore, modern approaches such as neural network-based techniques can also be explored to enhance the analysis and solution of the model.