Optical Multi-Peakon Dynamics in the Fractional Cubic–Quintic Nonlinear Pulse Propagation Model Using a Novel Integral Approach

Abstract

This study examines the soliton dynamics in the time-fractional cubic–quintic nonlinear non-paraxial propagation model, applicable to optical signal processing, nonlinear optics, fiber-optic communication, and biomedical laser–tissue interactions. The fractional framework exhibits a wide range of nonlinear effects, such as self-phase modulation, wave mixing, and self-focusing, arising from the balance between cubic and quintic nonlinearities. By employing the Multivariate Generalized Exponential Rational Integral Function (MGERIF) method, we derive an extensive catalog of analytic solutions, multi-peakon structures, lump solitons, kinks, and bright and dark solitary waves, while periodic and singular solutions emerge as special cases. These outcomes are systematically constructed within a single framework and visualized through 2D, 3D, and contour plots under both anomalous and normal dispersion regimes. The analysis also addresses modulation instability (MI), interpreted as a sideband amplification of continuous-wave backgrounds that generates pulse trains and breather-type structures. Our results demonstrate that cubic–quintic contributions substantially affect MI gain spectrum, broadening instability bands and permitting MI beyond the anomalous-dispersion regime. These findings directly connect the obtained solution classes to experimentally observed routes for solitary wave shaping, pulse propagation, and instability and instability-driven waveform formation in optical communication devices, photonic platforms, and laser technologies.

1. Introduction

In mathematical physics, nonlinear differential equations are indispensable for formulating models that describe the complexity of physical phenomena across various domains, including signal processing, communication systems, plasma physics, and fluid dynamics. Among such equations, the nonlinear Schrodinger equation (NLSE) serves as a central model for studying propagation in plasma waves, optical fibers, and quantum mechanical systems. The advent of optical fiber communication has revolutionized information transmission by dramatically increasing data transfer capacities and reshaping the landscape of modern telecommunication systems. A key element in advancing this technology is the ability to understand the propagation of a light pulse through fiber optics. Over the past few years, considerable progress has been made in designing propagation models that enhanced both efficiency and reliability of the optical fiber communication systems. Despite these advances, the standard NLSE does not account for various nonlinear complex phenomena that arise during pulse transmission in fiber optics. To overcome such limitations, numerous generalized and extended versions of the NLSE have been proposed, enriching the model with the terms that describe the higher-order dispersion, self-phase modulation, self-steepening, Raman scattering, and other nonlinear effects. By balancing the highest-order linear and nonlinear terms, these models admit solitary wave solutions—remarkable solitary waves that can preserve shape and velocity even after collisions. Solitons serve as an integral part in modern communication systems, enabling the long-distance transmission of data with minimal distortion, preserving pulse energy, amplitude, and velocity. The concept of solitary waves, first observed by Russell in 1845, was later given a theoretical foundation by Zabusky and Kruskal (1965) through the Korteweg–de Vries (KdV) equation. The remarkable advances in soliton dynamics have inspired researchers to explore a wide range of numerical and analytical models for studying soliton solutions in nonlinear models. A wide range of analytical and numerical methods has been introduced to explore nonlinear evolution equations (NLEEs), including the Kumar–Malik method [1], the $\varphi$6-model expansion [2], auxiliary equation methods [3], fractional variational iteration [4], Lie symmetry analysis [5], tanh–coth formulations [6], the generalized Arnous method [7], the new extended direct algebraic method [8], various (G’/G)-type expansion methods [9], the extended F-expansion method [10], and modified sub-equation method [11].

1.1. Formulation of Model

The present study investigates the pulse transmission in optical fiber within the framework of the nonlinear-paraxial pulse propagation (NPPP) model. The model is formulated through the following equation [12]:

$$i{\mathfrak{L}}_x+a{\mathfrak{L}}_{xx}+\left({\displaystyle \frac{b}{2}}\right){\mathfrak{L}}_{tt}+{K\left(\right|\mathfrak{L}|}^2)\mathfrak{L}=0.$$

(1)

Here, $\mathfrak{L}(x,t)$ is a complex wave function that describes the amplitude as well as the phase of the light pulse. In Equation (1), the first term $i{\mathfrak{L}}_x$ represents the longitudinal propagation of the light pulse $\mathfrak{L}(x,t)$ along the x-coordinate, the second term $a{\mathfrak{L}}_{xx}$ governs the paraxial spreading in the transverse direction and the parameter a control the strength of this effect, the third term $\left({\displaystyle \frac{b}{2}}\right){\mathfrak{L}}_{tt}$ describes the group-velocity dispersion (GVD) in the retarded time coordinate t and the constant b sets the dispersion either normal ($b=-1$) or anomalous ($b=1$), and nonlinear term $K\left(\right|\mathfrak{L}{|}^2)\mathfrak{L}$ characterizes the medium response, which may represent Kerr-type, cubic–quintic, or more general nonlinearities depending on the form of $K(\cdot )$. This equation provides a general model for analyzing soliton dynamics, pulse stability, and nonlinear wave interaction in optical media. By inserting

$$K\left(\xi \right)=c\mathfrak{L}+{\mathfrak{L}}^2.$$

(2)

Equation (1) converts into the the cubic–quintic non-paraxial pulse propagation model [13] as follows:

$$i{\mathfrak{L}}_x+a{\mathfrak{L}}_{xx}+\left({\displaystyle \frac{b}{2}}\right){\mathfrak{L}}_{tt}+\left({c\left|\mathfrak{L}\right|}^2+\sigma {\left|\mathfrak{L}\right|}^4\right)\mathfrak{L}=0.$$

(3)

In the above model, the cubic nonlinear ${c\left|\mathfrak{L}\right|}^2\mathfrak{L}$ accounts for self-phase modulation arising from Kerr law nonlinearity. The quintic term ${\sigma \left|\mathfrak{L}\right|}^2\mathfrak{L}$ represents a high-order nonlinear interaction that becomes significant at high pulse intensities. For $\sigma =0$, Equation (3) reduces to the cubic NPPP equation, while for $c=0$, it represents the quintic NPPP model. This study investigates the $\beta$-time fractional cubic–quintic nonlinear Schrodinger equation, which means a particular extension of the NLSE incorporating both Kerr-type and quintic effects. The mathematical form of the model is stated as follows:

$$i{\mathfrak{L}}_x+a{\mathfrak{L}}_{xx}+\left({\displaystyle \frac{b}{2}}\right)D_{tt}^{\beta }\mathfrak{L}+\left({c\left|\mathfrak{L}\right|}^2+\sigma {\left|\mathfrak{L}\right|}^4\right)\mathfrak{L}=0,0<\beta \le 1.$$

(4)

In this formulation, the third term represents a time-fractional derivative, where $\beta$ indicates the fractional order. The Kerr effect generates cubic nonlinearity, and depending on the sign of the coefficients, the resulting self-focusing can strengthen or weaken the effect. Quintic nonlinearities, serving as higher-order corrections, represent the intricate interaction processes between light and the medium. The model makes a substantial contribution to the study of physical processes, including quantum material dynamics, plasma dynamics, light transmission in optical fibers, and spectral behavior, while also being applicable in quantum mechanics. It is particularly relevant for assessing the interaction of strong laser pulses with optical media, enabling controlled manipulation of such interactions for the practical applications in nonlinear optics, including ultra-short pulse lasers, super-continuum sources, and high-power optical systems. In communication systems affected by the Kerr-type and high-order nonlinearities, as well as in signal processing across multiple fields, researchers continue to propose and investigate new mathematical formulations to provide effective solutions.

1.2. Literature Review

Several studies have been carried out on this equation and its variants, commonly referred to as the cubic–quintic nonlinear Helmholtz (CQ-HE) equation. Tariq et al. [13] employed the extended trial function method to analyze Equation (3) and obtained bright, dark, and optical soliton solutions. Rizi et al. [14] analyzed the chirped dynamics for Equation (3) using the Jacobian elliptic function and derived various solitary wave solutions. Khatar [15] examined the non-paraxial pulse propagation in a planar waveguide with Kerr-like and quintic nonlinearities and employed computational simulations using the Khatar II and generalized exponential methods. Khatar [16] carried out computational simulations of the cubic–quintic nonlinear Helmholtz model and presented numerical insight into the dynamics of optical pulse propagation. Zhong et al. [17] examined the evolution of finite-energy Airy pulses and the associated soliton generation in optical fibers with cubic–quintic nonlinearity using analytical modeling and numerical simulations. Jlali et al. [12] applied the complete discrimination system of the polynomial method (CDSPM) to investigate Equation (4), and obtained analytical soliton and quasi-periodic solutions along with sensitivity and chaos analyses. Akbar et al. [18] utilized the $(G^{\prime }/G,1/G)$-expansion method to derive various solitary wave solutions for Equation (4), analyzing the impact of fractional derivatives on soliton dynamics. Rizvi et al. [19] used the sub-ODE method to derive a wide range of soliton solutions for Equation (4), highlighting the effects of the impact of fractional derivatives on soliton dynamics. In addition to these earlier contributions, more recent studies have continued to advance the field of fractional solitons and peakon dynamics. Kaya Sağlam [20] investigated soliton solutions of the nonlinear time-fractional Schrödinger model using M-truncated and Atangana–Baleanu fractional operators. Iqbal et al. [21] reported new exact soliton solutions for nonlinear time-fractional PDEs based on the conformable fractional derivative, while Yasmin et al. [22] derived analytical soliton-like structures for a (2+1)-dimensional fractional Nizhnik–Novikov–Veselov system. In the context of peakon dynamics, Iqbal et al. [23] explored soliton molecules and solitary wave structures in the nonlinear damped KdV equation. Furthermore, Mayteevarunyoo and Malomed [24] analyzed the motion and stability of two-dimensional fractional cubic–quintic solitons, including vortex states. More recently, Murad et al. [25] investigated optical soliton structures in the nonlinear conformable Schrödinger equation with quadratic–cubic nonlinearity, extending the applicability of fractional calculus to hybrid nonlinear models. Collectively, these recent contributions underline the continuing interest in fractional soliton models and provide complementary insights to the present study.

1.3. Research Aim and Objectives

The objective of this study is to advance the understanding of light pulse dynamics within nonlinear optical systems, focusing on their interaction with the medium and the role of the fractional derivative. The investigation seeks to clarify the governing principle of such behavior while situating fractional calculus as an essential framework for its characterization. Analytical solutions of the model given in Equation (4) are obtained through the MGERIF method, a method widely regarded for its effectiveness in addressing nonlinear problems and leading to the formation of localized structures such as solitons, rogue, and lump waves. The research further addresses modulation instability, analyzing how small perturbations in continuous wave backgrounds can grow and lead to the formation of localized structures. Applicability and limitations of the method are discussed, highlighting the need for appropriate boundary conditions to produce solutions that correspond to real physical behavior in optical fibers and other nonlinear systems.

1.4. Contribution and Originality

This research introduces a novel analysis of the conformable cubic–quintic nonlinear Schrodinger-type model, with particular attention to the dynamics of multi-peakon solutions in fractional-order optical systems. By employing conformable fractional derivative, the study establishes a balance between mathematical simplicity and physical applicability, enabling a framework for the modeling of nonlinear optical pulse transmission. Unlike earlier research, limited to integer-order models or narrow solitary wave categories, the present work derives an extensive class of solutions, including multi-peakon structures, lump solitons, and kink-type solutions. Through the adoption of the MGERIF method, the research extends the solution space, allowing rational, trigonometric, exponential, and hyperbolic solution forms to be represented within a unified analytical framework. In addition to mathematical contributions, the study emphasizes physical applicability by evaluating their amplitude, localization, and stability in relation to ultrashort optical pulse transmission, modulation instability, and nonlinear interactions. These results advance the theoretical understandings of soliton behavior in fractional-order systems and provide new opportunities for promising applications in high-power optical communication, quantum material science, and plasma physics.

1.5. Advantages of MGERIF Method

The MGERIF method introduces key innovations beyond GERFM by incorporating integral operators into its solution structure. It enables the systematic generation of rational terms like csc, sech², and csch that are unobtainable with GERFM’s purely algebraic method. MGERIF can solve more complex physical problems, such as Equation (4), with both dispersion and dissipation terms, and capture advanced wave phenomena like singular solitons, breathers, and lump interactions. While GERFM solutions emerge as special cases when MGERIF’s integral terms are simplified, the full method produces strictly broader solutions that combine polynomial, trigonometric, hyperbolic, exponential, and rational functions in a unified framework. Unlike the Hirota bilinear method, Darboux transformation, or Bell-polynomial methods, which typically require integrable structures, Lax pairs, or specific bilinear forms, the MGERIF method does not rely on the equation being integrable. The MGERIF method is capable of solving these ODEs even when they include mixed polynomial, trigonometric, and rational structures.

1.6. Limitations of MGERIF Method

The MGERIF method offers an effective method for solving FNLEEs, but it exhibits significant drawbacks. Its framework relies heavily on a strict balance principle that involves irregular structures or non-polynomial nonlinear terms. Moreover, the technique tends to generate redundant or physically irrelevant solutions, which require additional scrutiny to confirm validity. In multi-dimensional contexts, the method produces highly complex algebraic systems, whose resolution typically requires symbolic computations. For these reasons, the method remains limited in scope and cannot be regarded as universally applicable. In higher-dimensional or strongly nonlinear cases, the MGERIF method often produces very large algebraic systems and may produce redundant or physically irrelevant solutions, which reduces its efficiency and restricts its applicability beyond one-dimensional or moderately nonlinear models.

1.7. Summarization

The comprehensive document is summarized as follows. Section 2 discusses the modeling of $\beta$-fractional derivative. Section 3 examines the mathematical methodology used to construct NLPDE into an ODE. Section 4 details the mathematical steps of the MGRIF method and constructs the solitary wave solutions. Section 5 discusses the framework of modulation instability analysis. Section 6 describes the graphical description of the obtained solution. After that, in Section 7, a comparison with previous literature is presented, and in the Nomenclature part, the nomenclature is discussed. Finally, Section 8 concludes the paper.

2. Fractional-Order Calculus

Fractional calculus constitutes a fundamental advancement in mathematical physics, furnishing a generalized framework for analyzing nonlinear models. In particular, the formulation of fractional nonlinear evolution equations (FNLEEs) has significantly broadened the scope of research on soliton dynamics. In contrast to integer-order models, FNLEEs admit derivatives of arbitrary real or complex order, thereby increasing flexibility in describing diverse physical phenomena. Several rigorous definitions of fractional differentiation have been introduced in the literature, including the Riemann–Liouville operator [26], the modified R–L operator [27], the Caputo operator [28], the M-Truncated derivative [29], and the $\beta$-operator [30]. The present work adopts the $\beta$-fractional derivative, whose definition is given as follows.

Definition 1.

Definition of β-derivative: Suppose $f:[0,1]\to \mathbb{R}$, then the β-derivative of $f(x,t)$ of order β is defined as

$$D_t^{\beta }f(x,t)=\underset{h\to 0}{lim}{\displaystyle \frac{f\left(x,t+h{\left(t+\frac{1}{\alpha \beta }\right)}^{1-\beta }\right)-f(x,t)}{h}},\beta \in (0,1],t>0.$$

(5)

Here, $f(x,t)$ denotes the function to which the differentiation operator is applied, β represents the order of the fractional derivative, $h=ϵ{\left(t+{\displaystyle \frac{1}{a\beta }}\right)}^{\beta -1}$ and α is the scaling parameter.

Theorem 1.

Suppose $\beta \in (0,1]$; let $f(x,t),g(x,t)$ be β-differentiable at point t. Under these conditions, the following relation holds.

$$D_t^{\beta }(af(x,t)+bg(x,t))=a{D}_t^{\beta }\left(f(x,t)\right)+b{D}_t^{\beta }\left(g(x,t)\right),\mathit{for}a,b\in \mathbb{R},$$

$$D_t^{\beta }\left(c\right)=0,\mathit{for}c\in \mathbb{R},$$

$$D_t^{\beta }\left(f(x,t)g(x,t)\right)=f(x,t)D_t^{\beta }\left(g(x,t)\right)+g(x,t)D_t^{\beta }\left(f(x,t)\right),$$

$$D_t^{\beta }\left({\displaystyle \frac{f(x,t)}{g(x,t)}}\right)={\displaystyle \frac{f(x,t)D_t^{\beta }\left(g(x,t)\right)-g(x,t)D_t^{\beta }\left(f(x,t)\right)}{g^2\left(t\right)}},$$

$$D_t^{\beta }f(x,t)={\left(t+{\displaystyle \frac{1}{\alpha \left(\beta \right)}}\right)}^{1-\beta }{\displaystyle \frac{df(x,t)}{dt}}.$$

$$D_t^{\beta }\left({\displaystyle \frac{1}{f(x,t)}}\right)={\displaystyle \frac{-D_t^{\beta }f(x,t)}{{\left(f(x,t)\right)}^2}}.$$

Theorem 2.

Suppose $f(x,t):[0,1]\to \mathbb{R}$ is a function possessing both classical differentiability and β-differentiability. Let $g(x,t)$ be a differentiable function defined on the range of $f(x,t)$. Under these conditions, the following relation holds.

$$D_t^{\beta }(f(x,t)◦g(x,t))={\left(t+{\displaystyle \frac{1}{\alpha \beta }}\right)}^{1-\beta }{g}^{\prime }(x,t)f^{\prime }(x,t)\left(g(x,t)\right).$$

The recently introduced $\beta$-fractional derivative is the only formulation that satisfies the complete set of axioms of the integer-order derivative. Other commonly used definitions, however, present challenges when applied in mathematical modeling. In the case of the conformable derivative, the differential coefficients become zero at the origin. Moreover, the ability of the Riemann–Liouville derivative incorrectly assigns a nonzero value to the derivative of constants. Given that the $beta$-fractional derivative consistently preserves the properties of integer-order calculus, it is adopted in the present study as the governing operator in the fractional model formulation.

3. Mathematical Investigation

Consider an NLEE defined in terms of the wave function $\mathfrak{L}=\mathfrak{L}(x,t)$, along with its differential coefficients with respect to the spatial variable x and the temporal variable t. The governing equation is expressed as follows:

$$U\left(\mathfrak{L},{\mathfrak{L}}_x,{\mathfrak{L}}_t,{\mathfrak{L}}_{xx},{\mathfrak{L}}_{tx},\dots ,D_t^{\beta }\mathfrak{L},D_x^{\beta }\mathfrak{L},\dots \right)=0,0<\beta \le 1.$$

(6)

Here, $U\left(\mathfrak{L}\right(x,t\left)\right)$ denotes the polynomial form of $\mathfrak{L}(x,t)$.

The reduction of Equation (7) to an ordinary differential equation (ODE) is a prerequisite for the application of analytical methods. In the present work, $\beta$-wave transformation is employed to obtain the governing differential equation corresponding to the given model. The $\beta$-wave transformation is formulated as:

$$\mathfrak{L}(x,t)=\mathfrak{R}\left(\xi \right)e^{i\mathsf{\Phi }},$$

(7)

together with, $\xi =\pm {\displaystyle \frac{k}{\beta }}{\left(x+{\displaystyle \frac{1}{a^{\beta }}}\right)}^{\beta }\pm {\displaystyle \frac{d}{\beta }}{\left(t+{\displaystyle \frac{1}{a^{\beta }}}\right)}^{\beta },\mathsf{\Phi }=\pm {\displaystyle \frac{\omega }{\beta }}{\left(x+{\displaystyle \frac{1}{a^{\beta }}}\right)}^{\beta }\pm {\displaystyle \frac{\delta }{\beta }}{\left(t+{\displaystyle \frac{1}{a^{\beta }}}\right)}^{\beta }.$ In Equation (7), the quantities $\mathfrak{R}\left(\xi \right)$ and $\mathsf{\Phi }(x,t)$ correspond to the wave amplitude and phase, respectively, whereas $\delta$ and $\omega$ signify the velocity and wave number of the newly introduced wave function. Upon employing the transformation (7), the associated governing differential equation is derived in the following form:

$$F\left(\mathfrak{R},{\mathfrak{R}}^{\prime },{\mathfrak{R}}^{″},\dots \right)=0,$$

(8)

In this context, the prime symbol is employed to represent the ordinary differentiation of the function $\mathfrak{R}\left(\xi \right)$ with respect to the new variable $\xi$.

Applying the $\beta$-wave transformation (7) to the fractional model yields the following reduced equations:

$$\begin{array}{cc}\hfill & c{\mathfrak{R}}^3\left(\xi \right)+\sigma {\mathfrak{R}}^5\left(\xi \right)-\frac{1}{2}\mathfrak{R}\left(\xi \right)\left(b{\delta }^2+2\omega (1+a\omega )\right)+i\left(1+bd\delta +2a\omega \right){\mathfrak{R}}^{\prime }\left(\xi \right)\hfill \\ & +\frac{1}{2}(2a+b{d}^2){\mathfrak{R}}^{″}\left(\xi \right)=0.\hfill \end{array}$$

(9)

Equation (9) represents the reduced nonlinear system after applying the $\beta$-wave transformation. The cubic and quintic terms express nonlinear self-phase modulation, while the derivative terms depict dispersive effects and their balance with the wave velocity and frequency parameters. Here $\sigma ,a,b,c$ are arbitrary constants of the system. $\delta ,\omega$ correspond to frequency and velocity parameters.

Here,

$$\mathfrak{L}(x,t)=\mathfrak{R}\left(\xi \right)e^{i\mathsf{\Phi }(x,t)},$$

(10)

along with

$$\xi =x-{\displaystyle \frac{d}{\beta }}{\left(t+{\displaystyle \frac{1}{a\beta }}\right)}^{\beta },\mathsf{\Phi }(x,t)=\omega x-{\displaystyle \frac{\delta }{\beta }}{\left(t+{\displaystyle \frac{1}{a\beta }}\right)}^{\beta }.$$

(11)

Here, $\mathfrak{R}\left(\xi \right)$ defines the amplitude and $\mathsf{\Phi }\left(\xi \right)$ the phase. The parameters $\delta ,\omega$, and d correspond to frequency, wave number, and velocity of the transformed wave, respectively. Substituting this form into Equation (9) and separating real and imaginary parts gives the following reduced system:

$$c{\mathfrak{R}}^3\left(\xi \right)+\sigma {\mathfrak{R}}^5\left(\xi \right)-\frac{1}{2}\mathfrak{R}\left(\xi \right)\left(b{\delta }^2+2\omega (1+a\omega )\right)+\frac{1}{2}(2a+b{d}^2){\mathfrak{R}}^{″}\left(\xi \right)=0,$$

(12)

$$\left(1+bd\delta +2a\omega \right){\mathfrak{R}}^{\prime }\left(\xi \right)=0.$$

(13)

Equations (12) and (13) represent the real and imaginary parts of the reduced system, respectively. Equation (12) characterizes the role of cubic–quintic nonlinearities and second-order dispersion in shaping the amplitude profile, while Equation (13) represents the wave frequency parameter $\omega$ through the balance condition. Extract the value of $\omega$ from Equation (13):

$$\omega ={\displaystyle \frac{(-1-b\delta d)}{2a}}.$$

(14)

Inserting the value of $\omega$ from Equation (14) into Equation (12) and applying the homogeneous balance principle yields N = ${\displaystyle \frac{1}{2}}$. Consequently, the wave function is reformulated as:

$$\mathfrak{R}\left(\xi \right)=\sqrt{\mathfrak{B}\left(\xi \right)}.$$

(15)

Substituting $\mathfrak{R}\left(\xi \right)$ back into the Equation (12), the following equation is obtained:

$$\begin{array}{cc}\hfill & 2\left(1-b\left(2a+b{d}^2\right){\delta }^2\right){\mathfrak{B}}^2\left(\xi \right)+8ac{\mathfrak{B}}^3\left(\xi \right)+8a\sigma {\mathfrak{B}}^4\left(\xi \right)-a\left(2a+b{d}^2{\delta }^2\right){\left({\mathfrak{B}}^{\prime }\left(\xi \right)\right)}^2\hfill \\ & +2a\left(2a+b{d}^2{\delta }^2\right)\mathfrak{B}\left(\xi \right){\mathfrak{B}}^{″}\left(\xi \right)=0.\hfill \end{array}$$

(16)

Equation (16) is a higher-order NODE for the amplitude function $\mathfrak{B}\left(\xi \right)$. The first three terms account for cubic, quintic, and dispersive effects, while the remaining terms describe nonlinear interactions involving derivatives of $\mathfrak{B}\left(\xi \right)$.

4. Multivariate Generalized Exponential Rational Integral Function Method

A new and efficient method, the multivariate generalized exponential rational integral function method, is introduced in this section. The MGERIF [31] method stands out for its superpower to offer distinctive and brilliant solutions to nonlinear partial differential equations (NLPDEs). This method’s foundation is the generalized exponential rational function [32].

The MGERIF method proceeds through the following steps:

• Step 01: The solution of Equation (16) is derived through the MGERIF as follows:

  $$\mathfrak{B}\left(\xi \right)={\lambda }_0+\sum _{i=1}^N{\lambda }_i{\left(\underset{i}{\underbrace{\int \dots \int }}K\left(\xi \right)d\xi d\xi \dots d\xi \right)}^i+\sum _{i=1}^N{\gamma }_j{\left(\underset{i}{\underbrace{\int \dots \int }}K\left(\xi \right)d\xi d\xi \dots d\xi \right)}^{-i}.$$

  (17)

  Here, ${\lambda }_0$, ${\gamma }_i$, ${\lambda }_i$ (with i = 1, 2, …, N) are arbitrary constants, and the function $K\left(\xi \right)$ satisfies the following relation:

  $$K\left(\xi \right)={\displaystyle \frac{u_1{e}^{v_1\xi }+u_2{e}^{v_2\xi }}{u_3{e}^{v_3\xi }+u_4{e}^{v_4\xi }}}.$$

  (18)

  Here, $u_i,v_i$ (for i = 1, 2, 3, 4) are parameters. Setting their values allows Equation (18) to be rewritten in familiar representations in

  Table 1. Special cases of $K\left(\xi \right)$ under specific parameter values.

  $[{\mathit{u}}_1,{\mathit{u}}_2,{\mathit{u}}_3,{\mathit{u}}_3]$	$[{\mathit{v}}_1,{\mathit{v}}_2,{\mathit{v}}_3,{\mathit{v}}_3]$	$\mathit{K}\left(\mathit{\xi }\right)$
  $[1,-1,i,i]$	$[i,-i,0,0]$	$sin\left(\xi \right)$
  $[1,1,1,1]$	$[i,-i,0,0]$	$cos\left(\xi \right)$
  $[2,2,2,2]$	$[{\displaystyle \frac{2}{5}},{\displaystyle \frac{2}{5}},0,0]$	$e^{{\displaystyle \frac{2\xi }{5}}}$
  $[i,i,i,i]$	$[1,-1,0,0]$	$cosh\left(\xi \right)$
  $[2i,-2i,4i,4i]$	$[{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}},0,0]$	${\displaystyle \frac{1}{2}}sinh\left({\displaystyle \frac{\xi }{2}}\right)$

  .
• Step 02: The homogeneous balance principle yields the positive integer N by equating the highest-order derivative and nonlinear terms in Equation (16).

  $$D\left[{\displaystyle \frac{d^p\mathfrak{B}\left(xi\right)}{d^p\mathfrak{B}\left(\xi \right)}}\right]=N+p,D\left[{\mathfrak{B}}^p{\left({\displaystyle \frac{d^p\mathfrak{B}\left(\xi \right)}{d^p\mathfrak{B}\left(\xi \right)}}\right)}^s\right]=qN+s(N+P).$$

  (19)
• Step 03: Substituting Equations (17) and (18) in Equation (16) yields the polynomial in term of $e^{mi\xi }$ for $1\le i\le 4$. Equating the coefficients of identical powers equal to zero results in a system of nonlinear algebraic equations. Solving this system, in conjunction with Equation (18), provides the solutions to Equation (4).
• Step 04: Mathematical reductions facilitated by the software Mathematica enable the exact determination of the variables ${\lambda }_0,{\lambda }_i,$ and ${\gamma }_i$$(1\le i\le N)$. Substituting these values into Equations (17,18) then leads to explicit soliton solutions of Equation (4).

4.1. Solutions by MGERIF Method

First, set $N=1$ based on the balancing principle. Then, insert this value into Equation (17) to proceed with solving Equation (16):

$$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\lambda }_1\left(\int K(\xi )d\xi \right)+{\displaystyle \frac{{\gamma }_1}{\int K\left(\xi \right)d\xi }}.$$

(20)

A sequence of solutions for Equation (16) is derived by substituting the constructed expressions, along with Equation (11), using the MGERIF method with the aid of Mathematica.

4.2. The Familiar Sin Description

By setting the values of parameters to $[u_1,u_2,u_3,u_4]=[1,-1,i,i]$ and $[v_1,v_2,v_3,v_4]=[i,-i,0,0]$, Equation (18) transformed into its equivalent standard sine representation:

$$K\left(\xi \right)=sin\left(\xi \right).$$

(21)

The following expression is obtained by substituting Equation (21) into Equation (17):

$$\mathfrak{L}\left(\xi \right)={\lambda }_0-{\lambda }_{1}cos\left(\xi \right)-{\gamma }_{1}sec\left(\xi \right).$$

(22)

• Case 1.1:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1=0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (22).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0-{\gamma }_{1}sec\left(\xi \right).$$

  (23)

  Hence, applying Equation (23) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $${\mathfrak{B}}_1(x,t)={\lambda }_0-{\displaystyle \frac{2{\gamma }_1{e}^{i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}{1+e^{2i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}}.$$

  (24)
• Case 1.2:
  ${\lambda }_0=0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (22).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_1(-cos\left(\xi \right))-{\gamma }_{1}sec\left(\xi \right).$$

  (25)

  Hence, applying Equation (25) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $$\begin{array}{c}\hfill {\mathfrak{B}}_2(x,t)=-{\displaystyle \frac{1}{2}}{\lambda }_1\left(e^{-i\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}+e^{i\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}\right)-{\displaystyle \frac{2{\gamma }_1{e}^{i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}{1+e^{2i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}}.\end{array}$$

  (26)
• Case 1.3:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (22).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0-{\lambda }_{1}cos\left(\xi \right)-{\gamma }_{1}sec\left(\xi \right).$$

  (27)

  Hence, applying Equation (27) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $$\begin{array}{cc}\hfill {\mathfrak{B}}_3(x,t)& =-{\displaystyle \frac{1}{2}}{\lambda }_1\left(e^{-i\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}+e^{i\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}\right)-{\displaystyle \frac{2{\gamma }_1{e}^{i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}{1+e^{2i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}}+{\lambda }_0.\hfill \end{array}$$

  (28)
• Case 1.4:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1=0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (22).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0-{\lambda }_{1}cos\left(\xi \right).$$

  (29)

  Hence, applying Equation (29) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $${\mathfrak{B}}_4(x,t)={\lambda }_0-{\displaystyle \frac{1}{2}}{\lambda }_1\left(e^{-i\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}+e^{i\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}\right).$$

  (30)

4.3. The Familiar Cosine Description

By setting the values of parameters to $[u_1,u_2,u_3,u_4]=[1,1,1,1]$ and $[v_1,v_2,v_3,v_4]=[i,-i,0,0]$, Equation (18) transforms into its equivalent standard cosine representation:

$$K\left(\xi \right)=cos\left(\xi \right).$$

(31)

The following expression is obtained by substituting Equation (31) into Equation (17):

$$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\lambda }_{1}sin\left(\xi \right)+{\gamma }_{1}csc\left(\xi \right).$$

(32)

• Case 1.1:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1=0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (32).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\gamma }_{1}csc\left(\xi \right).$$

  (33)

  Hence, applying Equation (33) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $${\mathfrak{B}}_5(x,t)={\lambda }_0+{\displaystyle \frac{2i{\gamma }_1{e}^{i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}{-1+e^{2i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}}.$$

  (34)
• Case 1.2:
  ${\lambda }_0=0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (32).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_{1}sin\left(\xi \right)+{\gamma }_{1}csc\left(\xi \right).$$

  (35)

  Hence, applying Equation (35) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $$\begin{array}{c}\hfill {\mathfrak{B}}_6(x,t)={\displaystyle \frac{i{e}^{-i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}\left({\lambda }_1{\left(-1+e^{2i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}\right)}^2-4{\gamma }_1{e}^{2i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}\right)}{2\left(-1+e^{2i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}\right)}}.\end{array}$$

  (36)
• Case 1.3:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (32).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\lambda }_{1}sin\left(\xi \right)+{\gamma }_{1}csc\left(\xi \right).$$

  (37)

  Hence, applying Equation (37) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $${\mathfrak{B}}_7(x,t)=-{\displaystyle \frac{1}{2}}i{\lambda }_1{e}^{-i\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}\left(-1+e^{2i\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}\right)+{\displaystyle \frac{2i{\gamma }_1{e}^{i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}{-1+e^{2i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}}+{\lambda }_0.$$

  (38)
• Case 1.4:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1=0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (32).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\lambda }_{1}sin\left(\xi \right).$$

  (39)

  Hence, applying Equation (39) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $$\begin{array}{c}\hfill {\mathfrak{B}}_8(x,t)={\lambda }_0-{\displaystyle \frac{1}{2}}i{\lambda }_1{e}^{-i\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}\left(-1+e^{2i\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}\right).\end{array}$$

  (40)

4.4. The Familiar Exponential Description

By setting the values of parameters to $[u_1,u_2,u_3,u_4]=[2,2,2,2]$ and $[v_1,v_2,v_3,v_4]=[2/5,2/5,0,0]$, Equation (18) transforms into its equivalent standard exponential representation:

$$K\left(\xi \right)=e^{{\displaystyle \frac{2\xi }{5}}}.$$

(41)

The following expression is obtained by substituting Equation (41) into Equation (17):

$$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\displaystyle \frac{5}{2}}{e}^{{\displaystyle \frac{2\xi }{5}}}{\lambda }_1+{\displaystyle \frac{2}{5}}{e}^{-{\displaystyle \frac{2\xi }{5}}}{\gamma }_1.$$

(42)

• Case 1.1:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1=0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (42).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\displaystyle \frac{2}{5}}{e}^{-{\displaystyle \frac{2\xi }{5}}}{\gamma }_1.$$

  (43)

  Hence, applying Equation (43) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $${\mathfrak{B}}_9(x,t)={\displaystyle \frac{2}{5}}{\gamma }_1{e}^{{\displaystyle \frac{1}{5}}(-2)\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}+{\lambda }_0.$$

  (44)
• Case 1.2:
  ${\lambda }_0=0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (42).

  $$\mathfrak{L}\left(\xi \right)={\displaystyle \frac{5}{2}}{e}^{{\displaystyle \frac{2\xi }{5}}}{\lambda }_1+{\displaystyle \frac{2}{5}}{e}^{-{\displaystyle \frac{2\xi }{5}}}{\gamma }_1.$$

  (45)

  Hence, applying Equation (45) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $${\mathfrak{B}}_{10}(x,t)={\displaystyle \frac{5}{2}}{\lambda }_1{e}^{{\displaystyle \frac{2}{5}}\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}+{\displaystyle \frac{2}{5}}{\gamma }_1{e}^{{\displaystyle \frac{1}{5}}(-2)\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}.$$

  (46)
• Case 1.3:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (42).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\displaystyle \frac{5}{2}}{e}^{{\displaystyle \frac{2\xi }{5}}}{\lambda }_1+{\displaystyle \frac{2}{5}}{e}^{-{\displaystyle \frac{2\xi }{5}}}{\gamma }_1.$$

  (47)

  Hence, applying Equation (47) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $${\mathfrak{B}}_{11}(x,t)={\displaystyle \frac{5}{2}}{\lambda }_1{e}^{{\displaystyle \frac{2}{5}}\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}+{\displaystyle \frac{2}{5}}{\gamma }_1{e}^{{\displaystyle \frac{1}{5}}(-2)\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}+{\lambda }_0.$$

  (48)
• Case 1.4:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1=0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (42).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\displaystyle \frac{5}{2}}{e}^{{\displaystyle \frac{2\xi }{5}}}{\lambda }_1.$$

  (49)

  Hence, applying Equation (49) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $${\mathfrak{B}}_{12}(x,t)={\displaystyle \frac{5}{2}}{\lambda }_1{e}^{{\displaystyle \frac{2}{5}}\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}+{\lambda }_0.$$

  (50)

4.5. The Familiar Cosine Hyperbolic Description

By setting the values of parameters to $[u_1,u_2,u_3,u_4]=[i,i,i,i]$ and $[v_1,v_2,v_3,v_4]=[1,-1,0,0]$, Equation (18) transforms into its equivalent standard cosine hyperbolic representation:

$$K\left(\xi \right)=cosh\left(\xi \right).$$

(51)

The following expression is obtained by substituting Equation (51) into Equation (17):

$$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\lambda }_{1}sinh\left(\xi \right)+{\gamma }_1\csc \mathrm{h}\left(\xi \right).$$

(52)

• Case 1.1:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1=0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (52).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\gamma }_1\csc \mathrm{h}\left(\xi \right).$$

  (53)

  Hence, applying Equation (53) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $${\mathfrak{B}}_{13}(x,t)={\lambda }_0+{\displaystyle \frac{2i{\gamma }_1{e}^{i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}{-1+e^{2i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}}.$$

  (54)
• Case 1.2:
  ${\lambda }_0=0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (52).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_{1}sinh\left(\xi \right)+{\gamma }_1\csc \mathrm{h}\left(\xi \right).$$

  (55)

  Hence, applying Equation (55) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $$\begin{array}{cc}\hfill {\mathfrak{B}}_{14}(x,t)& ={\displaystyle \frac{1}{2}}{\lambda }_1\left(e^{x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}}-e^{{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}-x}\right)+{\displaystyle \frac{2i{\gamma }_1{e}^{i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}{-1+e^{2i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}}.\hfill \end{array}$$

  (56)
• Case 1.3:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (52).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\lambda }_{1}sinh\left(\xi \right)+{\gamma }_1\csc \mathrm{h}\left(\xi \right).$$

  (57)

  Hence, applying Equation (57) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $$\begin{array}{c}\hfill {\mathfrak{B}}_{15}(x,t)={\displaystyle \frac{1}{2}}{\lambda }_1\left(e^{x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}}-e^{{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}-x}\right)+{\displaystyle \frac{2i{\gamma }_1{e}^{i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}{-1+e^{2i\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}}+{\lambda }_0.\end{array}$$

  (58)
• Case 1.4:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1=0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (52).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\lambda }_{1}sinh\left(\xi \right).$$

  (59)

  Hence, applying Equation (59) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $${\mathfrak{B}}_{16}(x,t)={\displaystyle \frac{1}{2}}{\lambda }_1{e}^{{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}-x}\left(e^{2\left(x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}-1\right)+{\lambda }_0.$$

  (60)

4.6. The Familiar Sin Hyperbolic Description

By setting the values of parameters to $[u_1,u_2,u_3,u_4]=[2i,-2i,4i,4i]$ and $[v_1,v_2,v_3,v_4]=[1/2,-1/2,0,0]$, Equation (18) transforms into its equivalent standard sin hyperbolic representation.

$$K\left(\xi \right)={\displaystyle \frac{1}{2}}sinh\left({\displaystyle \frac{\xi }{2}}\right).$$

(61)

The following expression is obtained by substituting Equation (61) into Equation (17):

$$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\lambda }_{1}cosh\left({\displaystyle \frac{\xi }{2}}\right)+{\gamma }_1\sec \mathrm{h}\left({\displaystyle \frac{\xi }{2}}\right).$$

(62)

• Case 1.1:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1=0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (62).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\gamma }_1\sec \mathrm{h}\left({\displaystyle \frac{\xi }{2}}\right).$$

  (63)

  Hence, applying Equation (63) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $${\mathfrak{B}}_{17}(x,t)={\displaystyle \frac{2{\gamma }_1{e}^{\frac{1}{2}\left(x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }\right)}}{e^{x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}+1}}+{\lambda }_0.$$

  (64)
• Case 1.2:
  ${\lambda }_0=0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (62).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_{1}cosh\left({\displaystyle \frac{\xi }{2}}\right)+{\gamma }_1\sec \mathrm{h}\left({\displaystyle \frac{\xi }{2}}\right).$$

  (65)

  Hence, applying Equation (65) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $$\begin{array}{c}\hfill {\mathfrak{B}}_{18}(x,t)={\displaystyle \frac{e^{\frac{1}{2}\left(\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }-x\right)}\left({\lambda }_1{\left(e^{x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}+1\right)}^2+4{\gamma }_1{e}^{x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}{2\left(e^{x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}+1\right)}}.\end{array}$$

  (66)
• Case 1.3:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1\ne 0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (62).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\lambda }_{1}cosh\left({\displaystyle \frac{\xi }{2}}\right)+{\gamma }_{1}sech\left({\displaystyle \frac{\xi }{2}}\right).$$

  (67)

  Hence, applying Equation (67) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $$\begin{array}{c}\hfill {\mathfrak{B}}_{19}(x,t)={\displaystyle \frac{e^{\frac{1}{2}\left(\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }-x\right)}\left({\lambda }_1{\left(e^{x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}+1\right)}^2+4{\gamma }_1{e}^{x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}\right)}{2\left(e^{x-\frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}+1\right)}}+{\lambda }_0.\end{array}$$

  (68)
• Case 1.4:
  ${\lambda }_0\ne 0,a=0,{\lambda }_1\ne 0,\delta =-{\displaystyle \frac{1}{bd}},{\gamma }_1=0.$ The corresponding result of Equation (16) is obtained by incorporating these parameter values into Equation (62).

  $$\mathfrak{L}\left(\xi \right)={\lambda }_0+{\lambda }_{1}cosh\left({\displaystyle \frac{\xi }{2}}\right).$$

  (69)

  Hence, applying Equation (69) in conjunction with Equation (11) enables the determination of the solution to Equation (4).

  $${\mathfrak{B}}_{20}(x,t)={\displaystyle \frac{1}{2}}{\lambda }_1{e}^{{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}-x\right)}\left(e^{x-{\displaystyle \frac{d{\left(\frac{1}{\alpha \beta }+t\right)}^{\beta }}{\beta }}}+1\right)+{\lambda }_0.$$

  (70)

5. Modulation Instability Analysis

Modulation Instability (MI) is a fundamental nonlinear phenomenon observed in various physical systems, most commonly in optics and fluid dynamics. It occurs when a continuous wave (CW) or plane wave becomes unstable due to small perturbations, leading to the exponential growth of these perturbations and the formation of localized structures, such as solitons or pulse trains.

Assume the steady state solution of Equation (3) as [33]

$$\mathfrak{L}(x,t)=e^{ix\eta }\left(\mathfrak{P}(x,t)+\sqrt{\eta }\right),$$

(71)

where $\eta$ is the normalized optical power. Upon putting Equation (71) into Equation (3) and linearizing, we obtain

$$\left({\mathfrak{P}}^{*}+\mathfrak{P}\right)\left(-a{\eta }^2+\sigma {\eta }^2+c\eta -\eta \right)+{\displaystyle \frac{b{\mathfrak{P}}_{\mathrm{tt}}}{2}}+(ia\eta +i)({\mathfrak{P}}_x+{\left(\mathfrak{P}x\right)}^{*})+a{\mathfrak{P}}_{\mathrm{xx}}.$$

(72)

The complex conjugate is represented by ∗. Assume the solution of Equation (72) to be of the form

$$\mathfrak{P}(x,t)=a_{1}exp\left(i(\theta x-\mathsf{\Psi }t)\right)+a_{2}exp(-i(\theta x-\mathsf{\Psi }t)).$$

(73)

where $\theta$ and $\mathsf{\Phi }$ are the normalized wave numbers, while $\omega$ is the frequency of perturbation. Substituting Equation (73) into Equation (72), split the coefficients of $a_1{e}^{i(\theta x-\mathsf{\Psi }t)}$ and $e^{-i(\theta x-\mathsf{\Psi }t)}$ and then solving the coefficients of the resulting determinant matrix, we obtain the following dispersion relation.

$$\begin{array}{cc}\hfill & -{\theta }^4{a}^2-2a^2{\theta }^2{\eta }^2-a{\theta }^{2}b{\mathsf{\Psi }}^2+2a{\theta }^2\sigma {\eta }^2+2a{\theta }^{2}c\eta -2a{\theta }^2\eta -ab{\mathsf{\Psi }}^2{\eta }^2-{\displaystyle \frac{b^2{\mathsf{\Psi }}^4}{4}}\hfill \\ & +b\sigma {\mathsf{\Psi }}^2{\eta }^2+bc{\mathsf{\Psi }}^2\eta -b{\mathsf{\Psi }}^2\eta =0.\hfill \end{array}$$

(74)

Equation (74) represents the dispersion relation obtained from the linear stability of the perturbed solution. Solving Equation (74), which is the dispersion relation for $\omega$, we obtain:

$$\mathsf{\Psi }=\pm {\displaystyle \frac{\sqrt{2}\sqrt{-{\theta }^{2}a-2a{\eta }^2+2\sigma {\eta }^2+2c\eta -2\eta }}{\sqrt{b}}}.$$

(75)

In a scenario where ${\displaystyle \frac{\sqrt{-{\theta }^{2}a-2a{\eta }^2+2\sigma {\eta }^2+2c\eta -2\eta }}{\sqrt{b}}}\ge 0$, $\mathsf{\Psi }$ is real for any real value of $\tau$ and the steady state is stable against small perturbations. In contrast, the constant state is unstable when ${\displaystyle \frac{\sqrt{-{\theta }^{2}a-2a{\eta }^2+2\sigma {\eta }^2+2c\eta -2\eta }}{\sqrt{b}}}<0$, that is, $\mathsf{\Psi }$ is imaginary, and hence, the perturbation grows exponentially, leading to an MI.

Under this condition, the gain spectrum may be given as:

$$G\left(\eta \right)=2Im\left(\mathsf{\Psi }\right)=2Im\left(\pm {\displaystyle \frac{\sqrt{2}\sqrt{-{\theta }^{2}a-2a{\eta }^2+2\sigma {\eta }^2+2c\eta -2\eta }}{\sqrt{b}}}\right).$$

(76)

Equation (76) provides the explicit gain spectrum for modulation instability. The growth rate of perturbations depends on the sign of the radicand, indicating the transition between stable and unstable regimes of continuous-wave propagation.

Stability of the Equation (24) Solution

To address the reviewer’s suggestion, we examine the stability of the localized solution given in Equation (24). As $\left|\xi \right|\to \infty$, this solution tends to the constant background $L\left(\xi \right)\to {\lambda }_0$. Identifying $\eta ={\lambda }_0^2$, the modulation instability (MI) of this background is governed by

$$E\left(\theta \right)=-a{\theta }^2-2a{\eta }^2+2\sigma {\eta }^2+2c\eta -2\eta \sqrt{b},$$

(77)

which follows directly from the dispersion relation in Equation (60). The continuous-wave state is stable for $E\left(\theta \right)\ge 0$ (real perturbation frequency $\mathsf{\Psi }$) and unstable when $E\left(\theta \right)<0$ (complex $\mathsf{\Psi }$, nonzero MI gain). The MI gain is therefore expressed as

$$G(\eta ,\theta )=2\mathrm{Im}\left\{\pm \sqrt{2}\sqrt{E\left(\theta \right)}\right\}.$$

(78)

Using the same parameter set as in the figures ($a=4$, $b=1$, $c=2$, $\sigma =2$) and choosing ${\lambda }_0=\sqrt{5}$ (so that $\eta =5$), Equation (77) reduces to

$$E\left(\theta \right)=-4{\theta }^2-90<0\mathrm{for}\mathrm{all}\mathrm{real}\theta .$$

Hence, the background of Equation (24) is modulation unstable, with nonzero MI gain according to Equation (78), which is consistent with the instability regions shown in Figure 1.

6. Graphical Description

This section explores the graphical analysis of the obtained solutions. To facilitate the visualization, specific numerical values are assigned to the unknown constants. The resulting solutions are illustrated using three-dimensional and contour plots. Subfigure (a) shows the real part of the solution in 3D, while subfigure (b) displays the imaginary part, and (c) represents the value of the solution in 3D. Similarly, (d) illustrates the 2D plot of the real part, subfigure (e) presents the 2D plot of the imaginary part, and (f) presents the 2D plot of the absolute value of the solution. Similarly, (g) illustrates the contour of the real part, subfigure (h) presents the contour of the imaginary part, and (i) presents the contour of the absolute value of the solution. Evaluating the real part of the solutions reveals their behavior, reflecting their directional behavior along the positive axis. The imaginary part reveals the structural pattern along the imaginary axis, while studying the absolute values highlights the areas of low and high intensities. All three together help us to understand how the solutions behave.

Figure 2 illustrates the real, imaginary, and absolute components of the solution (28) through 2D, 3D, and contour plots. Real and imaginary parts represent a lump-type solitary wave corresponding to the parameter values: ${\mathfrak{B}}_{03}(x,t):\alpha =-1,\beta =1,d=1.5I,$ ${\lambda }_0=0$, ${\lambda }_1=0,{\gamma }_1=-3$ with the phase variable defined as $\xi =x-{\displaystyle \frac{d}{\beta }}{\left(t+{\displaystyle \frac{1}{a\beta }}\right)}^{\beta }$. The 3D plots of both real and imaginary components are valued over the same range $-2\le x\le 2$ and $-0\le t\le 2$. For the real component, a 2D plot is shown for $-3\le y\le 3$ at time slices t = −1.3, −0.5, 0.2, 1.5, while the contour plot covers $-3\le x\le 3$ and $-1\le t\le 3.5$. For the imaginary part, a 2D plot is provided for $-4.5\le t\le 7.5$ at t = −1.5, −1, 1, 3, and the corresponding contour plot spans $-3\le x\le 3$ and $-1\le t\le 3.5$. The absolute component represents the multi-solitons under the same parameter setting. The 3D plot spans $-2\le x\le 2$ and $-0\le t\le 2$. The 2D plot is shown for $-3\le x\le 3$ at t = −1, −0.5, 0.5, 1.1, while the contour plot covers the domain $-3\le x\le 3$ and $-1\le t\le 3.5$.

Figure 3 illustrates the real, imaginary, and absolute components of the solution (36) through 2D, 3D, and contour plots. Real and imaginary parts represent a lump-type solitary wave corresponding to the parameter values: ${\mathfrak{B}}_{06}(x,t):\alpha =5,\beta =1,d=I,{\lambda }_1=0,{\gamma }_1=-3I$ with the phase variable defined as $\xi =x-{\displaystyle \frac{d}{\beta }}{\left(t+{\displaystyle \frac{1}{a\beta }}\right)}^{\beta }$. The 3D plots of both real and imaginary components are valued over the same range $-1\le x\le 1$ and $-1\le t\le 1$. For the real component, a 2D plot is shown for $-1.5\le x\le 1.5$ at time slices t = −1.5, −0.7, 0.4, 1, while the contour plot covers $-1\le x\le 1$ and $-1\le t\le 1$. For the imaginary part, a 2D plot is provided for $-3\le y\le 3$ at t = −1, −0.5, 0.5, 1, and the corresponding contour plot spans $-1\le x\le 1$ and $-1\le t\le 1$. The absolute component represents the multi-solitons under the same parameter setting. The 3D plot spans $-1\le x\le 1$ and $-1\le t\le 1$. The 2D plot is shown for $-1.5\le x\le 1.5$ at t = −2, −1, 0.5, 2, while the contour plot covers the domain $-1\le x\le 1$ and $-1\le t\le 1$.

Figure 4 illustrates the real, imaginary, and absolute components of the solution (38) through 2D, 3D, and contour plots. Real and imaginary parts represent a lump-type solitary wave corresponding to the parameter values: varying values of constants ${\mathfrak{B}}_{07}(x,t):\alpha =-2I,\beta =1,d=-I,{\lambda }_0=0,{\lambda }_1=0,{\gamma }_1=-6$, with the phase variable defined as $\xi =x-{\displaystyle \frac{d}{\beta }}{\left(t+{\displaystyle \frac{1}{a\beta }}\right)}^{\beta }$. The 3D plots of both real and imaginary components are valued over the same range $-5\le x\le 5$ and $-5\le t\le 5$. For the real component, a 2D plot is shown for $-4\le x\le 5$ at time slices t = −1.5, −1, 1, 3, while the contour plot covers $-4.2\le x\le 5$ and $-3\le t\le 3$. For the imaginary part, 2D plots are provided for $-5\le x\le 5$ at t = −1.4, −0.5, 0.2, 1.5, and the corresponding contour plot spans $-3.3\le x\le 5$ and $-3\le t\le 3$. The absolute component represents the multi-solitons under the same parameter setting. The 3D plot spans $-5\le x\le 5$ and $-5\le t\le 5$. The 2D plot is shown for $-4\le t\le 5$ at t = −1, −0.5, 0.5, 1.1, while the contour plot covers the domain $-4.3\le x\le 5$ and $-3\le t\le 3$.

Figure 5 provides the comprehensive visualization of the solution (58), illustrating the real, imaginary, and absolute components 3D surfaces, and contour plots. The real components represent the interaction of the kink soliton geometry under the parameter setting ${\mathfrak{B}}_{15}(x,t):\alpha =-1,\beta =0.4,d=0.01I,{\lambda }_0=0,{\lambda }_1=2,{\gamma }_1=1$, with the phase variable defined as $\xi =x-{\displaystyle \frac{d}{\beta }}{\left(t+{\displaystyle \frac{1}{a\beta }}\right)}^{\beta }$. The plot is generated over the domain $-1\le x\le 1$ and $-1\le t\le 1$, and the contour plot spans $-1\le x\le 1$ and $-1\le t\le 1$. The 2D slices are plotted along $x\in [-0.2,0.2]$ at t = −2, −1, 0, 1. The geometry of the imaginary part reflects the bright profile under the same parameter settings. The 3D plots span $-1\le x\le 1$ and $-1\le t\le 1$ and contour map covers $-1\le x\le 1$ and $-1\le t\le 1$. The 2D slices are plotted along $x\in [-0.2,0.2]$ at t = −2, −1, 2, 3. The absolute component illustrates the distinct bright structure under the same parameter settings. The 3D visualization is plotted over $-1\le x\le 1$ and $-1\le t\le 1$. The contour plot spans $-1\le x\le 1$ and $-1\le t\le 1$. The 2D slices are plotted along $x\in [-0.2,0.2]$ at t = −2, −1, 1, 2.

Figure 1 displays the modulation Instability gain spectrum for three fixed values of $\eta \in [5,10,15]$ with the same parameter choice as $a=4,b=1,\sigma =2,c=2$. The MGERIF method introduces key innovations beyond GERFM by incorporating integral operators into its solution structure. It enables the systematic generation of rational terms, such as csc, sech², and csch, that are unobtainable with GERFM’s purely algebraic method. MGERIF can solve more complex physical problems, such as Equation (4), capturing advanced wave phenomena like singular solitons, breathers, and lump interactions. While GERFM solutions emerge as special cases when MGERIF’s integral terms are simplified, the full method produces strictly broader solutions that combine polynomial, trigonometric, hyperbolic, exponential, and rational functions in a unified framework.

The localized soliton and multi-peakon profiles (e.g., Equations (24), (26) and (28)) correspond to solitary pulse propagation in nonlinear optical fibers, where dispersion is balanced by Kerr-type cubic nonlinearity. The inclusion of the quintic term reflects higher-order nonlinear effects that arise in highly nonlinear fibers, photonic crystal fibers, and semiconductor waveguides. In addition, the rogue-type and breather solutions capture transient spikes and localized structures, which have been observed experimentally in mode-locked fiber lasers and during super-continuum generation. Therefore, the present results not only enrich the theoretical description but also provide qualitative guidance for interpreting optical experiments in fibers and laser cavities.

7. Comparison with Previous Literature

To situate our findings within the broader research context, we summarize key contributions from earlier studies alongside the present work. This comparative overview highlights methodological differences, solution types, and advancements achieved in the current analysis.

The comparative details are presented in

Table 2. Chronological comparison of existing literature with the present study.

Reference	Year	Method/Model Used	Solution Type	Limitations	Advancement in Present Work
Zhong et al. [17]	2015	Analytical & numerical study of Airy pulses	Soliton generation and finite-energy Airy beams	Focused on Airy-type beams only	Captures lump, kink, and peakon solutions
Tariq et al. [13]	2021	Extended trial function method on cubic–quintic NPPP	Bright, dark, and optical solitons	Focused on integer-order PDE, limited wave classes	Includes fractional-order effects and multi-peakon solutions
Khater [15]	2021	Khater II and generalized exponential methods	Optical pulse propagation in planar waveguides	Relies on computational simulations only	Analytical multi-peakon solutions derived in fractional domain
Khater [16]	2022	Numerical simulation of CQ-HE	Optical pulse dynamics	Purely numerical, lacks analytical insight	Provides exact analytical solutions validated graphically
Rizi et al. [14]	2022	Jacobian elliptic function method	Chirped solitary waves	Restricted to periodic/elliptic forms	Extends to rational, trigonometric, and exponential solutions
Akbar et al. [18]	2024	$(G^{\prime }/G,1/G)$-expansion method	Fractional solitary wave solutions	Limited to basic trigonometric/hyperbolic forms	Integrates rational, exponential, and hybrid solutions
Rizvi et al. [19]	2025	Sub-ODE method	Wide range of soliton solutions	Cannot unify different solution structures	Unifies polynomial, exponential, trigonometric, and rational solutions
Jlali et al. [12]	2025	Complete discrimination system of polynomial method (CDSPM)	Analytical soliton & quasi-periodic solutions; chaos analysis	Restricted to algebraic polynomial balance	MGERIF generates mixed rational–integral structures beyond polynomial scope
Present Study	2025	Conformable cubic–quintic nonlinear Schrödinger-type model using MGERIF	Multi-peakon, lump, kink, and bright/dark solitons	Some redundant/irrelevant solutions need filtering; complex algebraic systems in higher dimensions	Introduces unified framework with fractional derivatives, provides physical interpretation for optical pulse propagation

, which outlines prior approaches and the improvements introduced in this study. This table serves to highlight the progression from earlier results toward the broader solution space achieved here.

8. Conclusions

In conclusion, this research examined the fractional cubic–quintic nonlinear non-paraxial pulse propagation model and established multiple families of solitary wave solutions, using the Multivariate Generalized Exponential Rational Integral Function (MGERIF) method. By employing $\beta$-fraction wave transform, the governing fractional model is effectively reduced to an integer-order framework, yielding a broad range of solutions expressed in rational, trigonometric, and hyperbolic functions with adjustable parameters. By carefully adjusting the parameters, we demonstrate a diverse range of nonlinear profiles, including bright and dark solitons, kink and anti-kink structures, periodic waves, anti-peakons, and singular states. The comparative analysis across anomalous and normal GVD regimes highlights the role of fractional order in shaping amplitude localization, waveform stability, and interaction dynamics. Furthermore, the inclusion of modulation instability provides additional insight into the onset of sideband amplification, pulse-train formation, and breather-type dynamics, revealing how cubic–quintic effects broaden MI gain spectra and permit instability beyond the conventional anomalous-dispersion setting. As prior work primarily considered the integer-order model without detailed dispersion-dependent analysis, this study introduces new perspectives and expands the catalog of soliton solutions. These results not only extend the mathematical theory of fractional nonlinear systems but also hold relevance for applications in fiber-optic communications, ultrafast photonics, and advanced laser technologies.