Exact Solutions and Soliton Transmission in Relativistic Wave Phenomena of Klein–Fock–Gordon Equation via Subsequent Sine-Gordon Equation Method

Abstract

This study explores the (1+1)-dimensional Klein–Fock–Gordon equation, a distinct third-order nonlinear differential equation of significant theoretical interest. The Klein–Fock–Gordon equation (KFGE) plays a pivotal role in theoretical physics, modeling high-energy particles and providing a fundamental framework for simulating relativistic wave phenomena. To find the exact solution of the proposed model, for this purpose, we utilized two effective techniques, including the sine-Gordon equation method and a new extended direct algebraic method. The novelty of these approaches lies in the form of different solutions such as hyperbolic, trigonometric, and rational functions, and their graphical representations demonstrate the different form of solitons like kink solitons, bright solitons, dark solitons, and periodic waves. To illustrate the characteristics of these solutions, we provide two-dimensional, three-dimensional, and contour plots that visualize the magnitude of the (1+1)-dimensional Klein–Fock–Gordon equation. By selecting suitable values for physical parameters, we demonstrate the diversity of soliton structures and their behaviors. The results highlighted the effectiveness and versatility of the sine-Gordon equation method and a new extended direct algebraic method, providing analytical solutions that deepen our insight into the dynamics of nonlinear models. These results contribute to the advancement of soliton theory in nonlinear optics and mathematical physics.

1. Introduction

During previous decades, nonlinear physical phenomena peculiarly shaped by nonlinear partial differential equations (NLPDEs) have demanded much attention from researchers, physicists, mathematicians, engineers, and scientists. These NLPDEs offer an essential mathematical foundation for comprehending how physical quantities or fields change over time in systems where nonlinearity is a crucial feature. These equations are used in many primary fields and are of great importance in comprehending and simulating complicated occurrences in a variety of fields. In plasma physics, the generalized long wave (GRLW) equation [1] is crucial due to its primary application in the study of dispersion wave phenomena, such as magneto-hydrodynamic waves in plasma physics. In addition, the GRLW equation is utilized to simulate the formation of undular bores and transverse waves in shallower water. Likewise, in optical physics, the generalized Schrödinger equation [2] describes the propagation of optical pulses in media. Nonlinear models are also used in population dynamics, particularly in ecology and epidemiology to explore disease transmission mechanisms, in predator–prey relationships, and population increases. For example, to express the dynamics of interacting species in ecosystems, the Lotka–Volterra [3,4] population models are extensively used. The complex Ginzburg–Landau equation [5] is another well-known nonlinear model which represents a broad variety of physical phenomena like Bose–Einstein condensation (BEC), super fluidity, second-order phase transitions, superconductivity, liquid crystals, and strings in field theory.

These formulas provide a thorough and clear comprehension of the physical processes under study, allowing for precise forecasts of how they will evolve. In order to obtain a more thorough understanding of the behavior shown by the physical events being studied, several researchers have dedicated their time to investigating a variety of nonlinear models. Some of the nonlinear models that exist in this scope are the Einstein vacuum field equation [6], the mKdV equation [7], novel (2 + 1) and (3 + 1) forms of the Biswas–Milovic equations [8], the Riemann wave model [9], the BBMP equation [10], and many more models [11,12,13,14,15,16,17].

Executing a precise solution of such nonlinear models is very challenging. Soliton wave analysis is an area that is experiencing a significant increase in the application of NLPDEs. Referred to as soliton waves, these waves are distinct bundles of energy called localized wave packets that move at a steady speed and maintain their shape. Researchers are utilizing a range of nonlinear physical frameworks to analyze and predict the characteristics of soliton waves systematically. As soliton theory has advanced, numerous genuine techniques have been established and presented to look at soliton solutions. These techniques or methods involve the unified Riccati equation expansion method [18], the extended $\left({\displaystyle \frac{G^{\prime }}{G^2}}\right)$ expansion method, the modified auxiliary equation method [10], the generalized Arnous method [14,19], the Sardar sub-equation method [20], the new Kudryashov method [14,21], the extended $exp(-\mathsf{\Psi }(\chi \left)\right)$ expansion method [14], the modified F-expansion method [22], the Kumar Malik method [23], the modified Kudryashov method [24], bifurcation and chaotic behavior [25,26,27], the modified Jacobi elliptic expansion method [28], the Riccati–Bernoulli sub-optimal differential equation approach [29], the generalized exponential rational function method [30], and many more [31,32,33,34,35,36,37].

The primary objective of this research is to perform an exhaustive analysis of the third-order Klein–Fock–Gordon equation (K-F-G-E) [38,39], which is characterized as a nonlinear model. This model has potential applicability across a wide range of material systems and has significant implications for high-energy particle physics and quantum mechanics. It characterizes the quantum amplitude for probing the behavior of a particle in varying fields as it propagates in both the forward and reverse temporal directions [40]. The relativistic wave equation linked with the Schrödinger equation is sometimes called the K-F-G-E. Several articles examine a variety of various forms of the K-F-G-E [41,42,43,44]. It is used to simulate many types of matter, such as the characteristics of elementary particles and the distribution of deviations in crystals. We begin our investigation by using the following form for KFGE:

$${\mathcal{U}}_{\mathcal{TT}}+\alpha {\mathcal{U}}_{\mathcal{XX}}+\beta \mathcal{U}+\delta {\mathcal{U}}^3=0.$$

(1)

Here $\mathcal{U}=\mathcal{U}(\mathcal{X},\mathcal{T})$ is the wave profile of the particle and $\alpha$, $\beta$, and $\delta$ are real constants. This model equation accounts for the propagation of magnetic flux along a Josephson line, the distribution of splay waves, the theoretical framework of elementary particles, the dispersal of dislocations within crystalline structures, etc. [45]. A variety of methods, encompassing the 2021 study by Alam et al. [46], studied the K-F-G model and found some stable and functional solutions like kink-type shapes, bright and dark lump shapes, bright and dark singular kinky shapes, periodic bright and dark lump shapes, multiple bright and dark lump shapes, a lump with a rough wave shape, and the rough wave shape by using the modified ${\displaystyle \frac{G^{\prime }}{G}}$ expansion method and generalized Kudryashov method. In 2020, Aly et al. [47] investigated the K-F-G equation using two mathematical approaches, namely the simple equation method and the modified F-expansion method, to obtain advanced analytical solutions. In 2022, Hamood et al. [39] studied the K-F-G equation using a newly developed technique known as the Sardar sub-equation method. They obtained various types of solutions, including singular, periodic singular, combined dark–singular, and combined dark–bright solitons. Furthermore, they also conducted a stability analysis. In 2024, Mohammad et al. [48] studied the fractional form of the (1+1)-dimensional K-F-G equation. By applying the Galilean transformation, the nonlinear ordinary differential equations (NLODEs) were converted into a dynamical system. They then performed both perturbed and unperturbed analyses to investigate bifurcation and chaotic behavior, respectively. In 2021, Ghazala et al. [49] investigated the time-fractional (1+1)-dimensional K-F-G equation using an effective technique known as the generalized projective Riccati equation method. These methods have already been used to derive the soliton solution from the K-F-G equation. The novelty of this study lies in solving the K-F-G equation using two innovative analytical techniques, namely the sine-Gordon equation method and a newly developed extended direct algebraic method. To the best of our knowledge, these methods have not been previously applied to this model. By employing these approaches, we successfully derived a wide variety of exact solutions, including kink solitons, bright solitons, dark solitons, anti-kink solitons, and periodic and solitonic structures, which provide deeper insight into the dynamics of the K-F-G equation. The application of this method allows researchers to explore and analyze complex nonlinear optical systems with greater confidence, contributing to a deeper understanding of their behavior and opening avenues for practical applications in optical communication, signal processing, and nonlinear optics.

The main goal of this study is to find stable and functional solutions to the nonlinear third-order K-F-G model by applying the NEDAM and the sine-Gordon expansion approach. The sine-Gordon equation expansion technique, especially for equations with sine and cosine nonlinearities, offers a methodical and adaptable way to get precise and approximate answers. In contrast to other approaches that might just provide numerical approximations, the sine-Gordon technique can produce analytically meaningful solutions, including soliton solutions that are significant in a variety of physical applications by handling complicated nonlinearities. Similarly, NEDAM produces easier and more straightforward answers by avoiding the need for complex numerical simulations or approximations through the use of an expansion technique. It has a wide range of solutions, including soliton-like ones, and it is especially useful for handling high-order nonlinear variables. The solutions obtained from these two techniques have never been derived before and will provide a thorough and deep understanding of the behavior of the nonlinear model. Furthermore, researchers may use these approximate findings to study the behavior of relativistic scalar fields, the behavior of spin-0 particles, QFT, etc.

The rest of this study is organized as follows:

• In Section 2, the mathematical analysis of the suggested models will be explained.
• In Section 3, the soliton solutions of the KFGE will be extracted via the SGE formula.
• In Section 4, the solutions of same model will be extracted via the NEDAM technique.
• In Section 5, conclusions will be explained on the basis of investigations on the proposed model.

2. Mathematical Evaluation of Proposed Models

Generally, an NLPDE has the following form:

$$M(\mathcal{U},{\mathcal{U}}_{\mathcal{X}},{\mathcal{U}}_{\mathcal{XX}},{\mathcal{U}}_{\mathcal{TT}},{\mathcal{U}}_{\mathcal{XT}},{\mathcal{U}}_{\mathcal{XXX}}\dots )=0.$$

(2)

as the unknown function $\mathcal{U}=M(\mathcal{X},\mathcal{T})$.

We impose the following traveling wave transformation:

$$\mathcal{U}(\mathcal{X},\mathcal{T})=u\left(\xi \right),\xi =\eta \mathcal{X}-\omega \mathcal{T}.$$

(3)

By substituting Equation (3) into Equation (2) with the help of Maple, an NLODE will be collected in the following form:

$$M(u,u^{\prime },u^{″},u^{‴}\dots )=0.$$

(4)

Utilizing the traveling wave transformation described in Equation (3) in Equation (1), we obtained the following ODE:

$$\delta u^3+({\omega }^2+\alpha {\eta }^2)u^{″}+\beta u=0.$$

(5)

2.1. Algorithm of Sine-Gordon Equation Expansion Method

This segment provides an overview and analysis of the process of the sine-Gordon expansion method. We consider the sine-Gordon equation of the following form [50]:

$${\mathcal{U}}_{\mathcal{TT}}-{\mathcal{U}}_{\mathcal{XX}}-{\gamma }^{2}sin\left(\mathcal{U}\right)=0.$$

(6)

Here $\mathcal{U}(\mathcal{X},\mathcal{T})$ is an anonymous function and $\gamma$ is a real nonzero constant. We may use the following transformation:

$$\mathcal{U}(\mathcal{X},\mathcal{T})=u\left(\xi \right),\xi =k\mathcal{X}-l\mathcal{T}.$$

(7)

We use Equation (6) to get the relevant ODE of the following form:

$$u^{″}={\displaystyle \frac{{\gamma }^2}{-k^2+l^2}}sinu\left(\xi \right).$$

(8)

where after simplification, we can write

$${\left({\displaystyle \frac{u^{\prime }}{2}}\right)}^2={\displaystyle \frac{{\gamma }^2}{l^2-k^2}}{\left(sin\left({\displaystyle \frac{u}{2}}\right)\right)}^2+K.$$

(9)

Here K is an integration constant and setting it equal to zero, also assuming that ${\displaystyle \frac{{\gamma }^2}{l^2-k^2}}=P^2$ and $z\left(\xi \right)={\displaystyle \frac{u\left(\xi \right)}{2}}$, we get

$${\left(z^{\prime }\left(\xi \right)\right)}^2=P^{2}sinz{\left(\xi \right)}^2.$$

(10)

or

$$z^{\prime }\left(\xi \right)=Psinz\left(\xi \right).$$

(11)

Assuming P = 1, we get the simplified form of the sine-Gordon equation as

$$z^{\prime }\left(\xi \right)=sinz\left(\xi \right).$$

(12)

By separating the variables, one may get the following solutions in the form of two remarkable equations for Equation (12):

$$\left.\begin{array}{c}sin\left(z\right)=sinz\left(\xi \right)={\displaystyle \frac{2B{e}^{\xi }}{1+B^2{e}^{2\xi }}}.\hfill \\ cos\left(z\right)=cosz\left(\xi \right)={\displaystyle \frac{-1+B^2{e}^{2\xi }}{1+B^2{e}^{2\xi }}}.\hfill \end{array}\right\}$$

(13)

Here B is a constant of integration and if we let B = 1, then

$$\left.\begin{array}{c}sin\left(z\right)=sinz\left(\xi \right)={\displaystyle \frac{2e^{\xi }}{1+e^{2\xi }}}=sech\left(\xi \right).\hfill \\ cos\left(z\right)=cosz\left(\xi \right)={\displaystyle \frac{-1+e^{2\xi }}{1+e^{2\xi }}}=tanh\left(\xi \right).\hfill \end{array}\right\}$$

(14)

Remember that

$$\left.\begin{array}{c}sinz\left(\xi \right)=sech\left(\xi \right)andsinz\left(\xi \right)=\iota cosech\left(\xi \right).\hfill \\ cosz\left(\xi \right)=tanh\left(\xi \right)andcosz\left(\xi \right)=coth\left(\xi \right).\hfill \end{array}\right\}$$

(15)

Now we consider a polynomial of Equation (2) which can be converted to an ODE of Equation (4) by using the traveling wave solution i.e., Equation (3). As per the sine-Gordon expansion method, one may set the solution of Equation (4) to the following form:

$$u\left(\xi \right)=m_0+\sum _{r=1}^{N}tan{h}^{r-1}\left(\xi \right)[n_{r}sech\left(\xi \right)+m_{r}tanh\left(\xi \right)].$$

(16)

We rewrite Equation (16) by joining the findings from Equations (14) and (15) as follows:

$$u\left(\xi \right)=m_0+\sum _{r=1}^{N}cosz{\left(\xi \right)}^{r-1}\left(\xi \right)[n_{r}sinz\left(\xi \right)+m_{r}cosz\left(\xi \right)].$$

(17)

By applying the homogeneous balancing principle to the NLODE, we find that the value of N occurs in Equation (16) or Equation (17). Afterwards, by setting the various coefficients of [$si{n}^r\left(z\left(\xi \right)\right)co{s}^s\left(z\left(\xi \right)\right)$] to zero, an algebraic system of equations is generated. The values of $m_r$, $n_r$, k, and l are determined by solving this system of algebraic equations, ultimately leading to the attainment of the requisite solutions to Equation (2).

2.2. Algorithm of New Extended Direct Algebraic Method

The key steps of the new extended direct algebraic technique will be covered in this section. According to NEDAM [5], the solution to Equation (4) can be written in form of a polynomial as follows:

$$u\left(\xi \right)=\sum _{i=0}^N{b}_i{V}^i\left(\xi \right).$$

(18)

where $b_i$ will be determined later, as they are constant coefficients. The polynomial $V\left(\xi \right)$ in Equation (18) is represented as

$$V^{\prime }\left(\xi \right)=ln\left(D\right)(ϵ+\pi V\left(\xi \right)+\tau V^2\left(\xi \right)),D\ne 0,1.$$

(19)

Here $ϵ$, $\pi$, and $\tau$ are real constants, and we let $\mathsf{\Omega }={\pi }^2-4ϵ\tau$. The following categories apply to the answers of Equation (19):

• $\mathbf{Solution}\mathbf{I}$: When $\mathsf{\Omega }<0$ and $\tau \ne 0$, then

  $$V_1\left(\xi \right)=-{\displaystyle \frac{\pi }{2\tau }}+{\displaystyle \frac{\sqrt{-\left(\mathsf{\Omega }\right)}}{2\tau }}ta{n}_D\left({\displaystyle \frac{\sqrt{-\left(\mathsf{\Omega }\right)}\xi }{2}}\right).$$

  (20)

  $$V_2\left(\xi \right)=-{\displaystyle \frac{\pi }{2\tau }}+{\displaystyle \frac{\sqrt{-\left(\mathsf{\Omega }\right)}}{2\tau }}co{t}_D\left({\displaystyle \frac{\sqrt{-\left(\mathsf{\Omega }\right)}\xi }{2}}\right).$$

  (21)

  $$V_3\left(\xi \right)=-{\displaystyle \frac{\pi }{2\tau }}+{\displaystyle \frac{\sqrt{-\left(\mathsf{\Omega }\right)}}{2\tau }}\left(ta{n}_D\left(\sqrt{-\left(\mathsf{\Omega }\right)}\xi \right)\pm \sqrt{rs}se{c}_D\left(\sqrt{-\left(\mathsf{\Omega }\right)}\xi \right)\right).$$

  (22)

  $$V_4\left(\xi \right)=-{\displaystyle \frac{\pi }{2\tau }}-{\displaystyle \frac{\sqrt{-\left(\mathsf{\Omega }\right)}}{2\tau }}\left(co{t}_D\left(\sqrt{-\left(\mathsf{\Omega }\right)}\xi \right)\pm \sqrt{rs}cose{c}_D\left(\sqrt{-\left(\mathsf{\Omega }\right)}\xi \right)\right).$$

  (23)

  $$V_5\left(\xi \right)=-{\displaystyle \frac{\pi }{2\tau }}+{\displaystyle \frac{\sqrt{-\left(\mathsf{\Omega }\right)}}{4\tau }}\left(ta{n}_D\left({\displaystyle \frac{\sqrt{-\left(\mathsf{\Omega }\right)}\xi }{4}}\right)-co{t}_D\left({\displaystyle \frac{\sqrt{-\left(\mathsf{\Omega }\right)}\xi }{4}}\right)\right).$$

  (24)
• $\mathbf{Solution}\mathbf{II}$: When $\mathsf{\Omega }>0$ and $\tau \ne 0$, then

  $$V_6\left(\xi \right)=-{\displaystyle \frac{\pi }{2\tau }}-{\displaystyle \frac{\sqrt{\mathsf{\Omega }}}{2\tau }}tan{h}_D\left({\displaystyle \frac{\sqrt{\mathsf{\Omega }}\xi }{2}}\right).$$

  (25)

  $$V_7\left(\xi \right)=-{\displaystyle \frac{\pi }{2\tau }}-{\displaystyle \frac{\sqrt{\mathsf{\Omega }}}{2\tau }}cot{h}_D\left({\displaystyle \frac{\sqrt{\mathsf{\Omega }}\xi }{2}}\right).$$

  (26)

  $$V_8\left(\xi \right)=-{\displaystyle \frac{\pi }{2\tau }}-{\displaystyle \frac{\sqrt{\mathsf{\Omega }}}{2\tau }}\left(tan{h}_D\left(\sqrt{\mathsf{\Omega }}\xi \right)\pm i\sqrt{rs}sec{h}_D\left(\sqrt{\mathsf{\Omega }}\xi \right)\right).$$

  (27)

  $$V_9\left(\xi \right)=-{\displaystyle \frac{\pi }{2\tau }}-{\displaystyle \frac{\sqrt{\mathsf{\Omega }}}{2\tau }}\left(cot{h}_D\left(\sqrt{\mathsf{\Omega }}\xi \right)\pm \sqrt{rs}cosec{h}_D\left(\sqrt{\mathsf{\Omega }}\xi \right)\right).$$

  (28)

  $$V_{10}\left(\xi \right)=-{\displaystyle \frac{\pi }{2\tau }}-{\displaystyle \frac{\sqrt{\mathsf{\Omega }}}{4\tau }}\left(tan{h}_D\left({\displaystyle \frac{\sqrt{\mathsf{\Omega }}\xi }{4}}\right)+cot{h}_D\left({\displaystyle \frac{\sqrt{\mathsf{\Omega }}\xi }{4}}\right)\right).$$

  (29)
• $\mathbf{Solution}\mathbf{III}$: When $ϵ\tau >0$ and $\pi =0$, then

  $$V_{11}\left(\xi \right)=\sqrt{{\displaystyle \frac{ϵ}{\tau }}}ta{n}_D\left(\sqrt{ϵ\tau }\xi \right).$$

  (30)

  $$V_{12}\left(\xi \right)=-\sqrt{{\displaystyle \frac{ϵ}{\tau }}}co{t}_D\left(\sqrt{ϵ\tau }\xi \right).$$

  (31)

  $$V_{13}\left(\xi \right)=\sqrt{{\displaystyle \frac{ϵ}{\tau }}}\left(ta{n}_D\left(2\sqrt{ϵ\tau }\xi \right)\pm \sqrt{rs}se{c}_D\left(2\sqrt{ϵ\tau }\xi \right)\right).$$

  (32)

  $$V_{14}\left(\xi \right)=-\left(\sqrt{{\displaystyle \frac{ϵ}{\tau }}}\left(co{t}_D\left(2\sqrt{ϵ\tau }\xi \right)\pm \sqrt{rs}cose{c}_D\left(2\sqrt{ϵ\tau }\xi \right)\right)\right).$$

  (33)

  $$V_{15}\left(\xi \right)={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{ϵ}{\tau }}}\left(ta{n}_D\left({\displaystyle \frac{\sqrt{ϵ\tau }}{2}}\xi \right)-co{t}_D\left({\displaystyle \frac{\sqrt{ϵ\tau }}{2}}\xi \right)\right).$$

  (34)
• $\mathbf{Solution}\mathbf{IV}$: When $ϵ\tau <0$ and $\pi =0$, then

  $$V_{17}\left(\xi \right)=-\sqrt{-{\displaystyle \frac{ϵ}{\tau }}}tan{h}_D\left(\sqrt{-ϵ\tau }\xi \right).$$

  (35)

  $$V_{17}\left(\xi \right)=-\sqrt{-{\displaystyle \frac{ϵ}{\tau }}}cot{h}_D\left(\sqrt{-ϵ\tau }\xi \right).$$

  (36)

  $$V_{18}\left(\xi \right)=-\sqrt{-{\displaystyle \frac{ϵ}{\tau }}}\left(tan{h}_D\left(2\sqrt{-ϵ\tau }\xi \right)\pm i\sqrt{rs}sec{h}_D\left(2\sqrt{-ϵ\tau }\xi \right)\right).$$

  (37)

  $$V_{19}\left(\xi \right)=-\left(\sqrt{-{\displaystyle \frac{ϵ}{\tau }}}\left(cot{h}_D\left(2\sqrt{-ϵ\tau }\xi \right)\pm \sqrt{rs}cosec{h}_D\left(2\sqrt{-ϵ\tau }\xi \right)\right)\right).$$

  (38)

  $$V_{20}\left(\xi \right)=-{\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{ϵ}{\tau }}}\left(tan{h}_D\left({\displaystyle \frac{\sqrt{-ϵ\tau }}{2}}\xi \right)+cot{h}_D\left({\displaystyle \frac{\sqrt{-ϵ\tau }}{2}}\xi \right)\right).$$

  (39)
• $\mathbf{Solution}\mathbf{V}$: When $\pi =0$ and $ϵ=\tau$, then

  $$V_{21}\left(\xi \right)=ta{n}_D\left(ϵ\xi \right).$$

  (40)

  $$V_{22}\left(\xi \right)=-co{t}_D\left(ϵ\xi \right).$$

  (41)

  $$V_{23}\left(\xi \right)=\left(ta{n}_D\left(2ϵ\xi \right)\pm \sqrt{rs}se{c}_D\left(2ϵ\xi \right)\right).$$

  (42)

  $$V_{24}\left(\xi \right)=-\left(co{t}_D\left(2ϵ\xi \right)\pm \sqrt{rs}cose{c}_D\left(2ϵ\xi \right)\right).$$

  (43)

  $$V_{25}\left(\xi \right)={\displaystyle \frac{1}{2}}\left(ta{n}_D\left({\displaystyle \frac{ϵ\xi }{2}}\right)-co{t}_D\left({\displaystyle \frac{ϵ\xi }{2}}\right)\right).$$

  (44)
• $\mathbf{Solution}\mathbf{VI}$: When $\pi =0$ and $\tau =-ϵ$, then

  $$V_{26}\left(\xi \right)=-tan{h}_D\left(ϵ\xi \right).$$

  (45)

  $$V_{27}\left(\xi \right)=-cot{h}_D\left(ϵ\xi \right).$$

  (46)

  $$V_{28}\left(\xi \right)=-\left(tan{h}_D\left(2ϵ\xi \right)\pm i\sqrt{rs}sec{h}_D\left(2ϵ\xi \right)\right).$$

  (47)

  $$V_{29}\left(\xi \right)=-\left(cot{h}_D\left(2ϵ\xi \right)\pm \sqrt{rs}cosec{h}_D\left(2ϵ\xi \right)\right).$$

  (48)

  $$V_{30}\left(\xi \right)=-{\displaystyle \frac{1}{2}}\left(tan{h}_D\left({\displaystyle \frac{ϵ\xi }{2}}\right)+cot{h}_D\left({\displaystyle \frac{ϵ\xi }{2}}\right)\right).$$

  (49)
• $\mathbf{Solution}\mathbf{VII}$: When $\mathsf{\Omega }=0$ or ${\pi }^2=4ϵ\tau$, then

  $$V_{31}\left(\xi \right)=-{\displaystyle \frac{2ϵ(\pi \xi LnD+2)}{{\pi }^2\xi LnD}}.$$

  (50)
• $\mathbf{Solution}\mathbf{VIII}$: When $\pi =\lambda$, $ϵ=n\lambda (n\ne 0)$, and $\tau =0$, then

  $$V_{32}\left(\xi \right)=D^{\lambda \xi }-n.$$

  (51)
• $\mathbf{Solution}\mathbf{IX}$: When $\pi =\tau =0$, then

  $$V_{33}\left(\xi \right)=ϵ\xi LnD.$$

  (52)
• $\mathbf{Solution}\mathbf{X}$: When $\pi =ϵ=0$, then

  $$V_{34}\left(\xi \right)=-{\displaystyle \frac{1}{\tau \xi LnD}}.$$

  (53)
• $\mathbf{Solution}\mathbf{XI}$: When $ϵ=0$ and $\pi \ne 0$, then

  $$V_{35}\left(\xi \right)=-{\displaystyle \frac{r\pi }{\tau (cos{h}_D\left(\pi \xi \right)-sin{h}_D\left(\pi \xi \right)+r)}}.$$

  (54)

  $$V_{36}\left(\xi \right)=-{\displaystyle \frac{\pi (sin{h}_D\left(\pi \xi \right)+cos{h}_D\left(\pi \xi \right)+r)}{\tau (sin{h}_D\left(\pi \xi \right)+cos{h}_D\left(\pi \xi \right)+s)}}.$$

  (55)
• $\mathbf{Solution}\mathbf{XII}$: When $\pi =\lambda$, $\tau =n\lambda$ and $ϵ=0$, then

  $$V_{37}\left(\xi \right)={\displaystyle \frac{r{D}^{\lambda \xi }}{s-nr{D}^{\lambda \xi }}}.$$

  (56)
  With reference to the above solutions, the generalized trigonometric and hyperbolic functions are [5] $si{n}_D\left(\xi \right)={\displaystyle \frac{r{D}^{i\xi }-q{D}^{-i\xi }}{2i}}$, $co{s}_D\left(\xi \right)={\displaystyle \frac{r{D}^{i\xi }+q{D}^{-i\xi }}{2}}$, $ta{n}_D\left(\xi \right)=-i{\displaystyle \frac{r{D}^{i\xi }-s{D}^{-i\xi }}{r{D}^{i\xi }+s{D}^{-i\xi }}}$. $cose{c}_D\left(\xi \right)={\displaystyle \frac{1}{si{n}_D\left(\xi \right)}}$, $se{c}_D\left(\xi \right)={\displaystyle \frac{1}{co{s}_D\left(\xi \right)}}$, $co{t}_D\left(\xi \right)={\displaystyle \frac{1}{ta{n}_D\left(\xi \right)}}$. $sin{h}_D\left(\xi \right)={\displaystyle \frac{r{D}^{\xi }-q{D}^{-\xi }}{2}}$, $cos{h}_D\left(\xi \right)={\displaystyle \frac{r{D}^{\xi }+q{D}^{-\xi }}{2}}$, $tan{h}_D\left(\xi \right)={\displaystyle \frac{r{D}^{\xi }-s{D}^{-\xi }}{r{D}^{\xi }+s{D}^{\xi }}}$. $cosec{h}_D\left(\xi \right)={\displaystyle \frac{1}{sin{h}_D\left(\xi \right)}}$, $sec{h}_D\left(\xi \right)={\displaystyle \frac{1}{cos{h}_D\left(\xi \right)}}$, $cot{h}_D\left(\xi \right)={\displaystyle \frac{1}{tan{h}_D\left(\xi \right)}}$. where $r,s>0$, known as deformation parameters.

The value of N in Equation (18) is found by using the homogeneous balance principle. Then, by using Equations (18) and (5) in Equation (4), we will achieve a polynomial equation in term of $V\left(\xi \right)$. By extracting all the coefficients of different powers of $V\left(\xi \right)$, we will get a set of algebraic equations which can be solved further simultaneously via Maple to provide $b_i(b_N\ne 0)$ and $\omega$. After figuring out these factors, we can use Equation (18) to get an analytical solution $\mathcal{U}(\mathcal{X},\mathcal{T})$.

3. Solution Extraction Through SGE Formula

To get the solutions of Equation (1), we first find the value of N in Equation (16) or Equation (17) using the homogeneous principle in Equation (5), which leads to $N=1$. When putting $N=1$ into Equation (17), we get

$$u\left(z\left(\xi \right)\right)=m_0+n_{1}sinz\left(\xi \right)+m_{1}cosz\left(\xi \right).$$

(57)

Here $m_0$, $n_1$, and $m_1$ are unknown constants to be assessed later. It is possible for both $n_1$ and $m_1$ to be zero, but they cannot be zero simultaneously. The extensions of Equation (57) together with Equation (12), lead the following results:

$${\displaystyle \frac{du\left(z\right(\xi \left)\right)}{d\xi }}=n_{1}sinz\left(\xi \right)cosz\left(\xi \right)-m_{1}si{n}^{2}z\left(\xi \right);{\displaystyle \frac{d^{2}u\left(z\left(\xi \right)\right)}{d{\xi }^2}}=n_1[-si{n}^{3}z\left(\xi \right)+sinz\left(\xi \right)co{s}^{2}z\left(\xi \right)]-2m_{1}si{n}^{2}z\left(\xi \right)cosz\left(\xi \right).$$

(58)

Using the set of Equation (58) along with Equations (14) and (15) in Equation (5), and then setting the coefficients of [$si{n}^r\left(z\left(\xi \right)\right)co{s}^s\left(z\left(\xi \right)\right)$] to zero after equating them with the combination of constant terms yields the following equations:

$$\left.\begin{array}{ccc}constant\hfill & :\hfill & \delta m_0^3+3\delta m_0{n}_1^2+\beta m_0=0,\hfill \\ sinz\left(\xi \right)\hfill & :\hfill & -n_1\alpha {\eta }^2+3\delta m_0^2{n}_1+\delta n_1^3-{\omega }^2{n}_1+\beta n_1=0,\hfill \\ cosz\left(\xi \right)\hfill & :\hfill & -2m_1\alpha {\eta }^2+3\delta m_0^2{m}_1+3\delta n_1^2{m}_1-2{\omega }^2{m}_1+\beta m_1=0,\hfill \\ sinz\left(\xi \right)cosz\left(\xi \right)\hfill & :\hfill & 6\delta m_0{m}_1{n}_1=0,\hfill \\ sinz\left(\xi \right)co{s}^{2}z\left(\xi \right)\hfill & :\hfill & 2n_1\alpha {\eta }^2+3\delta n_1{m}_1^2-\delta n_1^3+2{\omega }^2{n}_1=0,\hfill \\ sinz\left(\xi \right)co{s}^{2}z\left(\xi \right)\hfill & :\hfill & 2n_1\alpha {\eta }^2+3\delta n_1{m}_1^2-\delta n_1^3+2{\omega }^2{n}_1=0,\hfill \\ co{s}^{2}z\left(\xi \right)\hfill & :\hfill & 3\delta m_0{m}_1^2-3\delta m_0{n}_1^2=0,\hfill \\ co{s}^{3}z\left(\xi \right)\hfill & :\hfill & 2\alpha {\eta }^2{m}_1+\delta m_1^3-3\delta m_1{n}_1^2+2{\omega }^2{m}_1=0.\hfill \end{array}\right\}$$

(59)

The unidentified parameters are the subject of the above set of equations. We then use computational skills in mathematical software like Maple 2025 to help us unravel the resultant system of algebraic equations in order to obtain the succeeding solution sets:

Collection 1. 

$$\begin{array}{c}\hfill \eta =\eta ,\omega =\sqrt{-{\eta }^2\alpha +2\beta },m_0=0,m_1=\sqrt{-{\displaystyle \frac{\beta }{\delta }}},n_1=\sqrt{{\displaystyle \frac{\beta }{\delta }}}.\end{array}$$

Collection 2. 

$$\begin{array}{ccc}\hfill \eta =\eta ,\omega =\sqrt{-{\eta }^2\alpha -\beta },m_0=0,m_1=0,& & n_1=\sqrt{-{\displaystyle \frac{2\beta }{\delta }}}.\hfill \end{array}$$

Collection 3. 

$$\begin{array}{cc}\hfill \eta =\eta ,\omega =\sqrt{-{\eta }^2\alpha +{\displaystyle \frac{\beta }{2}}},m_0=0,m_1=\sqrt{-{\displaystyle \frac{\beta }{\delta }}},& n_1=0.\end{array}$$

Graphical representations of the solution ${\mathcal{U}}_2$ described in Figure 1, and solurion ${\mathcal{U}}_3$ described in Figure 2. The explicit solutions in terms of trigonometric functions may be obtained using the values of unknown parameters gathered in Collections 1–3. For these, we first use the values listed in Collection 1, inserting them into Equation (57) to get the following answers:

$$u_1\left(\xi \right)=\sqrt{{\displaystyle \frac{\beta }{\delta }}}sin\left(z\left(\xi \right)\right)+\sqrt{-{\displaystyle \frac{\beta }{\delta }}}cos\left(z\left(\xi \right)\right).$$

(60)

By considering Equation (15), we might get the following:

$$u_1\left(\xi \right)=\sqrt{{\displaystyle \frac{\beta }{\delta }}}sech\left(z\left(\xi \right)\right)+\sqrt{-{\displaystyle \frac{\beta }{\delta }}}tanh\left(z\left(\xi \right)\right).$$

(61)

and

$$u_1\left(\xi \right)=i\sqrt{{\displaystyle \frac{\beta }{\delta }}}cosech\left(z\left(\xi \right)\right)+\sqrt{-{\displaystyle \frac{\beta }{\delta }}}coth\left(z\left(\xi \right)\right).$$

(62)

In a (1+1) spatial and temporal system, we can express Equation (60) as (similarly, we can assume this for Equations (61) and (62)):

$${\mathcal{U}}_1(\mathcal{X},\mathcal{T})=\sqrt{{\displaystyle \frac{\beta }{\delta }}}sin\left(\eta x-\sqrt{-{\eta }^2\alpha +2\beta }t\right)+\sqrt{-{\displaystyle \frac{\beta }{\delta }}}cos\left(\eta x-\sqrt{-{\eta }^2\alpha +2\beta }t\right).$$

(63)

We then use the values listed in Collection 2, inserting them into Equation (57) to get the following answers:

$$u_2\left(\xi \right)=\sqrt{-{\displaystyle \frac{2\beta }{\delta }}}sin\left(z\left(\xi \right)\right).$$

(64)

By considering Equation (15), we might get the following:

$$u_2\left(\xi \right)=\sqrt{-{\displaystyle \frac{2\beta }{\delta }}}sech\left(z\left(\xi \right)\right).$$

(65)

and

$$u_2\left(\xi \right)=i\sqrt{-{\displaystyle \frac{2\beta }{\delta }}}cosech\left(z\left(\xi \right)\right).$$

(66)

In a (1+1) spatial and temporal system, we can express Equation (64) as (similarly, we can assume the same for Equations (65) and (66)):

$${\mathcal{U}}_2(\mathcal{X},\mathcal{T})=\sqrt{{\displaystyle \frac{\beta }{\delta }}}sin\left(\eta x-\sqrt{-{\eta }^2\alpha +2\beta }t\right).$$

(67)

Finally, the values listed in Collection 3 are used, inserting them into Equation (57) to get the following answer:

$$u_3\left(\xi \right)=\sqrt{-{\displaystyle \frac{\beta }{\delta }}}cos\left(z\left(\xi \right)\right).$$

(68)

By considering Equation (15), we might get the following:

$$u_3\left(\xi \right)=\sqrt{-{\displaystyle \frac{\beta }{\delta }}}tanh\left(z\left(\xi \right)\right).$$

(69)

and

$$u_3\left(\xi \right)=\sqrt{-{\displaystyle \frac{2\beta }{\delta }}}coth\left(z\left(\xi \right)\right).$$

(70)

In a (1+1) spatial and temporal system, we can express Equation (68) as (similarly, we can assume the same for Equations (69) and (70)):

$${\mathcal{U}}_3(\mathcal{X},\mathcal{T})=\sqrt{{\displaystyle \frac{\beta }{\delta }}}cos\left(\eta x-{\displaystyle \frac{\sqrt{-4{\eta }^2\alpha +2\beta }}{2}}t\right).$$

(71)

Notably, the analytic solutions (60)–(71) derived from the K-F-G system using the SGE technique are easy, comprehensive, and standard. The significant wave profiles are extracted from the above soliton solutions, which may be utilized to analyze various events like the behavior of spin-zero particles, the relativistic nature of particles, etc. Despite being a useful model, there are some limitations attached to the KFGE model. It does not directly explain particles with spin, and it might result in negative probability densities, which need to be interpreted carefully.

4. Solution Extraction Through NEDAM Formula

Graphical representations of the solution ${\mathcal{U}}_2$ described in Figure 3, solurion ${\mathcal{U}}_5$ described in Figure 4, plots of the solution ${\mathcal{U}}_{21}$ described in Figure 5, Figure 6 represent the solution ${\mathcal{U}}_{22}$, Figure 7 represent the solution ${\mathcal{U}}_{24}$, Figure 8 represent the solution ${\mathcal{U}}_{25}$, Figure 9 represent the solution ${\mathcal{U}}_{26}$ and Figure 10 represent the solution ${\mathcal{U}}_{34}$. It is possible to acquire the analytical precise soliton solutions for Equation (1) by using the recently developed new extended direct algebraic method. The homogeneous balancing principle applied to Equation (5) results in $N=1$. Equation (18) will now become

$$u\left(\xi \right)=\sum _{i=0}^1{b}_i{V}^i\left(\xi \right)=b_0+b_{1}V\left(\xi \right).$$

(72)

where

$$V^{\prime }\left(\xi \right)=ln\left(D\right)(ϵ+\pi V\left(\xi \right)+\tau V^2\left(\xi \right)),D\ne 0,1.$$

(73)

Embedding Equation (72) along with Equation (73) into Equation (5) results in the following equation:

$$\begin{array}{c}\hfill \delta b_0^3+3\delta b_0^2{b}_{1}V\left(\xi \right)+3\delta b_0{b}_1^{2}V{\left(\xi \right)}^2+\delta b_1^{3}V{\left(\xi \right)}^3+\beta b_0+\beta b_{1}V\left(\xi \right)+b_{1}ln{\left(D\right)}^2\pi ϵ\alpha {\eta }^2+b_{1}ln{\left(D\right)}^2\pi ϵ{\omega }^2\\ \hfill +b_{1}ln{\left(D\right)}^2{\pi }^{2}V\left(\xi \right)\alpha {\eta }^2+b_{1}ln{\left(D\right)}^2{\pi }^{2}V\left(\xi \right){\omega }^2+3b_{1}ln{\left(D\right)}^2\pi \tau V{\left(\xi \right)}^2\alpha {\eta }^2+3b_{1}ln{\left(D\right)}^2\pi \tau V{\left(\xi \right)}^2{\omega }^2\\ \hfill +2b_{1}ln{\left(D\right)}^2\tau V\left(\xi \right)ϵ\alpha {\eta }^2+2b_{1}ln{\left(D\right)}^2\tau V\left(\xi \right)ϵ{\omega }^2+2b_{1}ln{\left(D\right)}^2{\tau }^{2}V{\left(\xi \right)}^3\alpha {\eta }^2+2b_{1}ln{\left(D\right)}^2{\tau }^{2}V{\left(\xi \right)}^3{\omega }^2=0.\end{array}$$

(74)

At this point, we extract every coefficient of various powers of $V\left(\xi \right)$ occurring in Equation (74) to get a set of the following algebraic equations:

$$\left.\begin{array}{ccc}V{\left(\xi \right)}^3\hfill & :\hfill & 2ln{\left(D\right)}^2\alpha {\eta }^2{\tau }^2{b}_1+2ln{\left(D\right)}^2{\omega }^2{\tau }^2{b}_1+\delta b_1^3=0,\hfill \\ V{\left(\xi \right)}^2\hfill & :\hfill & 3ln{\left(D\right)}^2\pi \alpha {\eta }^2\tau b_1+3ln{\left(D\right)}^2\pi {\omega }^2\tau b_1+3\delta b_0{b}_1^2=0,\hfill \\ V\left(\xi \right)\hfill & :\hfill & ln{\left(D\right)}^2{\pi }^2\alpha {\eta }^2{b}_1+2ln{\left(D\right)}^2\alpha {\eta }^2\tau ϵb_1+ln{\left(D\right)}^2{\pi }^2{\omega }^2{b}_1+2ln{\left(D\right)}^2{\omega }^2\tau ϵb_1\hfill \\ & \hfill & +3\delta b_0^2{b}_1+\beta b_1=0,\hfill \\ V{\left(\xi \right)}^0\hfill & :\hfill & b_{1}ln{\left(D\right)}^2\pi ϵ\alpha {\eta }^2+b_{1}ln{\left(D\right)}^2\pi ϵ{\omega }^2+\delta b_0^3+\beta b_0=0.\hfill \end{array}\right\}$$

(75)

Figure 3. Graphical visualization of the derived solution of Equation (78) of the NEDAM method, resulting in periodic wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for ${\mathcal{U}}_2(\mathcal{X},\mathcal{T}):$$B=1.71,\pi =0.5,ϵ=1,\tau =1,r=4,s=3,\eta =2,\alpha =1,\beta =2,\delta =-1,\omega =1.59,$$m_0=0.36i,$$m_1=1.46i,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

Figure 3. Graphical visualization of the derived solution of Equation (78) of the NEDAM method, resulting in periodic wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for ${\mathcal{U}}_2(\mathcal{X},\mathcal{T}):$$B=1.71,\pi =0.5,ϵ=1,\tau =1,r=4,s=3,\eta =2,\alpha =1,\beta =2,\delta =-1,\omega =1.59,$$m_0=0.36i,$$m_1=1.46i,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

When solving the above set of equations (75) concurrently, Maple will provide the following collection for values of $b_0$, $b_1$, and $\omega$.

Collection 4. 

$$\begin{array}{c}\hfill \omega ={\displaystyle \frac{\sqrt{-\frac{-ln{\left(D\right)}^2{\pi }^2\alpha {\eta }^2+4ln{\left(D\right)}^2\alpha {\eta }^2\tau ϵ+2\beta }{-{\pi }^2+4ϵ\tau }}}{ln\left(D\right)}},b_0=-{\displaystyle \frac{\pi \beta }{({\pi }^2-4ϵ\tau )\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}},b_1=2\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau .\end{array}$$

When the values of $b_0$ and $b_1$ are embedded in Equation (72), the generic solution obtained is

$$u\left(\xi \right)=-{\displaystyle \frac{\pi \beta }{({\pi }^2-4ϵ\tau )\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}+2\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau V\left(\xi \right).$$

(76)

Figure 4. Graphical visualization of the derived solution of Equation (81) of the NEDAM method, resulting in periodic wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_5\left(\xi \right):$$B=2.71,\pi =0.5,ϵ=1,\tau =1,r=2,s=3,\eta =2,\alpha =1,\beta =2,\delta =-1,\omega =2.27,$$m_0=0.36i,$$m_1=1.46i,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

Figure 4. Graphical visualization of the derived solution of Equation (81) of the NEDAM method, resulting in periodic wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_5\left(\xi \right):$$B=2.71,\pi =0.5,ϵ=1,\tau =1,r=2,s=3,\eta =2,\alpha =1,\beta =2,\delta =-1,\omega =2.27,$$m_0=0.36i,$$m_1=1.46i,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

The use of Equations (20)–(56) can therefore yield a wide variety of the following analytical solutions:

• $\mathbf{Solution}\mathbf{I}$: When $\mathsf{\Omega }<0$ and $\tau \ne 0$, then

  $${\mathcal{U}}_1(\mathcal{X},\mathcal{T})=-\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\pi +\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\sqrt{-\mathsf{\Omega }}ta{n}_D\left({\displaystyle \frac{\sqrt{-\mathsf{\Omega }}}{2}}(\eta \mathcal{X}-\omega \mathcal{T})\right)-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (77)

  $${\mathcal{U}}_2(\mathcal{X},\mathcal{T})=-\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\pi +\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\sqrt{-\mathsf{\Omega }}co{t}_D\left({\displaystyle \frac{\sqrt{-\mathsf{\Omega }}}{2}}(\eta \mathcal{X}-\omega \mathcal{T})\right)-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (78)

  $$\begin{array}{c}\hfill {\mathcal{U}}_3(\mathcal{X},\mathcal{T})=-\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\pi +\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\sqrt{-\mathsf{\Omega }}ta{n}_D\left(\sqrt{-\mathsf{\Omega }}(\eta \mathcal{X}-\omega \mathcal{T})\right)\pm \\ \hfill \sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\sqrt{-\mathsf{\Omega }}\sqrt{rs}se{c}_D\left(\sqrt{-\mathsf{\Omega }}(\eta \mathcal{X}-\omega \mathcal{T})\right)-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (79)

  $$\begin{array}{c}\hfill {\mathcal{U}}_4(\mathcal{X},\mathcal{T})=-\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\pi +\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\sqrt{-\mathsf{\Omega }}co{t}_D\left(\sqrt{-\mathsf{\Omega }}(\eta \mathcal{X}-\omega \mathcal{T})\right)\pm \\ \hfill \sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\sqrt{-\mathsf{\Omega }}\sqrt{rs}cose{c}_D\left(\sqrt{-\mathsf{\Omega }}(\eta \mathcal{X}-\omega \mathcal{T})\right)-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (80)

  $$\begin{array}{c}\hfill {\mathcal{U}}_5(\mathcal{X},\mathcal{T})=-\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\pi +{\displaystyle \frac{\sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}\sqrt{-\mathsf{\Omega }}ta{n}_D\left(\frac{\sqrt{-\mathsf{\Omega }}}{4}(\eta \mathcal{X}-\omega \mathcal{T})\right)}{2}}-\\ \hfill {\displaystyle \frac{\sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}\sqrt{-\mathsf{\Omega }}co{t}_D\left(\frac{\sqrt{-\mathsf{\Omega }}}{4}(\eta \mathcal{X}-\omega \mathcal{T})\right)}{2}}-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (81)
• $\mathbf{Solution}\mathbf{II}$: When $\mathsf{\Omega }>0$ and $\tau \ne 0$, then

  $$\begin{array}{c}\hfill {\mathcal{U}}_6(\mathcal{X},\mathcal{T})=-\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\pi +\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\sqrt{-\mathsf{\Omega }}tan{h}_D\left({\displaystyle \frac{\sqrt{-\mathsf{\Omega }}}{2}}(\eta \mathcal{X}-\omega \mathcal{T})\right)-\\ \hfill {\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (82)

  $$\begin{array}{c}\hfill {\mathcal{U}}_7(\mathcal{X},\mathcal{T})=-\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\pi +\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\sqrt{-\mathsf{\Omega }}cot{h}_D\left({\displaystyle \frac{\sqrt{-\mathsf{\Omega }}}{2}}(\eta \mathcal{X}-\omega \mathcal{T})\right)-\\ \hfill {\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (83)

  $$\begin{array}{c}\hfill {\mathcal{U}}_8(\mathcal{X},\mathcal{T})=-\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\pi +\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\sqrt{-\mathsf{\Omega }}tan{h}_D\left(\sqrt{-\mathsf{\Omega }}(\eta \mathcal{X}-\omega \mathcal{T})\right)\pm \\ \hfill i\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\sqrt{-\mathsf{\Omega }}\sqrt{rs}sec{h}_D\left(\sqrt{-\mathsf{\Omega }}(\eta \mathcal{X}-\omega \mathcal{T})\right)-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (84)

  Figure 5. Graphical visualization of the derived solution of Equation (97) of the NEDAM method, providing periodic wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_{21}\left(\xi \right):$$B=2.71,\pi =0,ϵ=0.3,\tau =0.3,r=2,s=3,\eta =2,\alpha =1,\beta =2,\delta =-1,\omega =9.72,$$m_0=0,$$m_1=1.41i,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

  Figure 5. Graphical visualization of the derived solution of Equation (97) of the NEDAM method, providing periodic wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_{21}\left(\xi \right):$$B=2.71,\pi =0,ϵ=0.3,\tau =0.3,r=2,s=3,\eta =2,\alpha =1,\beta =2,\delta =-1,\omega =9.72,$$m_0=0,$$m_1=1.41i,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

  Figure 6. Graphical visualization of the derived solution of Equation (98) of the NEDAM method, providing periodic wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_{22}\left(\xi \right):$$B=2.71,\pi =0,ϵ=0.6,\tau =0.6,r=2,s=3,\eta =2,\alpha =1,\beta =2,\delta =-1,\omega =4.54,$$m_0=0,$$m_1=1.41i,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

  Figure 6. Graphical visualization of the derived solution of Equation (98) of the NEDAM method, providing periodic wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_{22}\left(\xi \right):$$B=2.71,\pi =0,ϵ=0.6,\tau =0.6,r=2,s=3,\eta =2,\alpha =1,\beta =2,\delta =-1,\omega =4.54,$$m_0=0,$$m_1=1.41i,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

  Figure 7. Graphical visualization of the derived solution of Equation (100) of the NEDAM method, providing periodic wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_{24}\left(\xi \right):$$B=2.71,\pi =0,ϵ=0.7,\tau =0.7,r=2,s=3,\eta =2,\alpha =1,\beta =2,\delta =-1,\omega =3.75,$$m_0=0,$$m_1=1.41i,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

  Figure 7. Graphical visualization of the derived solution of Equation (100) of the NEDAM method, providing periodic wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_{24}\left(\xi \right):$$B=2.71,\pi =0,ϵ=0.7,\tau =0.7,r=2,s=3,\eta =2,\alpha =1,\beta =2,\delta =-1,\omega =3.75,$$m_0=0,$$m_1=1.41i,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

  $$\begin{array}{c}\hfill {\mathcal{U}}_9(\mathcal{X},\mathcal{T})=-\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\pi +\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\sqrt{-\mathsf{\Omega }}cot{h}_D\left(\sqrt{-\mathsf{\Omega }}(\eta \mathcal{X}-\omega \mathcal{T})\right)\pm \\ \hfill \sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\sqrt{-\mathsf{\Omega }}\sqrt{rs}cosec{h}_D\left(\sqrt{-\mathsf{\Omega }}(\eta \mathcal{X}-\omega \mathcal{T})\right)-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (85)

  $$\begin{array}{c}\hfill {\mathcal{U}}_{10}(\mathcal{X},\mathcal{T})=-\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\pi +{\displaystyle \frac{\sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}\sqrt{-\mathsf{\Omega }}tan{h}_D\left(\frac{\sqrt{-\mathsf{\Omega }}}{4}(\eta \mathcal{X}-\omega \mathcal{T})\right)}{2}}-\\ \hfill {\displaystyle \frac{\sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}\sqrt{-\mathsf{\Omega }}cot{h}_D\left(\frac{\sqrt{-\mathsf{\Omega }}}{4}(\eta \mathcal{X}-\omega \mathcal{T})\right)}{2}}-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (86)
• $\mathbf{Solution}\mathbf{III}$: When $ϵ\tau >0$ and $\pi =0$, then

  $${\mathcal{U}}_{11}(\mathcal{X},\mathcal{T})=2\sqrt{{\displaystyle \frac{ϵ}{\tau }}}ta{n}_D\left(\sqrt{ϵ\tau }(\eta \mathcal{X}-\omega \mathcal{T})\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (87)

  $${\mathcal{U}}_{12}(\mathcal{X},\mathcal{T})=-2\sqrt{{\displaystyle \frac{ϵ}{\tau }}}co{t}_D\left(\sqrt{ϵ\tau }(\eta \mathcal{X}-\omega \mathcal{T})\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (88)

  $$\begin{array}{c}\hfill {\mathcal{U}}_{13}(\mathcal{X},\mathcal{T})=2\sqrt{{\displaystyle \frac{ϵ}{\tau }}}\left(ta{n}_D\left(2\sqrt{ϵ\tau }(\eta \mathcal{X}-\omega \mathcal{T})\right)\pm \sqrt{rs}se{c}_D\left(2\sqrt{ϵ\tau }(\eta \mathcal{X}-\omega \mathcal{T})\right)\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -\\ \hfill {\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (89)

  $$\begin{array}{c}\hfill {\mathcal{U}}_{14}(\mathcal{X},\mathcal{T})=2\sqrt{{\displaystyle \frac{ϵ}{\tau }}}\left(co{t}_D\left(2\sqrt{ϵ\tau }(\eta \mathcal{X}-\omega \mathcal{T})\right)\pm \sqrt{rs}cose{c}_D\left(2\sqrt{ϵ\tau }(\eta \mathcal{X}-\omega \mathcal{T})\right)\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -\\ \hfill {\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (90)

  $$\begin{array}{c}\hfill {\mathcal{U}}_{15}(\mathcal{X},\mathcal{T})=\sqrt{{\displaystyle \frac{ϵ}{\tau }}}\left(ta{n}_D\left({\displaystyle \frac{\sqrt{ϵ\tau }}{2}}(\eta \mathcal{X}-\omega \mathcal{T})\right)-co{t}_D\left({\displaystyle \frac{\sqrt{ϵ\tau }}{2}}(\eta \mathcal{X}-\omega \mathcal{T})\right)\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -\\ \hfill {\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (91)

  Figure 8. Graphical visualization of the derived solution of Equation (101) of the NEDAM method, providing periodic wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_{25}\left(\xi \right):$$B=2.71,\pi =0,ϵ=0.9,\tau =0.9,r=2,s=3,\eta =2,\alpha =1,\beta =2,\delta =-1,\omega =2.63,$$m_0=0,$$m_1=1.41i,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

  Figure 8. Graphical visualization of the derived solution of Equation (101) of the NEDAM method, providing periodic wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_{25}\left(\xi \right):$$B=2.71,\pi =0,ϵ=0.9,\tau =0.9,r=2,s=3,\eta =2,\alpha =1,\beta =2,\delta =-1,\omega =2.63,$$m_0=0,$$m_1=1.41i,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

  Figure 9. Graphical visualization of the derived solution of Equation (102) of the NEDAM method, providing kink wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_{26}\left(\xi \right):$$B=2.71,$$\pi =0,$$ϵ=-1.7,\tau =1.7,r=4,s=3,\eta =2.7,\alpha =1.6,\beta =2,\delta =-1,\omega =4.68i,m_0=0,$$m_1=1.41i,$$\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

  Figure 9. Graphical visualization of the derived solution of Equation (102) of the NEDAM method, providing kink wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_{26}\left(\xi \right):$$B=2.71,$$\pi =0,$$ϵ=-1.7,\tau =1.7,r=4,s=3,\eta =2.7,\alpha =1.6,\beta =2,\delta =-1,\omega =4.68i,m_0=0,$$m_1=1.41i,$$\xi =\eta \mathcal{X}-\omega \mathcal{T}.$
• $\mathbf{Solution}\mathbf{IV}$: When $ϵ\tau <0$ and $\pi =0$, then

  $${\mathcal{U}}_{16}(\mathcal{X},\mathcal{T})=-2\sqrt{-{\displaystyle \frac{ϵ}{\tau }}}tan{h}_D\left(\sqrt{-ϵ\tau }(\eta \mathcal{X}-\omega \mathcal{T})\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (92)

  $${\mathcal{U}}_{17}(\mathcal{X},\mathcal{T})=-2\sqrt{-{\displaystyle \frac{ϵ}{\tau }}}cot{h}_D\left(\sqrt{-ϵ\tau }(\eta \mathcal{X}-\omega \mathcal{T})\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (93)

  $$\begin{array}{c}\hfill {\mathcal{U}}_{18}(\mathcal{X},\mathcal{T})=-2\sqrt{-{\displaystyle \frac{ϵ}{\tau }}}\left(tan{h}_D\left(2\sqrt{-ϵ\tau }(\eta \mathcal{X}-\omega \mathcal{T})\right)\pm i\sqrt{rs}sec{h}_D\left(2\sqrt{-ϵ\tau }(\eta \mathcal{X}-\omega \mathcal{T})\right)\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -\\ \hfill {\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (94)

  $$\begin{array}{c}\hfill {\mathcal{U}}_{19}(\mathcal{X},\mathcal{T})=-2\sqrt{-{\displaystyle \frac{ϵ}{\tau }}}\left(cot{h}_D\left(2\sqrt{-ϵ\tau }(\eta \mathcal{X}-\omega \mathcal{T})\right)\pm \sqrt{rs}cosec{h}_D\left(2\sqrt{-ϵ\tau }(\eta \mathcal{X}-\omega \mathcal{T})\right)\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -\\ \hfill {\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (95)

  $$\begin{array}{c}\hfill {\mathcal{U}}_{20}(\mathcal{X},\mathcal{T})=-\sqrt{-{\displaystyle \frac{ϵ}{\tau }}}\left(tan{h}_D\left({\displaystyle \frac{\sqrt{-ϵ\tau }}{2}}(\eta \mathcal{X}-\omega \mathcal{T})\right)-cot{h}_D\left({\displaystyle \frac{\sqrt{-ϵ\tau }}{2}}(\eta \mathcal{X}-\omega \mathcal{T})\right)\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -\\ \hfill {\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (96)

  Figure 10. Graphical visualization of the derived solution of Equation (110) of the NEDAM method, providing dark-bright wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_{34}\left(\xi \right):$$B=2.71,\pi =0,ϵ=0,n=3,\tau =1.5,r=2,s=3,\eta =-2,\alpha =1,\beta =2,\delta =-1,$$\omega =1.89i,$$m_0=undefined,m_1=undefined,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$

  Figure 10. Graphical visualization of the derived solution of Equation (110) of the NEDAM method, providing dark-bright wave solutions such as a (a) 3D surface, (b) 2D surface, and (c) contour plot for $u_{34}\left(\xi \right):$$B=2.71,\pi =0,ϵ=0,n=3,\tau =1.5,r=2,s=3,\eta =-2,\alpha =1,\beta =2,\delta =-1,$$\omega =1.89i,$$m_0=undefined,m_1=undefined,\xi =\eta \mathcal{X}-\omega \mathcal{T}.$
• $\mathbf{Solution}\mathbf{V}$: When $\pi =0$ and $ϵ=\tau$, then

  $${\mathcal{U}}_{21}(\mathcal{X},\mathcal{T})=2ta{n}_D\left(ϵ(\eta \mathcal{X}-\omega \mathcal{T})\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (97)

  $${\mathcal{U}}_{22}(\mathcal{X},\mathcal{T})=-2co{t}_D\left(ϵ(\eta \mathcal{X}-\omega \mathcal{T})\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (98)

  $$\begin{array}{c}\hfill {\mathcal{U}}_{23}(\mathcal{X},\mathcal{T})=2\left(ta{n}_D\left(2ϵ(\eta \mathcal{X}-\omega \mathcal{T})\right)\pm \sqrt{rs}se{c}_D\left(2ϵ(\eta \mathcal{X}-\omega \mathcal{T})\right)\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (99)

  $${\mathcal{U}}_{24}(\mathcal{X},\mathcal{T})=2\left(-co{t}_D\left(2ϵ(\eta \mathcal{X}-\omega \mathcal{T})\right)\pm \sqrt{rs}cose{c}_D\left(2ϵ(\eta \mathcal{X}-\omega \mathcal{T})\right)\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (100)

  $$\begin{array}{c}\hfill {\mathcal{U}}_{25}(\mathcal{X},\mathcal{T})=2\left({\displaystyle \frac{ta{n}_D\left(\frac{ϵ(\eta \mathcal{X}-\omega \mathcal{T})}{2}\right)}{2}}-{\displaystyle \frac{co{t}_D\left(\frac{ϵ(\eta \mathcal{X}-\omega \mathcal{T})}{2}\right)}{2}}\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (101)
• $\mathbf{Solution}\mathbf{VI}$: When $\pi =0$ and $\tau =-ϵ$, then

  $${\mathcal{U}}_{26}(\mathcal{X},\mathcal{T})=2tan{h}_D\left(ϵ(\eta \mathcal{X}-\omega \mathcal{T})\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (102)

  $${\mathcal{U}}_{27}(\mathcal{X},\mathcal{T})=-2cot{h}_D\left(ϵ(\eta \mathcal{X}-\omega \mathcal{T})\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (103)

  $${\mathcal{U}}_{28}(\mathcal{X},\mathcal{T})=2\left(-tan{h}_D\left(2ϵ(\eta \mathcal{X}-\omega \mathcal{T})\right)\pm i\sqrt{rs}sec{h}_D\left(2ϵ(\eta \mathcal{X}-\omega \mathcal{T})\right)\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (104)

  $${\mathcal{U}}_{29}(\mathcal{X},\mathcal{T})=2\left(-cot{h}_D\left(2ϵ(\eta \mathcal{X}-\omega \mathcal{T})\right)\pm \sqrt{rs}cosec{h}_D\left(2ϵ(\eta \mathcal{X}-\omega \mathcal{T})\right)\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (105)

  $$\begin{array}{c}\hfill {\mathcal{U}}_{30}(\mathcal{X},\mathcal{T})=2\left(-{\displaystyle \frac{tan{h}_D\left(\frac{ϵ(\eta \mathcal{X}-\omega \mathcal{T})}{2}\right)}{2}}-{\displaystyle \frac{cot{h}_D\left(\frac{ϵ(\eta \mathcal{X}-\omega \mathcal{T})}{2}\right)}{2}}\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (106)
• $\mathbf{Solution}\mathbf{VII}$: When $\mathsf{\Omega }=0$ or ${\pi }^2=4ϵ\tau$, then

  $$\begin{array}{c}\hfill {\mathcal{U}}_{31}(\mathcal{X},\mathcal{T})={\displaystyle \frac{-4ϵ(\pi ϵLn\left(D\right)+2)\sqrt{\frac{\beta }{-{\pi }^2+4\delta \tau ϵ}}\tau }{{\pi }^2ϵLn\left(D\right)}}-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.\end{array}$$

  (107)
• $\mathbf{Solution}\mathbf{VIII}$: When $\pi =\lambda$, $ϵ=n\lambda (n\ne 0)$ and $\tau =0$, then

  $${\mathcal{U}}_{32}(\mathcal{X},\mathcal{T})=2\left(D^{\lambda (\eta \mathcal{X}-\omega \mathcal{T})-n}\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2+4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (108)
• $\mathbf{Solution}\mathbf{IX}$: When $\pi =\tau =0$, then

  $${\mathcal{U}}_{33}(\mathcal{X},\mathcal{T})=2ϵ(\eta \mathcal{X}-\omega \mathcal{T})Ln\left(D\right)\sqrt{{\displaystyle \frac{\beta }{-{\pi }^2+4\delta \tau ϵ}}}\tau -{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (109)
• $\mathbf{Solution}\mathbf{X}$: When $\pi =ϵ=0$, then

  $${\mathcal{U}}_{34}(\mathcal{X},\mathcal{T})={\displaystyle \frac{-2\sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}{(\eta \mathcal{X}-\omega \mathcal{T})Ln\left(D\right)}}-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (110)
• $\mathbf{Solution}\mathbf{XI}$: When $ϵ=0$ and $\pi \ne 0$, then

  $${\mathcal{U}}_{35}(\mathcal{X},\mathcal{T})=-{\displaystyle \frac{2r\pi \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}{cos{h}_D\left(\pi (\eta \mathcal{X}-\omega \mathcal{T})\right)-sin{h}_D\left(\pi (\eta \mathcal{X}-\omega \mathcal{T})\right)+r}}-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (111)

  $${\mathcal{U}}_{36}(\mathcal{X},\mathcal{T})=-{\displaystyle \frac{2\pi (sin{h}_D\left(\pi (\eta \mathcal{X}-\omega \mathcal{T})\right)+cos{h}_D\left(\pi (\eta \mathcal{X}-\omega \mathcal{T})\right))\sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}{sin{h}_D\left(\pi (\eta \mathcal{X}-\omega \mathcal{T})\right)-cos{h}_D\left(\pi (\eta \mathcal{X}-\omega \mathcal{T})\right)+s}}-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (112)
• $\mathbf{Solution}\mathbf{XII}$: When $\pi =\lambda$, $\tau =n\lambda$, and $ϵ=0$, then

  $${\mathcal{U}}_{37}(\mathcal{X},\mathcal{T})={\displaystyle \frac{2r{D}^{\lambda (\eta \mathcal{X}-\omega \mathcal{T})}\sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}{s-nr{D}^{\lambda (\eta \mathcal{X}-\omega \mathcal{T})}}}-{\displaystyle \frac{\pi \beta }{\left(\mathsf{\Omega }\right)\delta \sqrt{\frac{\beta }{-{\pi }^2\delta +4\delta \tau ϵ}}}}.$$

  (113)

In all the above solutions (from ${\mathcal{U}}_1(\mathcal{X},\mathcal{T})$ to ${\mathcal{U}}_{37}(\mathcal{X},\mathcal{T}))$, the value of wave speed $\omega$ is given already in Collection 4 of Section 4.

5. Conclusions

In summary, this study provides a comprehensive analysis of the nonlinear KFG equation, enhanced by the SGE and NEDAM methods, uncovering new soliton solutions with unique dynamic characteristics. The current study encompasses a comprehensive profile of stable and functional soliton solutions, encompassing the periodic wave structures with their 3D, 2D, and contour plots as shown in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10. The distinctive efficacy of these two methodologies in addressing the proposed model was substantiated through the application of both analytical and graphical instruments, facilitating the derivation and visualization of solutions that inherently exhibit periodic structures. The solutions were articulated through the utilization of exponential, trigonometric, and hyperbolic functions, along with their combinations, under specific parametric conditions. The soliton solutions presented in this study offer significant insights across various fields of theoretical physics such as particle physics, condensed matter physics, QFT, etc. The anticipated outcomes substantiate that the methodologies under scrutiny are indispensable, reliable, and robust in producing a diverse profile of stable and efficacious solutions for an extensive spectrum of nonlinear models, and one could apply these effective models in many other NLPDE models.