Analytical Solitary Wave Solutions of Fractional Tzitzéica Equation Using Expansion Approach: Theoretical Insights and Applications

Abstract

In this study, we investigate the fractional Tzitzéica equation, a nonlinear evolution equation known for modeling complex phenomena in various scientific domains such as solid-state physics, crystal dislocation, electromagnetic waves, chemical kinetics, quantum field theory, and nonlinear optics. Using the (G′/G, 1/G)-expansion approach, we derive different categories of exact solutions, like hyperbolic, trigonometric, and rational functions. The beta fractional derivative is used here to generalize the classical idea of the derivative, which preserves important principles. The derived solutions with broader nonlinear wave structures are periodic waves, breathers, peakons, W-shaped solitons, and singular solitons, which enhance our understanding of nonlinear wave dynamics. In relation to these results, the findings are described by showing the solitons’ physical behaviors, their stabilities, and dispersions under fractional parameters in the form of contour plots and 2D and 3D graphs. Comparisons with earlier studies underscore the originality and consistency of the (G′/G, 1/G)-expansion approach in addressing fractional-order evolution equations. It contributes new solutions to analytical problems of fractional nonlinear integrable systems and helps understand the systems’ dynamic behavior in a wider scope of applications.

1. Introduction

Nonlinear evolution equations (NLEEs) and their disorderly behaviors have received incredible attention across diverse disciplines during the last few decades [1]. These equations are crucial for understanding many phenomena occurring in different domains of science and technology, such as fluid dynamics, thermodynamics, heat conduction, plasma physics, electro-magneto dynamics, optics, signal processing, quantum physics, etc. Exact traveling wave solutions are one of the most important features of NLEEs, which can give a full understanding of the underlying phenomenon under different conditions and parameters. These solutions explain the underlying interactions between variables and make known fundamental properties that remain obscured in numerical approaches. Unlike usual traveling waves, solitary waves can propagate over long distances with minimal loss of energy, amplitude, and velocity [2]. A self-reinforcing solitary wave that propagates with a constant shape and speed regardless of its interaction with other waves is referred to as a soliton. Due to their remarkable stability, solitons play an important role in communication engineering and signal processing and enable long-distance data transmission [3]. In the last few decades, soliton solutions have been investigated in numerous domains, such as electromagnetic waves [4], plasma waves [5], fluid dynamics [6], communications through optical fibers [7], biological systems [8], ecological models [9], etc. In order to develop exact soliton solutions to different kinds of nonlinear evolution equations (NLEEs), a number of analytical techniques have been formulated, such as the exp(-$\varphi \left(\xi \right)$)-expansion method [10,11], the modified $\left(G^{\prime }/G^2\right)$-expansion approach [12,13], the $\left(m+\left(G^{\prime }/G\right)\right)$-expansion technique [14,15], the rational sine–Gordon expansion process [16,17], the extended Riccati equation expansion method [18,19], the improved $\tan \left(\varphi \left(\xi \right)/2\right)$-expansion technique [20,21], the improved F-expansion scheme [22,23], the modified homotopy perturbation mode [24,25], the modified extended direct algebraic method [26,27], the modified variational iteration method [28,29], the Kudryashov scheme [30,31], the $\left(G^{\prime }/G^2\right)$-expansion method [32,33], the modified auxiliary equation technique [34,35], the generalized $\left(G^{\prime }/G\right)$-expansion method [36,37], the $\left(G^{\prime }/G, 1/G\right)$-expansion method [38,39], the Darboux transformations method [40], the Hirota bilinear method [41], the sine–collocation method [42], the homotopy decomposition method [43], etc.

Fractional calculus is the modern branch of mathematics and is concerned with non-integer order differentiation and integration of functions. Fractional evolution equations extend the classical evolution equations by including fractional-order derivatives, which allow them to capture memory effects, hereditary properties, and non-locality. These equations predict the future behavior of a given system considering the entire past history, making them useful for long-range dependent processes. Over time, various well-known definitions of fractional derivatives have been developed through extensive research by mathematicians. For instance, these include the Riemann–Liouville (RL) fractional derivative [44], the Caputo definition [45], the Grunwald–Letnikov definition [46], the conformable definition [47], the beta definition [48], and some others. The RL and Caputo fractional derivatives fail to satisfy the Leibnitz and chain rules of classical derivatives. The conformable derivative of a function always yields zero value in zero domains, which contradicts the principle of classical derivatives. The beta fractional derivative is, therefore, a generalization of classical derivative concepts while preserving their foundational principles, including the principles of linearity, the Leibniz rule, the chain rule, etc. With this operator, key analytical properties can hold with no partner, offering expanded applicability in numerous applications in the area of anomalous diffusion, viscoelasticity, and complex dynamical systems. This new concept is similar to classical calculus, allowing the application of fractional operators into infinite dimensions, opening diverse perspectives in solving many problems where regular derivatives fall short of providing adequate solutions. The beta fractional derivative of a function, $g\left(y\right)$, is defined as [49]

$$D_y^{\beta }g\left(y\right)=\underset{\delta \to 0}{\lim }{\displaystyle \frac{g\left(y+\delta {\left(y+\frac{1}{\Gamma \left(\beta \right)}\right)}^{1-\beta }\right)-g\left(y\right)}{\delta }},$$

(1)

where $0<\beta \le 1$.

The beta derivative used in this study is a local fractional derivative that extends the classical derivative but holds all fundamental properties like linearity, the product rule, the chain rule, and others. Unlike the Caputo or Riemann–Liouville derivatives, the beta derivative is local in nature, meaning that it depends only on the neighborhood of a point. The locality allows a suitable physical interpretation in cases where long-term memory effects are not dominant but fractional behaviors are present. The beta derivative introduces a tunable parameter, $\beta$, that enables flexible modeling of complex systems, keeping the features of classical derivative. It reverts back to the classical derivative when $\beta \to 1$, which makes it possible to transition from fractional to classical dynamics. In wave propagation, it describes the impact of local structures on the velocity and shapes of waves. Thus, the beta derivative is useful for revealing the interplay between subtle dynamical behaviors, bridging the gap between local and non-local fractional derivatives.

The Tzitzéica (TZ) equation, introduced by Tzitzéica in 1908 during his study of integrable surfaces, is a significant contribution to the field of differential geometry. This equation is characterized by its nonlinear structure, which includes exponential terms, and can be formulated as follows [50,51]:

$$p_{tt}-p_{xx}-e^{p\left(x,t\right)}+e^{-2p\left(x,t\right)}=0,$$

(2)

where $p(x, t$) is a scalar function that depends on space, $x$, and time, $t$. It usually represents a physical field or geometric quantity, which is relative to the context, e.g., wave profile in physics and surface curvature in differential geometry. The second-order partial derivative $p_{tt}$ represents the temporal curvature or acceleration, and the second-order partial derivative $p_{xx}$ represents spatial curvature or dispersion. The exponential term $e^{p\left(x, t\right)}$ introduces nonlinearity into the equation, which plays a key role in the interaction dynamics. Another nonlinear term, $e^{-2p\left(x, t\right)}$, opposing $e^{p\left(x, t\right)}$ makes a nonlinear balance that prevents unbounded growth and allows the existence of soliton-like solutions. This structure is a characteristic of integrable systems and plays a central role in the dynamics described by the Tzitzéica equation.

The classical Tzitzéica equation originates in differential geometry, soliton theory, and the theory of surfaces with constant affine curvature. The fractional Tzitzéica equation is a generalization of the classical Tzitzéica equation using fractional derivatives. Its application involves nonlinear wave dynamics and complex media where memory effects and non-locality play a role. The fractional extension expands its scope to include physically realistic modeling. The fractional Tzitzéica equation is also applicable in optics, fluid dynamics, plasma physics, and nonlinear behavior in fluids with complex rheological properties, such as in shock waves or compactons.

Classical models use integer-order derivatives and have long been used to describe dynamical systems. However, fractional-order models include fractional derivatives and provide a more detailed representation of complex phenomena, such as viscoelastic materials, anomalous diffusion, and non-local effects. Classical derivatives concentrate only on local changes, while fractional derivatives account for long-term dependence and heredity. This characteristic makes them useful for modeling viscoelastic materials, biological systems, and chaotic processes. While classical models are simple and computationally efficient, fractional models provide deeper insights when traditional methods do not work well. Since fractional derivatives introduce additional degrees of freedom, we suppose that the nonlinear effects can be explained with the help of fractional derivatives. The Tzitzéica equation is a well-known NLEE, which is widely recognized to represent complex phenomena arising in solid-state physics, crystal dislocation, electromagnetic waves, chemical kinetics, quantum field theory, and nonlinear optics. Introducing the beta fractional derivative in Equation (2), the fractional Tzitzéica equation can be read as

$$D_{tt}^{2\beta }p-D_{xx}^{2\beta }p-e^p+e^{-2p}=0.$$

(3)

In previous years, several researchers studied the Tzitzéica equation and its solutions through some numerical and analytical techniques. For instance, these include the $\left(G^{\prime }/G\right)$-expansion method [51], the Sardar sub-equation method [52], the dressing method and Hirota method [53], the generalized exponential rational function method [54], the polynomial complete discriminant method [55], the improved $\tan \left(\varphi \left(\xi \right)/2\right)$-expansion method [56], the simple ansatz approach [57], the simplest equation method [58], the unified method [59], etc. Nonlinear effects and fractional derivatives play a key role in system behavior, but their impact and how they work are not well studied theoretically or experimentally in the existing literature. Moreover, how the changes in the key parameters impact system behavior, including stability and possible shifts in solutions, is not well measured. These gaps make it difficult to fully understand nonlinear systems and develop accurate models. To our knowledge, the closed-form wave solutions to the fractional Tzitzéica equation have not been studied yet using the $\left(G^{\prime }/G, 1/G\right)$-expansion approach. Therefore, the objective of this article is to study nonlinear effects in detail and analyze how different parameters affect solutions under various conditions. We derive periodic waves, breathers, peakons, W-shaped solitons, and singular manifold solitons for the fractional Tzitzéica equation. The $\left(G^{\prime }/G, 1/G\right)$-expansion method is a robust, reliable tool for constructing soliton solutions in both integer and fractional-order nonlinear evolution equations.

Thus, the originality of this article is in the development of diverse analytical soliton solutions, such as breathers, peakons, W-shaped solitons, etc., for the fractional Tzitzéica equation, which have previously been unexplored. In this study, we use the beta fractional derivative, which preserves the fundamental principles of classical calculus, and introduce solutions with hyperbolic, trigonometric, and rational forms. Graphical analyses reveal the impact of fractional parameters on soliton dynamics. Comparisons with previous studies confirm both consistency and new contributions, particularly in addressing fractional-order effects. These findings advance understanding of nonlinear wave phenomena in fields like optics and quantum physics, offering benchmarks for future theoretical and numerical studies.

The rest of this article is outlined as follows: Section 2 contains the method of this study, Section 3 deals with the application of the regarded method to find out exact soliton solutions of the considered model, Section 4 comprises the graphical representation of the obtained solitons along with their physical interpretations, and Section 5 presents comparisons between the present and prior results to withstand the novelty of our study. A summary of our study and findings is included in the Conclusions Section.

2. Outline of the Method

In this study, we adopt a straightforward and reliable approach, the $\left(G^{\prime }/G, 1/G\right)$-expansion approach, to seek out novel analytical soliton solutions of the beta fractional TZ equation. The outline of the regarded method to find out the exact soliton solution to NLEEs with beta fractional derivatives is discussed in the subsequent content.

Let us consider a general fractional nonlinear differential model in the sense of a beta derivative in the subsequent form:

$$Q\left(D_t^{\beta }p, D_x^{\beta }p, p^2{D}_{tt}^{2\beta }p, p^{2}p, {\left|p\right|}^2,\dots \right)=0, 0<\beta \le 1,$$

(4)

where the parameter $\beta$ signifies the differential order of the fractional equation of the wave function $p=p\left(x,t\right)$ with respect to the independent variables $x$ and $t$, and $Q$ is the polynomial of the wave function $p$ and its various fractional derivatives. As we are seeking traveling solutions, we introduce a beta wave transformation for Equation (4) to convert the governing fractional model to an integer order model. The considered beta wave transformation is

$$p\left(x,t\right)=\Omega \left(\eta \right), \mathrm{where} \eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }-{\displaystyle \frac{w}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta },$$

(5)

Here, $\Omega \left(\eta \right)$ is the amplitude of the transformed wave with the frequency $w$ and wave number $k$. Applying the indicated wave transformation to the governing model, the obtained nonlinear equation can be read as

$$L\left({\Omega }^{\prime },{\Omega }^{″},{\Omega }^{‴},\dots \right)=0,$$

(6)

where prime denotes the order of classical differentiation of the transformed wave function $\Omega =\Omega \left(\eta \right)$ with respect to $\eta$. Now, the next four steps, which are to be followed to extract exact soliton solutions of Equation (6), are discussed in the subsequent content.

Step 1: The regarded analytical approach provides a series solution to Equation (6) of the following form:

$$\Omega \left(\eta \right)={{\displaystyle \sum }}_{j=0}^n{m}_j\mathsf{\Phi }{\left(\eta \right)}^{\mathrm{j}}+{{\displaystyle \sum }}_{h=1}^n{n}_h\mathsf{\Phi }{\left(\eta \right)}^{h-1}\Psi \left(\eta \right),$$

(7)

wherein $\mathsf{\Phi }\left(\eta \right)=G^{\prime }/G$, $\Psi \left(\eta \right)=1/G$. The unknown parameters $m_j$ and $n_h$ are to be determined through mathematical calculations, and $n$ is the balancing parameter that is to be determined through the homogeneous principle of balance. The arbitrary function $G=G\left(\eta \right)$ is governed by the general solutions of the following second-order equation:

$$G^{″}\left(\eta \right)+\lambda G\left(\eta \right)=\mu ,$$

(8)

where $\lambda$ and $\mu$ are arbitrary parameters. Solving Equation (8), the derived general solutions are

$$G\left(\eta \right)=\left\{\begin{array}{c}{C}_1\sin \left(\sqrt{\lambda } \eta \right)+C_2\cos \left(\sqrt{\lambda } \eta \right)+{\displaystyle \frac{\mathsf{\mu }}{\lambda }}, \lambda >0,\\ {\mathrm{C}}_1\sinh \left(\sqrt{-\lambda } \eta \right)+{\mathrm{C}}_2\cosh \left(\sqrt{-\lambda } \eta \right)+{\displaystyle \frac{\mathsf{\mu }}{\lambda }}, \lambda <0\\ {\displaystyle \frac{\mu }{2}}{\eta }^2+{\mathrm{C}}_1\eta +{\mathrm{C}}_2, \lambda =0,\end{array}\right.,$$

(9)

where $C_1$ and $C_2$ are arbitrary parameters. Using the general solutions in (9) and Equation (8), the estimated values of ${\mathsf{\Psi }}^2\left(\eta \right)$ are as follows:

$${\Psi }^2\left(\eta \right)=\left\{\begin{array}{c}{\displaystyle \frac{\lambda \left({\mathsf{\Phi }}^2-2\Psi \mathsf{\mu }+\lambda \right)}{{\lambda }^2\mathsf{\Lambda }-{\mathsf{\mu }}^2}}, \Lambda =C_1^2+C_2^2, \lambda >0.\\ -{\displaystyle \frac{\lambda \left({\mathsf{\Phi }}^2-2\Psi \mathsf{\mu }+\lambda \right)}{{\lambda }^2\mathsf{\Lambda }+{\mathsf{\mu }}^2}}, \Lambda =C_1^2-C_2^2, \lambda <0.\\ {\displaystyle \frac{1}{{\mathrm{C}}_1^2-2{\mathsf{\mu }\mathrm{C}}_2}}\left({\mathsf{\Phi }}^2-2\mathsf{\mu }\Psi \right), \lambda =0.\end{array}\right.$$

(10)

Step 2: Inserting the trial solutions into the derived Equation (6), we obtain a new equation of the following form:

$$P\left({\mathsf{\Phi }}^{\prime },{\Psi }^{\prime }, {\Psi }^{″}\mathsf{\Phi }, {\mathsf{\Phi }}^{″}{\Psi }^2, {\mathsf{\Phi }}^3\Psi ,\dots \right)=0,$$

(11)

wherein $P$ stands for a polynomial of $\Psi$, $\mathsf{\Phi }$, and their derivatives with respect to $\eta$.

Step 3: To convert the derived Equation (11) into an algebraic equation, we replace the derivatives of $\mathsf{\Phi }\left(\eta \right)$ and $\Psi \left(\eta \right)$ by the following relations:

$${\mathsf{\Phi }}^{\prime }=-{\mathsf{\Phi }}^2+\mathsf{\mu }\Psi -\lambda , {\Psi }^{\prime }=-\mathsf{\Phi }\Psi .$$

(12)

Now, substituting the values of ${\Psi }^2\left(\eta \right)$, the degree of the function $\Psi \left(\eta \right)$ included in the algebraic equation must be restrained within one.

Step 4: Equating the coefficients of ${\mathsf{\Phi }}^i\left(\eta \right){\Psi }^l\left(\eta \right)$, $\left(i=0,1,\dots n;l=0,1\right)$, to zero, we obtain a system of algebraic equations. Solving the derived system of algebraic equations, a set of solutions for the chosen parameters can be estimated. Placing the assessed values of parameters in the trial solution in (7) and Equation (5), an exact traveling wave solution of the governing model can be generated.

3. Mathematical Analysis and Solutions

Using the transformation $p\left(x,t\right)=\ln u\left(x, t\right)$, Equation (3) is converted to

$$u{D}_{tt}^{2\beta }u-u{D}_{xx}^{2\beta }u-{\left(D_t^{\beta }u\right)}^2+{\left(D_x^{\beta }u\right)}^2-u^3+1=0.$$

(13)

We now introduce the beta fractional wave transformation $u\left(x,t\right)=\Omega \left(\eta \right)$, where $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }-{\displaystyle \frac{w}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$ in order to convert the Tzitzéica model given in Equation (13).

Now,

$$\begin{array}{c}{D}_x^{\beta }u\left(x, t\right)=D_x^{\beta }\Omega \left(\eta \right)={\displaystyle \frac{{\partial }^{\beta }}{\partial x^{\beta }}}\Omega \left(\eta \right)={\displaystyle \frac{\partial }{\partial \eta }}\Omega \left(\eta \right){\displaystyle \frac{{\partial }^{\beta }}{\partial x^{\beta }}}\left(\eta \right).\hfill \\ ={\Omega }^{\prime }\left(\eta \right){\displaystyle \frac{{\partial }^{\beta }}{\partial x^{\beta }}}\left({\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }-{\displaystyle \frac{w}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }\right).\hfill \\ ={\Omega }^{\prime }\left(\eta \right)\left\{{\displaystyle \frac{{\partial }^{\beta }}{\partial x^{\beta }}}\left({\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }\right)-{\displaystyle \frac{{\partial }^{\beta }}{\partial x^{\beta }}}\left({\displaystyle \frac{w}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }\right)\right\}.\hfill \\ ={\Omega }^{\prime }\left(\eta \right){\displaystyle \frac{k}{\beta }}\left\{{\displaystyle \frac{{\partial }^{\beta }}{\partial x^{\beta }}}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }-0\right\}.\hfill \end{array}$$

Taking the beta derivative of ${\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }$, we obtain

$$\begin{array}{c}{D}_x^{\beta }u\left(x, t\right)={\displaystyle \frac{k}{\beta }}{\Omega }^{\prime }\left(\eta \right)\underset{\delta \to \infty }{\lim }{\displaystyle \frac{1}{\delta }}\left\{{\left(x+\delta {\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{1-\beta }+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }-{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\right\}.\hfill \\ ={\displaystyle \frac{k}{\beta }}{\Omega }^{\prime }\left(\eta \right)\underset{\delta \to \infty }{\lim }{\displaystyle \frac{1}{\delta }}\left\{{\left(\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)+\delta {\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{1-\beta }\right)}^{\beta }-{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\right\}.\hfill \end{array}$$

Expanding with the binomial theorem, we obtain

$$\begin{array}{c}{D}_x^{\beta }u\left(x, t\right)={\displaystyle \frac{k}{\beta }}{\Omega }^{\prime }\left(\eta \right)\underset{\delta \to \infty }{\lim }{\displaystyle \frac{1}{\delta }}\{{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }+\beta {\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta -1}\delta {\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{1-\beta }.\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{1}{2}}\beta \left(\beta -1\right){\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta -2}{\delta }^2{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{2-2\beta }+\dots -{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\}.\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{k}{\beta }}{\Omega }^{\prime }\left(\eta \right)\underset{\delta \to \infty }{\lim }{\displaystyle \frac{\delta }{\delta }}\left\{\beta {\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta -1+1-\beta }+{\displaystyle \frac{\delta }{2}}\beta \left(\beta -1\right){\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta -2+2-2\beta }+\dots \right\}.\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{k}{\beta }}{\Omega }^{\prime }\left(\eta \right) \underset{\delta \to \infty }{\lim }\left\{\beta {\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^0+{\displaystyle \frac{\delta }{2}}\beta \left(\beta -1\right){\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{-\beta }+\dots \right\}.\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{k}{\beta }}{\Omega }^{\prime }\left(\eta \right)\left\{\beta +0+0+\dots \right\}.\hfill \end{array}$$

Therefore, $D_x^{\beta }u\left(x, t\right)=k{\Omega }^{\prime }\left(\eta \right)$.

Continuing this process iteratively, we obtain

$$D_{xx}^{2\beta }u\left(x, t\right)=k^2{\Omega }^{\prime }\left(\eta \right)$$

Again,

$$\begin{array}{c}{D}_t^{\beta }u\left(x, t\right)=D_t^{\beta }\Omega \left(\eta \right)={\displaystyle \frac{{\partial }^{\beta }}{\partial t^{\beta }}}\Omega \left(\eta \right)={\displaystyle \frac{\partial }{\partial \eta }}\Omega \left(\eta \right){\displaystyle \frac{{\partial }^{\beta }}{\partial t^{\beta }}}\left(\eta \right).\hfill \\ ={\Omega }^{\prime }\left(\eta \right){\displaystyle \frac{{\partial }^{\beta }}{\partial x^{\beta }}}\left({\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }-{\displaystyle \frac{w}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }\right).\hfill \\ ={\Omega }^{\prime }\left(\eta \right)\left\{{\displaystyle \frac{{\partial }^{\beta }}{\partial t^{\beta }}}\left({\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }\right)-{\displaystyle \frac{{\partial }^{\beta }}{\partial t^{\beta }}}\left({\displaystyle \frac{w}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }\right)\right\}.\hfill \\ ={\Omega }^{\prime }\left(\eta \right)\left\{0-{\displaystyle \frac{w}{\beta }} {\displaystyle \frac{{\partial }^{\beta }}{\partial t^{\beta }}}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }\right\}.\hfill \end{array}$$

$$D_t^{\beta }u\left(x, t\right)=-{\displaystyle \frac{w}{\beta }}{\Omega }^{\prime }\left(\eta \right) {\displaystyle \frac{{\partial }^{\beta }}{\partial t^{\beta }}}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }.$$

Taking the beta derivative of ${\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\beta \right)}}\right)}^{\beta }$, we obtain

$$\begin{array}{c}{D}_t^{\beta }u\left(x, t\right)=-{\displaystyle \frac{w}{\beta }}{\Omega }^{\prime }\left(\eta \right)\underset{\delta \to \infty }{\lim }{\displaystyle \frac{1}{\delta }}\left\{{\left(t+\delta {\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{1-\beta }+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }-{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\right\}.\hfill \\ =-{\displaystyle \frac{w}{\beta }}{\Omega }^{\prime }\left(\eta \right)\underset{\delta \to \infty }{\lim }{\displaystyle \frac{1}{\delta }}\left\{{\left(\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)+\delta {\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{1-\beta }\right)}^{\beta }-{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\right\}.\hfill \end{array}$$

Expanding in the binomial theorem, we get

$$\begin{array}{c}{D}_t^{\beta }u\left(x, t\right)=-{\displaystyle \frac{w}{\beta }}{\Omega }^{\prime }\left(\eta \right)\underset{\delta \to \infty }{\lim }{\displaystyle \frac{1}{\delta }}\{{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }+\beta {\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta -1}\delta {\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{1-\beta }\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +{\displaystyle \frac{1}{2}}\beta \left(\beta -1\right){\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta -2}{\delta }^2{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{2-2\beta }+\dots -{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\}.\hfill \\ =-{\displaystyle \frac{w}{\beta }}{\Omega }^{\prime }\left(\eta \right)\underset{\delta \to \infty }{\lim }{\displaystyle \frac{\delta }{\delta }} \left\{\beta {\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta -1+1-\beta }+{\displaystyle \frac{\delta }{2}}\beta \left(\beta -1\right){\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta -2+2-2\beta }+\dots \right\}.\hfill \\ =-{\displaystyle \frac{w}{\beta }}{\Omega }^{\prime }\left(\eta \right) \underset{\delta \to \infty }{\lim }\left\{\beta {\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^0+{\displaystyle \frac{\delta }{2}}\beta \left(\beta -1\right){\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{-\beta }+\dots \right\}.\hfill \\ =-{\displaystyle \frac{w}{\beta }}{\Omega }^{\prime }\left(\eta \right)\left\{\beta +0+0+\dots \right\}.\hfill \end{array}$$

Therefore, $D_t^{\beta }u\left(x, t\right)=-w{\Omega }^{\prime }\left(\eta \right)$.

Therefore, by iteratively applying the same procedure, we obtain

$$D_{tt}^{2\beta }u\left(x, t\right)=w^2{\Omega }^{\prime }\left(\eta \right).$$

Similarly, we obtain ${\left(D_x^{\beta }u\right)}^2={\left(k{\Omega }^{\prime }\left(\eta \right)\right)}^2=k^2{\left({\Omega }^{\prime }\left(\eta \right)\right)}^2$.

And ${\left(D_t^{\beta }u\right)}^2={\left(-w{\Omega }^{\prime }\left(\eta \right)\right)}^2=w^2{\left({\Omega }^{\prime }\left(\eta \right)\right)}^2$.

Therefore, substituting the expressions for $D_{tt}^{2\beta }u$, $u{D}_{xx}^{2\beta }u$, ${\left(D_t^{\beta }u\right)}^2$, and ${\left(D_x^{\beta }u\right)}^2$ into the Tzitzéica model (13) and simplifying yields the transformed nonlinear equation as

$$\left(w^2-k^2\right)\Omega \left(\eta \right){\Omega }^{″}\left(\eta \right)-\Omega {\left(\eta \right)}^3+\left(k^2-w^2\right){\Omega }^{\prime }{\left(\eta \right)}^2+1=0.$$

(14)

Here, $w$ and $k$ signify the wave velocity and wave number, respectively.

By means of the homogeneous balance principle, from Equation (14) we obtain $n=2$. Therefore, the trial solution of Equation (14) can be formulated as

$$\Omega \left(\eta \right)=m_0+m_1\mathsf{\Phi }\left(\eta \right)+m_2\mathsf{\Phi }{\left(\eta \right)}^2+n_1\Psi \left(\eta \right)+n_2\Psi \left(\eta \right)\mathsf{\Phi }\left(\eta \right),$$

(15)

where $m_0$, $m_1$, $m_2$, $n_1$, and $n_2$ are arbitrary parameters. Setting the value of $\Omega \left(\eta \right)$ and its derivatives into Equation (14) and following the steps described in the methodology section, we obtain several sets of solutions for the selected parameters included in trial solution. The assessed solution sets and corresponding exact soliton solutions are discussed in the succeeding content.

Case 1  $(\lambda >0)$: In this case, we obtain the following solution sets for the considered parameters.

Set 1:

$$\begin{array}{c}{m}_0=1\pm \mathrm{i}\sqrt{3}, m_1=0, n_1=\pm {\displaystyle \frac{3\mathrm{i}\left(\pm \mathrm{i}\mu +\sqrt{3}\mu \right)}{2\lambda }}, n_2=\pm {\displaystyle \frac{3\sqrt{\frac{{\mu }^2}{{\lambda }^2} \pm \frac{\mathrm{i}\sqrt{3}{\mu }^2}{{\lambda }^2} – \sigma \pm \mathrm{i}\sqrt{3}\sigma }}{\sqrt{2}\sqrt{\lambda }}},\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} m_2={\displaystyle \frac{3\left(1\pm \mathrm{i}\sqrt{3}\right)}{2\lambda }}, w=\pm \sqrt{k^2+{\displaystyle \frac{3}{2\lambda }}\pm {\displaystyle \frac{3\mathrm{i}\sqrt{3}}{2\lambda }}}.\hfill \end{array}$$

Set 2:

$$m_0=-2, m_1=0, n_1={\displaystyle \frac{3\mu }{\lambda }}, m_2=-{\displaystyle \frac{3}{\lambda }}, n_2=\pm {\displaystyle \frac{3\sqrt{-{\mu }^2+{\lambda }^2\sigma }}{{\lambda }^{3/2}}}, w=\pm {\displaystyle \frac{\sqrt{-3+k^2\lambda }}{\sqrt{\lambda }}}.$$

Set 3:

$$\begin{gathered} m_0={\displaystyle \frac{1}{4}}\left(1\pm \mathrm{i}\sqrt{3}\right), m_1=0, n_1=0, m_2={\displaystyle \frac{3\left(1\pm \mathrm{i}\sqrt{3}\right)}{4\lambda }}, n_2=0, \\ w=\pm \sqrt{k^2+{\displaystyle \frac{3}{8\lambda }}\pm {\displaystyle \frac{3\mathrm{i}\sqrt{3}}{8\lambda }}}, \mu =0. \end{gathered}$$

Set 4:

$$\begin{gathered} m_0=1\pm \mathrm{i}\sqrt{3}, m_1=0, n_1=0, m_2={\displaystyle \frac{3\left(1\pm \mathrm{i}\sqrt{3}\right)}{2\lambda }}, n_2=\pm {\displaystyle \frac{3\sqrt{-\frac{\sigma }{\lambda }\pm \frac{\mathrm{i}\sqrt{3}\sigma }{\lambda }}}{\sqrt{2}}}, \mu =0, \\ w=\pm \sqrt{k^2+{\displaystyle \frac{3}{2\lambda }}\pm {\displaystyle \frac{3\mathrm{i}\sqrt{3}}{2\lambda }}}. \end{gathered}$$

Set 5:

$$m_0=-2, m_1=0, n_1=0, m_2=-{\displaystyle \frac{3}{\lambda }}, n_2=\pm {\displaystyle \frac{3\sqrt{\sigma }}{\sqrt{\lambda }}}, w=\pm {\displaystyle \frac{\sqrt{-3+k^2\lambda }}{\sqrt{\lambda }}}, \mu =0.$$

Set 6:

$$m_0=-{\displaystyle \frac{1}{2}}, m_1=0, n_1=0, m_2=-{\displaystyle \frac{3}{2\lambda }}, n_2=0, w=\pm {\displaystyle \frac{\sqrt{-3+4k^2\lambda }}{2\sqrt{\lambda }}}, \mu =0.$$

Placing the estimated values of parameters in the trial solution in (15), we obtain an exact solution for the converted nonlinear Equation (14). Recognizing the solution set 1, we obtain

$${\Omega }_{11}\left(\eta \right)=1\pm \sqrt{-3}+{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right){\left(f_2\left(\eta \right)\right)}^2}{2{\left(f_1\left(\eta \right)\right)}^2}}\pm {\displaystyle \frac{3\sqrt{\left(1\pm \sqrt{-3}\right)\left(\frac{{\mu }^2}{{\lambda }^2}-\sigma \right)}{f}_2\left(\eta \right)}{\sqrt{2}{\left(f_1\left(\eta \right)\right)}^2}}\pm {\displaystyle \frac{3\mu \left(1\pm \sqrt{-3}\right)}{2\lambda f_1\left(\eta \right)}},$$

where $f_1\left(\eta \right)={\displaystyle \frac{\mu }{\lambda }}+C_1\sin \left(\eta \sqrt{\lambda }\right)+C_2\cos \left(\eta \sqrt{\lambda }\right)$, and $f_2\left(\eta \right)=C_1\cos \left(\eta \sqrt{\lambda }\right)-C_2\sin \left(\eta \sqrt{\lambda }\right)$. The subscripts in ${\Omega }_{11}\left(\eta \right)$ define the category and solution set, respectively, which are to be recognized to form the solution. Introducing the values of ${\Omega }_{11}\left(\eta \right)$ to the beta wave transformation, we attain the required exact soliton solution to the governing Tzitzéica model. The derived solution is

$$p_{11}\left(x,t\right)=\ln \left(1\pm \sqrt{-3}+{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right){\left(f_2\left(\eta \right)\right)}^2}{2{\left(f_1\left(\eta \right)\right)}^2}}\pm {\displaystyle \frac{3\sqrt{\left(1\pm \sqrt{-3}\right)\left(\frac{{\mu }^2}{{\lambda }^2}-\sigma \right)}{f}_2\left(\eta \right)}{\sqrt{2}{\left(f_1\left(\eta \right)\right)}^2}}\pm {\displaystyle \frac{3\mu \left(1\pm \sqrt{-3}\right)}{2\lambda f_1\left(\eta \right)}}\right),$$

(16)

with $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\pm {\displaystyle \frac{\sqrt{k^2+\frac{3}{2\lambda }\pm \frac{3\sqrt{-3}}{2\lambda }}}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$ and $\sigma =C_1^2+C_2^2$.

Analogously, the following exact soliton solutions are derived by recognizing the remaining solution sets.

$$p_{12}\left(x,t\right)=\ln \left(-2+{\displaystyle \frac{3\mu }{\lambda f_1\left(\eta \right)}}\pm {\displaystyle \frac{3\left(f_2\left(\eta \right)\right)}{\lambda {\left(f_1\left(\eta \right)\right)}^2}}\left\{\sqrt{-{\mu }^2+{\lambda }^2\sigma }-\lambda f_2\left(\eta \right)\right\}\right),$$

(17)

together with $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\pm {\displaystyle \frac{\sqrt{-3+k^2\lambda }}{\beta \sqrt{\lambda }}}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$ and $\sigma =C_1^2+C_2^2$.

$$p_{13}\left(x,t\right)=\ln \left({\displaystyle \frac{1}{4}}\left(1\pm \sqrt{-3}\right)+{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right){\left(f_2\left(\eta \right)\right)}^2}{4{\left(f_1\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}\right),$$

(18)

where $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\pm {\displaystyle \frac{\sqrt{k^2+\frac{3}{8\lambda }\pm \frac{3\sqrt{-3}}{8\lambda }}}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$.

$$p_{14}\left(x,t\right)=\ln \left(1\pm \mathrm{i}\pm {\displaystyle \frac{3\sqrt{\lambda }\sqrt{-\frac{\sigma }{\lambda }\left(1\pm \mathrm{i}\sqrt{3}\right)} f_2\left(\eta \right)}{\sqrt{2}{\left(f_1\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}+{\displaystyle \frac{3\left(1\pm \mathrm{i}\sqrt{3}\right){\left(f_2\left(\eta \right)\right)}^2}{2{\left(f_1\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}\right),$$

(19)

with $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\pm {\displaystyle \frac{\sqrt{k^2+\frac{3}{2\lambda }\pm \frac{3\sqrt{-3}}{2\lambda }}}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$ and $\sigma =C_1^2+C_2^2$.

$$p_{15}\left(x,t\right)=\ln \left(-2\pm {\displaystyle \frac{3\sqrt{\sigma }{f}_2\left(\eta \right)}{{\left(f_1\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}-{\displaystyle \frac{3{\left(f_2\left(\eta \right)\right)}^2}{{\left(f_1\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}\right),$$

(20)

where $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\pm {\displaystyle \frac{\sqrt{-3+k^2\lambda }}{\beta \sqrt{\lambda }}}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$ and $\sigma =C_1^2+C_2^2$.

$$p_{16}\left(x,t\right)=\ln \left(-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{3{\left(f_2\left(\eta \right)\right)}^2}{2{\left(f_1\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}\right),$$

(21)

along with $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\pm {\displaystyle \frac{\sqrt{-3+4k^2\lambda }}{2\beta \sqrt{\lambda }}}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$.

If we let $\mu =0$ and $C_1=0$, we get the following from the solutions stated from (16) to (21):

$$p_{11}^1\left(x,t\right)=\ln \left(1\pm \sqrt{-3}+{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right)}{2}}{\tan }^2\left(\eta \sqrt{\lambda }\right)\pm {\displaystyle \frac{3i\sqrt{\left(1\pm \sqrt{-3}\right)}}{\sqrt{2}}}\tan \left(\eta \sqrt{\lambda }\right)\sec \left(\eta \sqrt{\lambda }\right)\right).$$

(22)

$$p_{12}^1\left(x,t\right)=\ln \left(-2\pm {\displaystyle \frac{3}{\lambda }}\left\{\mathsf{\lambda }\tan \left(\eta \sqrt{\lambda }\right)\sec \left(\eta \sqrt{\lambda }\right)-\mathsf{\lambda }{\tan }^2\left(\eta \sqrt{\lambda }\right)\right\}\right).$$

(23)

$$p_{13}^1\left(x,t\right)=\ln \left({\displaystyle \frac{1}{4}}\left(1\pm \sqrt{-3}\right)+{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right)}{4}}{\tan }^2\left(\eta \sqrt{\lambda }\right)\right).$$

(24)

$$p_{14}^1\left(x,t\right)=\ln \left(1\pm \mathrm{i}\pm {\displaystyle \frac{3\sqrt{\lambda }\sqrt{-\frac{1}{\lambda }\left(1\pm \mathrm{i}\sqrt{3}\right)}}{\sqrt{2}}}\tan \left(\eta \sqrt{\lambda }\right)\sec \left(\eta \sqrt{\lambda }\right)+{\displaystyle \frac{3\left(1\pm \mathrm{i}\sqrt{3}\right)}{2}}{\tan }^2\left(\eta \sqrt{\lambda }\right)\right).$$

(25)

$$p_{15}^1\left(x,t\right)=\ln \left(-2\pm 3\tan \left(\eta \sqrt{\lambda }\right)\sec \left(\eta \sqrt{\lambda }\right)-3 {\tan }^2\left(\eta \sqrt{\lambda }\right)\right).$$

(26)

$$p_{16}^1\left(x,t\right)=\ln \left(-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{3}{2}}{\tan }^2\left(\eta \sqrt{\lambda }\right)\right).$$

(27)

Analogously, diverse forms of solutions can be derived by letting $\mu =0$ and $C_2=0$.

Case 2  $(\lambda <0)$: In this instance, the following solution sets for the selected parameters are estimated:

Set 1:

$$\begin{gathered} m_0=1\pm \mathrm{i}\sqrt{3}, m_1=0, n_1=\pm {\displaystyle \frac{3\mathrm{i}\left(\pm \mathrm{i}\mu +\sqrt{3}\mu \right)}{2\lambda }}, m_2={\displaystyle \frac{3\left(1\pm \mathrm{i}\sqrt{3}\right)}{2\lambda }}, \\ \hspace{1em}\hspace{1em}\hspace{1em}{n}_2=\pm {\displaystyle \frac{3\sqrt{\frac{{\mu }^2}{{\lambda }^2}\pm \frac{\mathrm{i}\sqrt{3}{\mu }^2}{{\lambda }^2}+\sigma \pm \mathrm{i}\sqrt{3}\sigma }}{\sqrt{2}\sqrt{\lambda }}}, w=\pm \sqrt{k^2+{\displaystyle \frac{3}{2\lambda }}\pm {\displaystyle \frac{3\mathrm{i}\sqrt{3}}{2\lambda }}}. \end{gathered}$$

Set 2:

$$m_0=-2, m_1=0, n_1={\displaystyle \frac{3\mu }{\lambda }}, m_2=-{\displaystyle \frac{3}{\lambda }}, n_2=\pm {\displaystyle \frac{3\sqrt{-{\mu }^2-{\lambda }^2\sigma }}{{\lambda }^{3/2}}}, w=\pm {\displaystyle \frac{\sqrt{-3+k^2\lambda }}{\sqrt{\lambda }}}.$$

Set 3:

$$\begin{gathered} m_0={\displaystyle \frac{1}{4}}\left(1\pm \mathrm{i}\sqrt{3}\right), m_1=0, n_1=0, m_2={\displaystyle \frac{3\left(1\pm \mathrm{i}\sqrt{3}\right)}{4\lambda }}, \\ \hspace{1em}\hspace{1em}{n}_2=0, w=\pm \sqrt{k^2+{\displaystyle \frac{3}{8\lambda }}\pm {\displaystyle \frac{3\mathrm{i}\sqrt{3}}{8\lambda }}}, \mu =0. \end{gathered}$$

Set 4:

$$\begin{gathered} m_0=1\pm \mathrm{i}\sqrt{3}, m_2={\displaystyle \frac{3\left(1\pm \mathrm{i}\sqrt{3}\right)}{2\lambda }}, n_2=\pm {\displaystyle \frac{3\sqrt{\frac{\sigma }{\lambda }\pm \frac{\mathrm{i}\sqrt{3}\sigma }{\lambda }}}{\sqrt{2}}}, w=\pm \sqrt{k^2+{\displaystyle \frac{3}{2\lambda }}\pm {\displaystyle \frac{3\mathrm{i}\sqrt{3}}{2\lambda }}}, \\ \hspace{1em}{m}_1=n_1=\mu =0. \end{gathered}$$

Set 5:

$$m_0=-2, m_1=0, n_1=0, m_2=-{\displaystyle \frac{3}{\lambda }}, n_2=\pm {\displaystyle \frac{3\mathrm{i}\sqrt{\sigma }}{\sqrt{\lambda }}}, w=\pm {\displaystyle \frac{\sqrt{-3+k^2\lambda }}{\sqrt{\lambda }}}, \mu =0.$$

Set 6:

$$m_0=-{\displaystyle \frac{1}{2}}, m_1=0, n_1=0, m_2=-{\displaystyle \frac{3}{2\lambda }}, n_2=0, w=\pm {\displaystyle \frac{\sqrt{-3+4k^2\lambda }}{2\sqrt{\lambda }}}, \mu =0.$$

In this case, the derived exact solutions comprise the hyperbolic functions, which are formulated by substituting the assessed values of parameters and introducing the values of the arbitrary functions $\mathsf{\Phi }\left(\eta \right)$ and $\Psi \left(\eta \right)$ in the trial solution (15). Considering the set 1 solution, we attain

$${\Omega }_{21}\left(\eta \right)=1\pm \sqrt{-3}\pm {\displaystyle \frac{3\mu \left(-1\pm \sqrt{-3}\right)}{2\lambda f_3\left(\eta \right)}}\pm {\displaystyle \frac{3\sqrt{-\lambda }\sqrt{\left(1\pm \sqrt{-3}\right)\left(\frac{{\mu }^2}{{\lambda }^2}+\sigma \right)}\left(f_4\left(\eta \right)\right)}{\sqrt{2}\sqrt{\lambda }{\left(f_3\left(\eta \right)\right)}^2}}-{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right){\left(f_4\left(\eta \right)\right)}^2}{2{\left(f_3\left(\eta \right)\right)}^2}},$$

where $f_3\left(\eta \right)={\displaystyle \frac{\mu }{\lambda }}+\sinh \left(\eta \sqrt{-\lambda }\right)C_1+\cosh \left(\eta \sqrt{-\lambda }\right)C_2$ and $f_4\left(\eta \right)=\cosh \left(\eta \sqrt{-\lambda }\right)C_1+\sinh \left(\eta \sqrt{-\lambda }\right)C_2$.

Substituting the values of ${\Omega }_{21}\left(\eta \right)$ in the beta wave transformation, we get

$$p_{21}\left(x,t\right)=\ln \left(1\pm \sqrt{-3}\pm {\displaystyle \frac{3\mu \left(-1\pm \sqrt{-3}\right)}{2\lambda f_3\left(\eta \right)}}\pm {\displaystyle \frac{3\sqrt{-\lambda }\sqrt{\left(1\pm \sqrt{-3}\right)\left(\frac{{\mu }^2}{{\lambda }^2}+\sigma \right)}\left(f_4\left(\eta \right)\right)}{\sqrt{2}\sqrt{\lambda }{\left(f_3\left(\eta \right)\right)}^2}}-{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right){\left(f_4\left(\eta \right)\right)}^2}{2{\left(f_3\left(\eta \right)\right)}^2}}\right),$$

(28)

with $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\pm {\displaystyle \frac{\sqrt{k^2+\frac{3}{2\lambda }\pm \frac{3\sqrt{-3}}{2\lambda }}}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$ and $\sigma =C_1^2-C_2^2$.

Similarly, inserting the assessed values from remaining solution sets, the obtained exact soliton solutions are as follows:

$$p_{22}\left(x,t\right)=\ln \left(-2+{\displaystyle \frac{3\mu }{\lambda f_3\left(\eta \right)}}\pm {\displaystyle \frac{3\sqrt{{\mu }^2+{\lambda }^2\sigma }\left(f_4\left(\eta \right)\right) }{\sqrt{\lambda }{\left(f_3\left(\eta \right)\right)}^2}}+{\displaystyle \frac{3{\left(f_4\left(\eta \right)\right)}^2}{{\left(f_3\left(\eta \right)\right)}^2}}\right),$$

(29)

together with $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\pm {\displaystyle \frac{\sqrt{-3+k^2\lambda }}{\beta \sqrt{\lambda }}}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$ and $\sigma =C_1^2-C_2^2$.

$$p_{23}\left(x,t\right)=\ln \left({\displaystyle \frac{1}{4}}\left(1\pm \sqrt{-3}\right)-{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right){\left(f_4\left(\eta \right)\right)}^2}{4{\left(f_3\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}\right),$$

(30)

where $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\pm {\displaystyle \frac{\sqrt{k^2+\frac{3}{8\lambda }\pm \frac{3\sqrt{-3}}{8\lambda }}}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$.

$$p_{24}\left(x,t\right)=\ln \left(1\pm \sqrt{-3}\pm {\displaystyle \frac{3\sqrt{-\lambda }\sqrt{\frac{\sigma }{\lambda }\pm \frac{\sqrt{-3}\sigma }{\lambda }}\left(f_4\left(\eta \right)\right)}{\sqrt{2}{\left(f_3\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}-{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right){\left(f_4\left(\eta \right)\right)}^2}{2{\left(f_3\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}\right),$$

(31)

with $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\pm {\displaystyle \frac{\sqrt{k^2+\frac{3}{2\lambda }\pm \frac{3\sqrt{-3}}{2\lambda }}}{\beta }}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$ and $\sigma =C_1^2-C_2^2$.

$$p_{25}\left(x,t\right)=\ln \left(-2\pm {\displaystyle \frac{3\left(f_4\left(\eta \right)\right)}{{\left(f_3\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}\left\{\sqrt{\sigma }+\left(f_4\left(\eta \right)\right)\right\}\right),$$

(32)

where $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\pm {\displaystyle \frac{\sqrt{-3+k^2\lambda }}{\beta \sqrt{\lambda }}}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$ and $\sigma =C_1^2-C_2^2$.

$$p_{26}\left(x,t\right)=\ln \left(-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{3{\left(f_4\left(\eta \right)\right)}^2}{2{\left(f_3\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}\right),$$

(33)

along with $\eta ={\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }\pm {\displaystyle \frac{\sqrt{-3+4k^2\lambda }}{2\beta \sqrt{\lambda }}}{\left(t+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mathsf{\beta }\right)}}\right)}^{\beta }$.

For $\mu =0$ and $C_2=0$, the generated solutions are

$$p_{21}\left(x,t\right)=\ln \left(1\pm \sqrt{-3}\pm {\displaystyle \frac{3\sqrt{-\lambda }\sqrt{\left(1\pm \sqrt{-3}\right)}}{\sqrt{2}\sqrt{\lambda }}}\coth \left(\eta \sqrt{-\lambda }\right)\csc \mathrm{h}\left(\eta \sqrt{-\lambda }\right)-{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right)}{2}}{\coth }^2\left(\eta \sqrt{-\lambda }\right)\right).$$

(34)

$$p_{22}\left(x,t\right)=\ln \left(-2\pm 3\coth \left(\eta \sqrt{-\lambda }\right)\csc \mathrm{h}\left(\eta \sqrt{-\lambda }\right)+3{\coth }^2\left(\eta \sqrt{-\lambda }\right)\right).$$

(35)

$$p_{23}\left(x,t\right)=\ln \left({\displaystyle \frac{1}{4}}\left(1\pm \sqrt{-3}\right)-{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right)}{4}}{\coth }^2\left(\eta \sqrt{-\lambda }\right)\right).$$

(36)

$$p_{24}\left(x,t\right)=\ln \left(1\pm \sqrt{-3}\pm {\displaystyle \frac{3\mathrm{i}\sqrt{1\pm \sqrt{-3}}}{\sqrt{2}}}\coth \left(\eta \sqrt{-\lambda }\right)\csc \mathrm{h}\left(\eta \sqrt{-\lambda }\right)-{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right)}{2}}{\coth }^2\left(\eta \sqrt{-\lambda }\right)\right).$$

(37)

$$p_{25}\left(x,t\right)=\ln \left(-2\pm 3\left\{\coth \left(\eta \sqrt{-\lambda }\right)\csc \mathrm{h}\left(\eta \sqrt{-\lambda }\right)+{\coth }^2\left(\eta \sqrt{-\lambda }\right)\right\}\right).$$

(38)

$$p_{26}\left(x,t\right)=\ln \left(-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{3}{2}}{\coth }^2\left(\eta \sqrt{-\lambda }\right)\right).$$

(39)

Similarly, a variety of solutions can be generated for $\mu =0$ and $C_1=0$.

Case 3  $\left(\lambda =0\right)$: For this case, this method yields only trivial solutions for the selected parameters.

4. Results and Discussion

This section of the present study investigates the physical features of the derived solutions along with their characteristics and applications. We plotted the three- and two-dimensional graphs and contour plots of the obtained soliton solutions for the selected parametric values. The contour plots represent the trajectories along which the amplitude of the wave function remains the same. The two-dimensional graphs were drawn for varied values of the fractional parameter to comprehend the impact of fractional derivatives on solitons.

Figure 1 is plotted for the values $\lambda =-10$, $\beta =0.99$, $k=0.001$, $\mu =-20$, $C_1=8.3$, and $C_2=-13.15$ and represents a W-shaped soliton for the modulus of solution $p_{21}\left(x,t\right)$. It shows a thrust at $t=0$ during propagation through the nonlinear medium. Its non-topological nature makes it suitable for data transmission over a long distance without any significant distortion of data. The continuous solution function becomes non-differentiable at $t=0$, as it has a cusp at its peak. With decreasing values of the fractional parameter $\beta$, it becomes dispersed, as shown in Figure 1b.

Figure 1. Diagrams of modulus of solution $p_{21}\left(x,t\right)$ for chosen parameters $\lambda =-10$, $\beta =0.99$, $k=0.001$, $\mu =-20$, $C_1=8.3$, and $C_2=-13.15$.

Figure 2 exhibits a breather soliton for the modulus of solution $p_{23}\left(x,t\right)$, which is plotted for the chosen values $\lambda =-10$, $\beta =0.99$, $k=0.15$, $C_1=7.05$, and $C_2=7.3$. A breather soliton is a localized oscillatory wave solution that remains confined to a certain region. It is particularly observed in the systems that are significantly influenced by the nonlinearity or dispersions. From the 2D diagrams of Figure 2, it can be notably observed that the obtained breather solution becomes dispersed with the decreasing values of $\beta$. It plays a significant role in mathematical modeling against instabilities and complexity.

Figure 2. Modulus plot of solution $p_{23}\left(x,t\right)$ for chosen parameters $\lambda =-10$, $\beta =0.99$, $k=0.15$, $C_1=7.05$, and $C_2=7.3$.

Figure 3 is drawn for the apt values of parameters, with $\lambda =-6.97$, $\beta =0.99$, $k=0.8$, $C_1=16.05$, and $C_2=13.4$, and exhibits a multi-peakon type soliton for the modulus of solution $p_{25}\left(x,t\right)$. A peakon soliton is a special type of soliton solution that has a sharp peak (a discontinuity in the first derivative) at its crest, making it different from smooth solitons. Despite the sharp peak, peakons are stable and follow well-defined dynamics. With the decreasing values of $\beta$, the amplitude of the obtained peakon soliton becomes augmented. As the velocity of the peakon soliton is proportional to the amplitude of the wave, the decreased value of $\beta$ provides an increased velocity.

Figure 3. Diagrams of modulus of solution $p_{25}\left(x,t\right)$ for chosen parameters $\lambda =-6.97$, $\beta =0.99$, $k=0.8$, $C_1=16.05$, and $C_2=13.4$.

The modulus plot of solution $p_{12}\left(x,t\right)$ represents a periodic soliton for the selected values, $\lambda =0.1$, $\beta =0.99$, $k=-6.95$, $\mu =6.35$, $C_1=17.7$, and $C_2=-20$, as shown in Figure 4. A periodic soliton oscillates in space and time during its propagation through a medium. It is a non-topological soliton and oscillates with inviolate velocity and amplitude over large regions. Its high stability and long life span make it absolutely applicable in transmitting data through nonlinear media. As depicted in Figure 4b, the wavelength of the periodic solutions spreads out for decreasing fractional parameters. Therefore, the fractional derivative has a great influence on the governing Tzitzéica model and its solutions.

Figure 4. Diagrams of modulus of solution $p_{12}\left(x,t\right)$ for chosen parameters $\lambda =0.1$, $\beta =0.99$, $k=-6.95$, $\mu =6.35$, $C_1=17.7$, and $C_2=-20$.

The wave solutions obtained from the fractional Tzitzéica equation correspond to physically observable phenomena such as solitons, shock waves, and compactons arising in various nonlinear and dispersive media. These solutions capture essential features of real-world wave behavior, including localization, amplitude preservation, memory, and long-range interaction effects influenced by spatial heterogeneity. For example, in optical fibers, similar pulse-like structures are observed during nonlinear light propagation, while in viscoelastic materials and fluid flows, stress waves and localized deformations resemble theoretical wave profiles. By aligning these solutions with measurable quantities such as wave speed, amplitude, and stability, the model provides meaningful insights into experimental observations, thereby further strengthening its physical relevance and practical applicability.

5. Comparison

In this section, we analyze the novelty of the obtained solutions through a systematic comparison test with the results of the previous literature. Previously, several researchers have established a good number of noteworthy findings in their studies to the Tzitzéica equation through numerous approaches. For example, Abazari [46] employed the $\left(G^{\prime }/G\right)$-expansion method, Chou et al. [47] applied the Sardar sub-equation technique, Ghanbari et al. [49] used the generalized exponential rational function method, Zhang and Li [50] employed the polynomial complete discrimination method, Manafian and Lakestani [51] adopted the improved $\tan \left(\varphi \left(\xi \right)/2\right)$-expansion method, and so on. In our recent study, we adopted the $\left(G^{\prime }/G, 1/G\right)$-expansion method to delve into the novel soliton dynamics for the fractional Tzitzéica equation, which has not yet been applied in any of the previous literature. The following tables contain the solutions that are identical with the results of previous studies.

From Table 1 and Table 2, it is notably observed that some of our derived solutions are identical with the prior results published in the previous articles. The rest of the solutions of the present study are novel and might have potential in numerical simulations and further theoretical investigations. The novel solutions might open new pathways to provide deep insight into the nonlinear phenomena arising in nonlinear optics and quantum field theory.

Table 1. Comparison of present results with results of [46].

The Solution Obtained in This Study	Solutions of [46]
Solution $p_{23}\left(x,t\right)$ is $p_{23}\left(x,t\right)=\ln \left({\displaystyle \frac{1}{4}}\left(1\pm \sqrt{-3}\right)-{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right){\left(f_4\left(\eta \right)\right)}^2}{4{\left(f_3\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}\right)$, with $f_3\left(\eta \right)={\displaystyle \frac{\mu }{\lambda }}+\sinh \left(\eta \sqrt{-\lambda }\right)C_1+\cosh \left(\eta \sqrt{-\lambda }\right)C_2$, and $f_4\left(\eta \right)=\cosh \left(\eta \sqrt{-\lambda }\right)C_1+\sinh \left(\eta \sqrt{-\lambda }\right)C_2$.	Solution $U_1\left(x,t\right)$ of this article is $u_1\left(x,t\right)=\ln \left({\displaystyle \frac{1}{4}}\left(1\pm \sqrt{-3}\right)+{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right){\left(g_4\left(\xi \right)\right)}^2}{4{\left(g_3\left(\xi \right)\right)}^2}}\right)$, where $g_3\left(\xi \right)=C_2\sinh \left(\sqrt{-4\mu }\xi \right)+C_1\cosh \left(\sqrt{-4\mu }\xi \right)$, and $g_4\left(\xi \right)=C_1\sinh \left(\sqrt{-4\mu }\xi \right)+C_2\cosh \left(\sqrt{-4\mu }\xi \right)$.
Solution $p_{13}\left(x,t\right)$ is $p_{13}\left(x,t\right)=\ln \left({\displaystyle \frac{1}{4}}\left(1\pm \sqrt{-3}\right)+{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right){\left({\mathrm{f}}_2\left(\eta \right)\right)}^2}{4{\left(f_1\left(\eta \right)-\frac{\mu }{\lambda }\right)}^2}}\right)$, where $f_1\left(x,t\right)={\displaystyle \frac{\mu }{\lambda }}+\sin \left(\eta \sqrt{\lambda }\right)C_1+\cos \left(\eta \sqrt{\lambda }\right)C_2$, and $f_2\left(\eta \right)=\cos \left(\eta \sqrt{\lambda }\right)C_1-\sin \left(\eta \sqrt{\lambda }\right)C_2$.	Solution $U_2\left(x,t\right)$ of this article is $u_2\left(x,t\right)=\ln \left({\displaystyle \frac{1}{4}}\left(1\pm \sqrt{-3}\right)+{\displaystyle \frac{3\left(1\pm \sqrt{-3}\right){\left({\mathrm{g}}_1\left(\xi \right)\right)}^2}{4{\left(g_1\left(\xi \right)\right)}^2}}\right)$, Here, $g_1\left(\xi \right)=C_2\sin \left(\sqrt{4\mu }\xi \right)+C_1\cos \left(\sqrt{4\mu }\xi \right)$, and $g_2\left(\xi \right)=-C_1\sin \left(\sqrt{4\mu }\xi \right)+C_2\cos \left(\sqrt{4\mu }\xi \right)$.

Table 2. Comparison of present results with results of [47].

The Solution Obtained in This Study	Solutions of [47]
$\mathrm{If} \mathrm{we} \mathrm{let} C_1=0$, solution $p_{26}\left(x,t\right)$ becomes $p_{26}\left(x,t\right)=\ln \left(-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{3}{2}}{\tanh }^2\left(\eta \sqrt{-\lambda }\right)\right)$.	Solution ${\mu }_{1,5}\left(x,t\right)$ of this paper is ${\mu }_{1,5}\left(x,t\right)=\ln \left({\mathsf{\omega }}_0-b{\tanh }^2\left({\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{b}{{\omega }_2}}}\chi \right)\right)$, where $b=\left(4{\omega }_0^3-1\right)/4{\omega }_0^2$.
If $\mathrm{we} \mathrm{let} C_2=0$, solution $p_{26}\left(x,t\right)$ becomes ${\mathrm{p}}_{26}\left(\mathrm{x},\mathrm{t}\right)=\ln \left(-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{3}{2}}{\coth }^2\left(\eta \sqrt{-\lambda }\right)\right)$.	Solution ${\mu }_{1,6}\left(x,t\right)$ of this study is ${\mu }_{1,5}\left(x,t\right)=\ln \left({\mathsf{\omega }}_0-b{\coth }^2\left({\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{b}{{\omega }_2}}}\chi \right)\right)$, where $b=\left(4{\omega }_0^3-1\right)/4{\omega }_0^2$.
For $C_1=0$, solution $p_{16}\left(x,t\right)$ can be expressed as $p_{16}\left(x,t\right)=\ln \left(-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{3}{2}}{\tan }^2\left(\eta \sqrt{\lambda }\right)\right)$.	Solution ${\mu }_{1,9}\left(x,t\right)$ of this article is ${\mu }_{1,9}\left(x,t\right)=\ln \left({\mathsf{\omega }}_0+b{\tan }^2\left({\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{b}{{\omega }_2}}}\chi \right)\right)$.
For $C_2=0$, solution $p_{16}\left(x,t\right)$ can be expressed as $p_{16}\left(x,t\right)=\ln \left(-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{3}{2}}{\cot }^2\left(\eta \sqrt{\lambda }\right)\right)$.	Solution ${\mu }_{1,10}\left(x,t\right)$ of this article is ${\mu }_{1,10}\left(x,t\right)=\ln \left({\mathsf{\omega }}_0+b{\cot }^2\left({\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{b}{{\omega }_2}}}\chi \right)\right)$.

6. Conclusions

In the present study, we attain assorted exact soliton solutions to the fractional Tzitzéica model, including breather, periodic, multi-peakon, and W-shaped solitons, through the use of the $\left(G^{\prime }/G, 1/G\right)$-expansion method. The obtained solutions, which include hyperbolic, trigonometric, and rational function forms, provide a deeper understanding of the wave dynamics described by the fractional Tzitzéica equation. We depict the 3D, 2D, and contour plots of the obtained solutions to demonstrate their physical topography. The 2D graphs present the impact of the fractional derivative on the obtained solutions. The results demonstrate the effectiveness and reliability of the $\left(G^{\prime }/G, 1/G\right)$-expansion method in handling nonlinear fractional differential equations. Given the importance of the Tzitzéica equation in mathematical physics and geometry, these solutions can be useful in exploring physical phenomena such as wave propagation in complex media. Moreover, the derived solutions can serve as benchmarks for numerical simulations and further theoretical investigations.