Bifurcation and chaos: Unraveling soliton solutions in a couple fractional-order nonlinear evolution equation

Muhammad Bilal Riaz; Adil Jhangeer; Jan Martinovic; Syeda Sarwat Kazmi

doi:10.1515/nleng-2024-0024

Artikel Open Access

Bifurcation and chaos: Unraveling soliton solutions in a couple fractional-order nonlinear evolution equation

Muhammad Bilal Riaz , Adil Jhangeer , Jan Martinovic und Syeda Sarwat Kazmi

Veröffentlicht/Copyright: 27. November 2024

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Abstract

Shallow water waves represent a significant and extensively employed wave type in coastal regions. The unconventional bidirectional transmission of extended waves across shallow water is elucidated through nonlinear fractional partial differential equations, specifically the space–time fractional-coupled Whitham–Broer–Kaup equation. The application of two distinct analytical methods, namely, the generalized logistic equation approach and unified approach, is employed to construct various solutions such as bright solitons, singular solitary waves, kink solitons, and dark solitons for the proposed equation. The physical behavior of calculated results is graphically represented through density, two- and three-dimensional plots. The obtained solutions could have significant implications across a range of fields including plasma physics, biology, quantum computing, fluid dynamics, optics, communication technology, hydrodynamics, environmental sciences, and ocean engineering. Furthermore, the qualitative assessment of the unperturbed planar system is conducted through the utilization of bifurcation theory. Subsequently, the model undergoes the introduction of an outward force with the aim of inducing disruption, resulting in the emergence of a perturbed dynamical system. The detection of chaotic trajectory in the perturbed system is accomplished through the utilization of a variety of tools designed for chaos detection. The execution of the Runge–Kutta method is employed to assess the sensitivity of the examined model. The results obtained serve to underscore the effectiveness and applicability of the proposed methodologies for the assessment of soliton structures within a broad spectrum of nonlinear models.

Keywords: coupled fractional Whitham–Broer–Kaup model; the generalized logistic equation method; unified method; Soliton structures; phenomena of bifurcation and chaos; sensitivity

1 Introduction

Many scientists consider the study of nature’s fascinating nonlinearity to be a significant commencement toward understanding the basic aspects of the natural world. The analysis of a variety of nonlinear partial differential equations (PDEs) is vital for the study of complex phenomena that vary over time. Numerous academic fields, including the natural sciences, neural networks, optics, fluid dynamics, and image processing, rely on these frameworks [1–3]. Numerous efficient methods have emerged, offering an array of choices for formulating precise solutions to nonlinear PDEs. This wide variety includes methods that include the extended modified rational expansion approach [4], simplified Hirota’s approach [5], auxiliary equation approach [6], Lie symmetry approach [7], Painlevé test [8], extended simple equation approach [9–11], logistic equation approach [12], and unified approach [13,14].

The generalized logistic equation (GLE) approach and the unified approach stand out among the many successful methods for identifying exact solutions to nonlinear PDEs. In particular, the unified approach is commended for being simple and easy to comprehend. This method makes it easier for academia to discover traveling wave patterns in two different formats: solutions that are rational and polynomial. On the other hand, employing the GLE approach yields outcomes presented in exponential and rational functions. Various scholars have extensively investigated these methods, alongside other analytical approaches, to address wave propagation challenges in PDEs. For instance, Chaffee–Infante model [15], fractional Schrödinger model [16], the Ivancevic option pricing equation [12], Schrödinger equation with Kerr law [17], Drinfeld–Sokolov–Wilson equation [18], and complex nonlinear Kuralay-IIA equation [19]. Currently, the exploration of fractional-order nonlinear PDEs has become a prominent topic in research circles.

In 1695, L’Hospital questioned Leibniz via a letter regarding the feasibility of extending the derivative from integer to fractional orders. This inquiry marked the inception of an enhanced calculus, eventually recognized as the arbitrary order calculus, and widely acknowledged today as fractional calculus [20,21]. In recent times, there has been a notable recognition of the substantial impact of fractional differential equations (FDEs) across various fields, including engineering, astrophysics, biology, control theory, geology, and signal processing [22–24].

To date, various fractional derivatives have been suggested, encompassing Riemann–Liouville, local M-truncated, Caputo, Hadamard, Beta, and Grunwald–Letnikov [25–27].

Each of these derivative presents advantages and limitations. The Riemann–Liouville derivative, in particular, faces a significant issue: it fails to provide the derivative of a constant function as zero. Additionally, when a function remains constant at the origin, its fractional derivative exhibits uniqueness at that point, as observed in exponential and Mittag–Leffler functions. These limitations restrict the practical applicability of Riemann–Liouville fractional derivatives. Similarly, the Caputo derivative struggles with problems involving singular kernels, further constraining its utility. Consequently, researchers have been motivated to seek more suitable definitions for fractional-order derivatives that are comprehensive in nature. One such innovative approach is the conformable derivative, introduced by Khalil et al. [28], which is a simple yet effective non-integer derivative based on the fundamental limit formulation. In the study by Abdelhakim [29], it was established by the author that, in specific cases, the Caputo definition does not produce outcomes of greater length compared to the conformable definition outlined in Khalil et al. [28]. Distinguished from both Caputo and Riemann–Liouville formulations, the conformable derivative satisfies many essential properties, making it a promising alternative. This unique derivative preserves key characteristics of integer-order calculus, including the linearity property, product rule, chain rule, quotient rule, and composition of functions. It finds applicability in numerous extensions of classical calculus theorems, including Rolle’s and the mean value theorem, conformable integration by parts, and fractional power series expansion [30,31].

The effectiveness of fractional derivatives in modeling complex systems across various fields of science and engineering stems from their ability to capture non-local dynamics, particularly memory effects [32,33]. Unlike ordinary derivatives, which are local in time, fractional operators possess global time characteristics, making them suitable for systems with memory. These memory effects can be categorized into two types: full memory, where fractional derivatives consider the entire past history of a system, and local memory, where only recent events influence the current state. Most existing fractional derivatives are defined over intervals with fixed real numbers a , which is typically denoted as [ a , t ] . As time progresses, these intervals can become exceedingly large due to constants like 0 or − ∞ , making them particularly suitable for capturing the full memory effect of a system. However, such derivatives may not adequately describe real-world systems with local memory properties, where memory effects occur within specific time segments. In such cases, it is more appropriate to employ a segment of time points to describe the status of the system at the current moment. In this study, our emphasis is on examining the space–time fractional-coupled Whitham–Broer–Kaup (WBK) equation, which describes the bi-directional propagation of long waves in shallow water. We consider the inclusion of the WBK equation in our analysis:

(1a) D t γ ℋ + ℋ D x γ ℋ + D x γ ℒ + λ D x 2 γ ℋ = 0 ,

(1b) D t γ ℒ + D x γ ( ℋ ℒ ) − λ D x 2 γ ℒ + δ D x 3 γ ℋ = 0 .

The representation of the horizontal velocity field is given by ℋ , while the elevation of the water surface above the horizontal reference point at the bottom is expressed as ℒ in Eq. (1). The operator D γ , which is characterized by its order γ , symbolizes the conformable derivative. Here, x and t refer to the spatial and temporal variables, respectively. Several approaches were employed to analyze the WBK model (1). Al-Shawba et al. [34] addressed this equation utilizing the ( Ψ ′ ∕ Ψ ) expansion approach and its extension, two variable ( Ψ ′ ∕ Ψ , 1 ∕ Ψ ) expansion approach for extracting various analytical solutions, while Cao et al. [35] explored it through the polynomial discriminant system. Additionally, Xu et al. [36] tackled this equation through the utilization of the scaling transforms approach. In the study by Sadat and Kassem [37], the Lie point symmetries were employed by the researchers to reduce the dimension of Eq. (1). The authors also utilized integrating factors and the Riccati sub-equation approaches to analyze soliton patterns. For a comprehensive understanding, the provided equation has not been explored utilizing unified approach and GLE approach. Consequently, this study seeks to elucidate the equation employing the mentioned methodologies. Various types of solitons, such as kink soliton, dark soliton, bright soliton, and singular solitons, have been recognized and visualized through three-dimensional, two-dimensional, and density plots.

Since the 1960s, pivotal findings in the realms of solitons, bifurcation, and chaos patterns have sparked significant interest in nonlinear dynamics [38,39]. Many continuous systems described by differential equations (DEs) involve parameters, where even a slight adjustment in a parameter can significantly impact the system’s response. This involves a qualitative analysis of dynamical systems under different values of the parameters inherent in the system, known as bifurcation [40]. On the contrary, chaos manifests as unpredictable patterns in even the most straightforward systems. An in-depth analysis of available literature indicates that chaos appears to be a common thread in various phenomena [39]. The examination of quasi-periodic phenomena, as well as the exploration of sensitivity analysis, has been thoroughly investigated [41]. Additionally, we carried out a system sensitivity analysis using the Runge–Kutta (RK) method, with varying initial conditions [40].

In this research, our focus is on the fractional-order WBK equation. Initially, we have successfully derived numerous soliton solutions through two distinct analytical methods: the unified approach and the GLE approach. Various types of solitons, such as kink solitons, dark solitons, bright solitons, and singular solitons, have been recognized and visualized through three-dimensional, two-dimensional, and density plots. These solutions play a significant role in various fields of nonlinear sciences such as physics, engineering, artificial intelligence, biology, image processing, and optical communication, among others. Following this, we conducted a thorough analysis of the qualitative aspects of the model at hand, integrating principles from bifurcation and chaos. In recent times, these phenomena have been explored across various scientific disciplines such as economics, engineering, and mathematics. The novel findings, previously unexplored in the examination of this equation, significantly advance our understanding of soliton dynamics and the diverse applications of dynamical systems in nonlinear science.

The article is structured as follows: Section 2 provides a necessary overview of fractional derivatives based on existing literature. Section 3 outlines the general methodology of the suggested approaches. In Section 4, soliton wave solutions are specifically addressed. The analysis results are deliberated in Section Section 5. Section Section 6 conducts a thorough examination of the dynamic characteristics of the proposed equation using phase portraits of bifurcation. The exploration of chaotic phenomena is undertaken in Section 7. Sensitive nature of the system is discussed in Section 8. Section 9 presents a conclusion.

2 Preliminaries

Khalil et al. [28] introduced a new concept known as the conformable derivative. In this discussion, we initially examine the proposed definition.

Consider the function h : [ 0 , ∞ ) → R . The γ th-order conformable derivative of h , as introduced by Khalil et al., is defined as

(2) T γ ( h ) ( t ) = lim η → 0 h ( t + η t 1 − γ ) − h ( t ) η ,

where γ ∈ ( 0 , 1 ] and t > 0 . As per the definition formulated by Khalil et al. [28], the conformable derivative adheres to the theorems outlined in the following.

Theorem 2.1

Assuming γ ∈ ( 0 , 1 ] and considering that both functions h and k are γ -differentiable for every t > 0 . Then,

T γ ( a h ( t ) + b k ( t ) ) = a T γ ( h ( t ) ) + b T γ ( k ( t ) ) , ∀ a , b ∈ R ,
T γ ( t e ) = e t e − γ , e ∈ R ,
T γ ( c ) = 0 , ∀ constant h ( t ) = c ,
T γ ( h ( t ) k ( t ) ) = h ( t ) T γ ( k ( t ) ) + k ( t ) T γ ( h ( t ) ) ,
T γ ( h ( t ) k ( t ) ) = k ( t ) T γ ( h ( t ) ) − h ( t ) T γ ( k ( t ) ) k ( t ) 2 .
As a result, if h undergoes differentiability, then T γ ( h ) ( t ) = t 1 − γ d h d t .

Theorem 2.2

If function h is assumed to be γ -differentiable in a conformable sense and function k is differentiable as well, then

T γ ( h ( t ) ∘ k ( t ) ) = t 1 − γ k ′ ( t ) h k ( t ) .

3 Analysis of the proposed approaches

In this section, two analytical techniques are examined: the GLE approach and the unified approach. The detail description of the proposed techniques is given in this segment.

Let us consider the nonlinear FDE as follows:

(3) P ( ℋ , D t γ ℋ , D x α ℋ , D t γ D t γ ℋ , D t γ D x α ℋ , D x α D x α ℋ , … ) = 0 ,

where γ ∈ ( 0 , 1 ] , α ∈ ( 0 , 1 ] are the fractional-order derivatives, and P is the polynomial including the unknown function ℋ . Then, using the following transformation as:

(4) ℋ = ϒ ( ξ ) , ℒ = Θ ( ξ ) , and ξ = p x α α − μ t γ γ ,

where the wave velocity is denoted as μ and the wave number as p , both parameters are non-zero. Eq. (3) is changed into an ODE as shown:

(5) O ( ϒ , ϒ ′ , ϒ ″ , … ) = 0 .

3.1 Analysis of the GLE approach

In this approach, the initial solution for Eq. (5) is outlined in the following:

(6) ϒ ( ξ ) = ∑ i = 0 j ϖ i Ω i ( ξ ) ,

where ϖ i , ( i = 0 , 1 , 2 , 3 , … , j ) , are the random constants to be resolved. The function Ω i ( ξ ) fulfils the following auxiliary equation:

(7) d Ω d ξ = r Ω ( ξ ) + s Ω κ ( ξ ) ,

where κ > 1 and its general solution is

(8) Ω ( ξ ) = − s + r d 1 exp ( − r ( κ − 1 ) ξ ) r ( − 1 κ − 1 ) ,

where r and s are the constants and d 1 is the integration constant.

3.2 Analysis of the unified approach

Through the utilization of this approach, outcomes can be derived as polynomial and rational functions. In the following, it has been addressed in detail.

3.2.1 Polynomial function solution

In this case, Eq. (5) has the initial solution as:

(9a) ϒ ( ξ ) = ∑ i = 0 j β i Λ i ( ξ ) ,

with satisfying

(9b) ( Λ ′ ( ξ ) ) ϑ = ∑ i = 0 ϑ ϱ σ i Λ i ( ξ ) , ξ = p x α α − μ t γ γ , ϑ = 1 , 2 ,

where β i and σ i are the constants that will be evaluated later. The determination of the degree of series solutions j is accomplished through the application of the principle of homogeneous balance, while ϱ can be calculated utilizing consistency criteria as j = ( ϱ − 1 ) for all ϱ ≥ 2 , and the value of ϱ is found to be 2. Here, we will consider the cases when ϑ = 1 , ϱ = 2 and ϑ = 2 , ϱ = 2 .

3.2.2 Rational function solution

In this case, Eq. (5) has the initial solution as

(10a) ϒ ( ξ ) = ∑ i = 0 j ε i Λ i ( ξ ) ∑ i = 0 q ω i Λ i ( ξ ) , j ≥ q ,

with satisfying

(10b) ( Λ ′ ( ξ ) ) ϑ = ∑ i = 0 ϑ ϱ σ i Λ i ( ξ ) , ξ = p x α α − μ t γ γ , ϑ = 1 , 2 ,

where ε i and ω i are the undetermined random constants. The parameter j in Eqs (6), (9a), and (10a) is a balancing number, and its value can be determined using the homogeneous balancing rule: it is employed to determine both the balancing number and coefficient within the assumed solution. In this rule, these values are obtained by equating the linear and nonlinear terms of the highest order in the ordinary DEs. To solve the PDE analytically, we employ a power series solution truncated at the balancing number derived from the equation. Balancing numbers serve a critical role in mathematical modeling by simplifying and analyzing complex nonlinear systems. They are particularly significant in the study of optical solitons, which are self-sustaining wave packets capable of propagating long distances in optical fibers without distortion. These numbers play a pivotal role in investigating soliton solutions across diverse disciplines, including fluid dynamics, plasma physics, optical fiber communication, neural science, signal processing, and beyond.

4 Extraction of soliton solutions

This segment encompasses the soliton solutions and its visual depictions related to the discussed model. Upon substituting the expression from Eq. (4) into Eq. (1), we obtain the resulting nonlinear equations:

(11a) − μ ϒ ′ + p ϒ ϒ ′ + p Θ ′ + λ p 2 ϒ ″ = 0 ,

(11b) − μ Θ ′ + p ( ϒ Θ ) ′ − λ p 2 Θ ″ + p 3 δ ϒ ‴ = 0 .

By integrating Eq. (11a) and Eq. (11b) once with respect to ξ , we acquire

(12a) − μ ϒ + 1 2 p ϒ 2 + p Θ + λ p 2 ϒ ′ = 0 ,

(12b) − μ Θ + p ( ϒ Θ ) − λ p 2 Θ ′ + p 3 δ ϒ ″ = 0 .

From Eq. (12a), we obtain the following form:

(13) Θ = μ p ϒ − 1 2 ϒ 2 − p λ ϒ ′ .

By substituting Eq. (13) into Eq. (12b), we obtain

(14) p 3 ( δ + λ 2 ) ϒ ″ + 3 μ 2 ϒ 2 − p 2 ϒ 3 − μ 2 p ϒ .

4.1 Solutions by GLE approach

In this segment, we employ the GLE approach with κ = 2 to obtain the precise solution for Eq. (14). Balancing the terms ϒ 3 and ϒ ″ in Eq. (14), as 3 j = j + 2 , yields j = 1 . Now, for j = 1 , the solution becomes

(15) ϒ ( ξ ) = ϖ 0 + ϖ 1 Ω ( ξ ) .

Substituting solution (15) and its associated derivative, provided in Eq. (7), into Eq. (14) and setting the coefficients of Ω i to zero result in a system of equations, as depicted in the following:

Ω 0 = − ϖ 0 3 p 2 + 3 ϖ 0 2 p μ − 2 ϖ 0 μ 2 = 0 , Ω 1 = − 3 ϖ 0 2 ϖ 1 p 2 + 2 ϖ 1 p 4 r 2 λ 2 + 2 ϖ 1 p 4 r 2 δ + 6 ϖ 0 ϖ 1 p μ − 2 ϖ 1 μ 2 = 0 , Ω 2 = 3 ϖ 0 ϖ 1 2 p 2 + 6 ϖ 1 p 4 r s ( λ 2 + δ ) + 3 ϖ 1 2 p μ = 0 , Ω 3 = − ϖ 1 3 p 2 + 4 ϖ 1 p 4 s 2 ( λ 2 + δ ) = 0 .

Solving the aforementioned system for ϖ 0 , ϖ 1 , and p yields the following solution sets:

Set 1:

(16) ϖ 0 = 2 ( − r 2 ( λ 2 + δ ) ) 1 4 μ , ϖ 1 = − 2 s ( r 2 ( λ 2 + δ ) ) 1 4 μ r , p = − μ ( r 2 ( λ 2 + δ ) ) 1 4 .

Set 2:

(17) ϖ 0 = 0 , ϖ 1 = 2 s ( r 2 ( λ 2 + δ ) ) 1 4 μ r , p = − μ ( r 2 ( λ 2 + δ ) ) 1 4 .

substituting the aforementioned values into Eq. (15), the solutions of Eq. (1) corresponding to Set 1 are as follows:

(18) ℋ = − 2 d 1 r e ( r ξ ) ( r 2 ( λ 2 + δ ) ) 1 4 μ r d 1 e ( r ξ ) − s ,

(19) ℒ = − 2 d 1 r e ( r ξ ) ( r 2 ( λ 2 + δ ) ) 1 4 μ p 2 r s λ + d 1 p r e ( r ξ ) ( r 2 ( λ 2 + δ ) ) 1 4 μ + d 1 r μ e ( r ξ ) − s μ p ( − d 1 r e ( r ξ ) + s ) 2 .

Solutions of Eq. (1) corresponding to Set 2 are as follows:

(20) ℋ = 2 s ( r 2 ( λ 2 + δ ) ) 1 4 μ r d 1 e ( r ξ ) − s ,

(21) ℒ = − 2 s ( r 2 ( λ 2 + δ ) ) 1 4 μ p s ( r 2 ( λ 2 + δ ) ) 1 4 μ + s μ − d 1 r e ( r ξ ) ( p 2 r λ + μ ) p ( − d 1 r e ( r ξ ) + s ) 2 ,

where ξ = p x α α − μ t γ γ .

4.2 Solutions by unified approach

In this segment, we aim to derive the soliton patterns described by Eq. (1) using the unified method. Utilizing this approach enables the extraction of results in the form of polynomial and rational functions.

4.2.1 Polynomial solutions

To obtain a polynomial solution, we substitute j = 1 into Eq. (9a), yielding the following result:

(22) ϒ ( ξ ) = β 0 + β 1 Λ ( ξ ) .

Following this, we explore two scenarios related to the parameter ϑ mentioned in Eq. (9b).

Case 1: Assuming ϑ = 1 , Eq. (9b) becomes:

(23) Λ ′ ( ξ ) = σ 0 + σ 1 Λ ( ξ ) + σ 2 Λ 2 ( ξ ) .

By substituting solution (22) and its associated derivative, provided in Eq. (23), into Eq. (14) and setting the coefficients of Λ i to zero result in a system of equations, as depicted in the following:

Λ 0 = 2 p 4 ( λ 2 + δ ) β 1 σ 1 σ 0 + ( β 0 ( − p 2 β 0 + 3 μ p ) ) 2 − 2 μ 2 β 0 = 0 , Λ 1 = 2 p 4 ( λ 2 + δ ) β 1 ( 2 σ 0 σ 2 + σ 1 2 ) + β 0 β 1 ( − 3 p 2 β 0 + 6 μ p ) − 2 μ 2 β 1 = 0 , Λ 2 = 6 p 4 ( λ 2 + δ ) β 1 σ 1 σ 2 + ( β 1 ( − 3 p 2 β 0 + 3 μ p ) ) 2 = 0 , Λ 3 = 4 p 4 β 1 σ 2 2 ( λ 2 + δ ) − p 2 β 1 3 = 0 .

Solving the aforementioned system for β 0 , β 1 , σ 0 , σ 1 , and σ 2 , yields the following solution set:

(24) β 0 = p 2 σ 1 ( λ 2 + δ ) + λ 2 + δ μ p λ 2 + δ , β 1 = 2 σ 2 p λ 2 + δ , σ 0 = p 4 ( σ 1 ( λ 2 + δ ) ) 2 − μ 2 4 p 4 σ 2 ( λ 2 + δ ) , σ 1 = σ 1 , σ 2 = σ 2 .

By addressing Eq. (23) for Λ ( ξ ) and substituting it into Eq. (22) along with Eq. (24), the solutions for Eq. (1) can be expressed as

(25) ℋ = p 2 − μ 2 p 4 ( λ 2 + δ ) ( λ 2 + δ ) tan 1 2 ξ − μ 2 p 4 ( λ 2 + δ ) + λ 2 + δ μ p λ 2 + δ ,

(26) ℒ = ( λ 2 + δ λ + λ 2 + δ ) μ 2 2 cos 1 2 ξ − μ 2 p 4 ( λ 2 + δ ) 2 ( λ 2 + δ ) p 2 ,

where ξ = p x α α − μ t γ γ .

Case 2: Assuming ϑ = 2 , Eq. (9b) becomes

(27) Λ ′ ( ξ ) = Λ ( ξ ) σ 0 + σ 1 Λ ( ξ ) + σ 2 Λ 2 ( ξ ) .

By substituting solution (22) and its associated derivative, provided in Eq. (27), into Eq. (14) and setting the coefficients of Λ i to zero result in a system of equations, as depicted in the following:

Λ 0 = β 0 ( − p 2 β 0 2 + 3 μ p β 0 − 2 μ 2 ) = 0 , Λ 1 = 2 p 4 β 1 σ 0 ( λ 2 + δ ) + β 0 β 1 ( − 3 p 2 β 0 + 6 μ p ) − 2 μ 2 β 1 = 0 , Λ 2 = 3 p 4 β 1 σ 1 ( λ 2 + δ ) + 3 β 1 2 p ( − p β 0 + μ ) = 0 , Λ 3 = 4 p 4 β 1 σ 2 ( λ 2 + δ ) − p 2 β 1 3 = 0 .

Solving the aforementioned system for β 0 , β 1 , σ 0 , σ 1 , and σ 2 , yields the following solution set:

(28) β 0 = 2 μ p , β 1 = p 3 σ 1 ( λ 2 + δ ) μ , σ 0 = μ 2 p 4 ( λ 2 + δ ) , σ 1 = σ 1 , σ 2 = p 4 σ 1 2 ( λ 2 + δ ) 4 μ 2 .

By addressing Eq. (27) for Λ ( ξ ) and substituting it into Eq. (22) along with Eq. (28), the solutions for Eq. (1) can be expressed as

(29) ℋ = 2 μ e − ξ μ 2 p 4 ( λ 2 + δ ) p e − ξ μ 2 p 4 ( λ 2 + δ ) − 2 σ 1 ,

(30) ℒ = − 4 μ σ 1 e − ξ μ 2 p 4 ( λ 2 + δ ) μ 2 p 4 ( λ 2 + δ ) λ p 2 + μ p 2 e − ξ μ 2 p 4 ( λ 2 + δ ) − 2 σ 1 2 ,

where ξ = p x α α − μ t γ γ .

4.2.2 Rational solutions

Utilizing balancing rule yields j − q = ( ϱ − 1 ) . Exploring the solutions for ϱ = 1 , we find j = q . Specifically examining the scenario, where ϱ = 1 and ϑ = 2 , leading to the transformation of Eq. (10)

(31a) ϒ ( ξ ) = ε 0 + ε 1 Λ ( ξ ) ω 0 + ω 1 Λ ( ξ ) ,

with satisfying

(31b) Λ ′ ( ξ ) = σ 0 + σ 1 Λ ( ξ ) + σ 2 Λ 2 ( ξ ) .

By substituting Eq. (31) into Eq. (14) and setting the coefficients of Λ i to zero result in a system of equations, as depicted in the following

Λ 0 = λ 2 p 4 ( − ε 0 ω 0 ω 1 σ 1 + 4 ε 0 ω 1 2 σ 0 + ε 1 ω 0 2 σ 1 − 4 ε 1 ω 0 ω 1 σ 0 ) + δ p 4 ( − ε 0 ω 0 ω 1 σ 1 + 4 ε 0 ω 1 2 σ 0 + ε 1 ω 0 2 σ 1 − 4 ε 1 ω 0 ω 1 σ 0 ) − 2 μ 2 ε 0 ω 0 2 + 3 μ p ε 0 2 ω 0 − p 2 ε 0 3 = 0 , Λ 1 = λ 2 p 4 ( − 2 ε 0 ω 0 ω 1 σ 2 + 3 ε 0 ω 1 2 σ 1 + 2 ε 1 ω 0 2 σ 2 − 3 ε 1 ω 0 ω 1 σ 1 ) + δ p 4 ( − 2 ε 0 ω 0 ω 1 σ 2 + 3 ε 0 ω 1 2 σ 1 + 2 ε 1 ω 0 2 σ 2 − 3 ε 1 ω 0 ω 1 σ 1 ) + μ 2 ( − 4 ε 0 ω 0 ω 1 − 2 ε 1 ω 0 2 ) + 3 μ p ε 0 2 ω 1 + 6 μ p ε 0 ε 1 ω 0 − 3 p 2 ε 0 2 ε 1 = 0 , Λ 2 = 2 λ 2 p 4 σ 2 ( ε 0 ω 1 2 − ε 1 ω 0 ω 1 ) + 2 δ p 4 σ 2 ( ε 0 ω 1 2 − ε 1 ω 0 ω 1 ) − 2 ( μ ( ε 0 ω 1 2 − 2 ε 1 ω 0 ω 1 ) ) 2 + 6 μ p ε 0 ε 1 ω 1 + 3 μ p ε 1 2 ω 0 − 3 p 2 ε 0 ε 1 2 = 0 , Λ 3 = − 2 μ 2 ε 1 ω 1 2 + 3 p μ ε 1 2 ω 1 − p 2 ε 1 3 = 0 .

Solving the aforementioned system for ε 0 , ε 1 , ω 0 , ω 1 , σ 0 , σ 1 , and σ 2 , yields the following solution set:

(32) ε 0 = p 3 ω 1 σ 1 ( λ 2 + δ ) μ , ε 1 = 2 μ ω 1 p , ω 0 = ω 0 , ω 1 = ω 1 , σ 0 = p 4 σ 1 2 ( λ 2 + δ ) 4 μ 2 , σ 1 = σ 1 , σ 2 = μ 2 p 4 ( λ 2 + δ ) .

By substituting Eq. (31) along with Eq. (32), the solutions for Eq. (1) can be expressed as

(33) ℋ = 2 μ 3 ω 1 e ξ μ 2 p 4 ( λ 2 + δ ) p − μ 2 p 4 ( λ 2 + δ ) ( p 4 ω 1 σ 1 ( λ 2 + δ ) − 2 μ 2 ω 0 ) + e ξ μ 2 p 4 ( λ 2 + δ ) μ 2 ω 1 ,

(34) ℒ = 2 μ 3 ω 1 μ 2 p 4 ( λ 2 + δ ) e ξ μ 2 p 4 ( λ 2 + δ ) μ 2 p 4 ( λ 2 + δ ) λ ( p ( p 4 ω 1 σ 1 ( λ 2 + δ ) − 2 μ 2 ω 0 ) ) 2 − σ 1 ( λ 2 + δ ) μ p 4 ω 1 + 2 μ 3 ω 0 p 2 μ 2 p 4 ( λ 2 + δ ) ( p 4 ω 1 σ 1 ( λ 2 + δ ) − 2 μ 2 ω 0 ) − e ξ μ 2 p 4 ( λ 2 + δ ) μ 2 ω 1 2 ,

where ξ = p x α α − μ t γ γ .

5 Findings and physical interpretation

This portion explores the characteristics of various closed-form solutions to the fractional WBK equation. Employing both the GLE approach and a unified method, the study introduces soliton wave solutions for the equation, illustrating the wave patterns inherent in the WBK model. To facilitate comprehension of their dynamics, visual depictions, including 3D, 2D, and density graphs, are provided to showcase some of the identified solutions.

Figure 1(a) demonstrates the behavior of the solution described in Eq. (18), portraying a kink soliton. The three-dimensional, two-dimensional, and density plots are shown using specific parameter values: r = λ = d 1 = μ = 1 , p = 0.5 , α = 0.9 , s = − 2 , δ = 17 , and γ = 0.85 . In Figure 1(b), a bell-shaped soliton is illustrated through three-dimensional, two-dimensional, and density plots for the solution given in Eq. (19) with the parameter values r = λ = d 1 = μ = 1 , p = − 0.5 , α = 0.9 , s = − 2 , δ = 8 , and γ = 0.85 . Figure 2(a) demonstrates the behavior of the solution described in Eq. (25), as kink soliton. The three-dimensional, two-dimensional, and density plots are shown using specific parameter values: μ = − 4 , p = 2 , α = 0.9 , λ = 1 , δ = 2 , and γ = 0.85 . In Figure 2(b), a bright soliton is illustrated through three-dimensional, two-dimensional, and density plots for the solution given in (26) with the parametric value μ = 1 , p = 1 , α = 0.9 , λ = 1 , δ = 2 , and γ = 0.85 .

Figure 1

Visual representation of solutions (18) and (19) via three-dimensional, two-dimensional and density plots: (a) 3D plot, (b) 3D plot, (c) 2D plot, (d) 2D plot, (e) density plot, and (f) density plot.

Figure 2

Visual representation of the solutions (25) and (26) via three-dimensional, two-dimensional, and density plots: (a) 3D plot, (b) 3D plot, (c) 2D plot, (d) 2D plot, (e) density plot, and (f) density plot.

Figure 3(a) portrays the behavior of the solution described in Eq. (29) in the form of kink soliton. The three-dimensional, two-dimensional, and density graphs are depicted using suitable parameter values: μ = 4 , p = 2 , α = 0.9 , λ = 1 , δ = 2 , and γ = 0.85 . In Figure 3(b)), dark soliton, through three-dimensional, two-dimensional, and density plots, is presented for solution (30) with the parametric value μ = 4.1 , p = 2 , α = 0.9 , λ = − 0.5 , δ = 5 , σ 1 = − 1 , and γ = 0.85 .

Figure 3

Visual representation of the solutions (29) and (30)via three-dimensional, two-dimensional, and density plots: (a) 3D plot, (b) 3D plot, (c) 2D plot, (d) 2D plot, (e) density plot, and (f) density plot.

Figure 4(a) demonstrates the behavior of the solution described in Eq. (33), portraying a singular periodic soliton. The three-dimensional, two-dimensional, and density graphs are depicted using suitable parameter values: μ = 3.1 , p = 1.8 , α = 0.9 , λ = − 0.5 , δ = 2 , σ 1 = 0.5 , ω 0 = 0.2 , ω 1 = − 1 , and γ = 0.85 . In Figure 4(b), dark soliton, through three-dimensional, two-dimensional, and density plots, is presented for solution (34) with the parametric value μ = 3.1 , p = 2 , α = 0.9 , λ = − 0.5 , δ = 5 , σ 1 = 0.5 , ω 0 = 0.2 , ω 1 = − 1 , and γ = 0.85 .

Figure 4

Visual representation of solutions (33) and (34) via three-dimensional, two-dimensional, and density plots: (a) 3D plot, (b) 3D plot, (c) 2D plot, (d) 2D plot, (e) density plot, and (f) density plot.

In Figure 5(a), solution (21) is depicted at various γ values, including 0.7, 0.8, 0.9, 0.98, and 1. Figures 5(b) and (c) showcases solutions (25) and (29), respectively, each plotted across different γ values such as 0.6, 0.7, 0.8, 0.9, 0.98, and 1. In Figure 5(d), solution (34) is illustrated at varying γ values, including 0.75, 0.85, 0.9, 0.98, and 1. The remaining parameters across all figures remain consistent with those discussed earlier in Figures 1(b), 2(a), 3(a), and 4(b). These visualizations offer valuable insights into how the behavior of the obtained solutions changes with variations in the γ parameter. Across all four figures, it becomes evident that varying γ from a non-integer to an integer order results in the solutions aligning closely with the classical outcomes derived from ordinary derivatives.

$Figure 5 (a) and (b) display the 2D-plots illustrating solutions (21) and (25), respectively, employing conformable derivatives with varying non-integer orders γ \gamma , while α \alpha remains fixed at 1. (c), and (d) present the 2D plots depicting solutions (29) and (34), respectively, also utilizing conformable derivatives with different non-integer orders γ \gamma , and α = 1 \alpha =1 : (a) 2D plot, (b) 2D plot, (c) 2D plot, and (d) 2D plot.$

Figure 5

(a) and (b) display the 2D-plots illustrating solutions (21) and (25), respectively, employing conformable derivatives with varying non-integer orders γ , while α remains fixed at 1. (c), and (d) present the 2D plots depicting solutions (29) and (34), respectively, also utilizing conformable derivatives with different non-integer orders γ , and α = 1 : (a) 2D plot, (b) 2D plot, (c) 2D plot, and (d) 2D plot.

Solitary waves with lower intensity than the surrounding background are referred to as dark solitons, while those with higher intensity are termed bright solitons. Singular solitons refer to solitary waves characterized by a lack of continuity. In the realm of nonlinear optics, a kink soliton refers to a shock front that travels through a dispersive nonlinear medium without experiencing distortion. Additionally, the utilization of bright solitons can play a crucial role in managing soliton interference. These solutions serve as a significant resource for controlling the disorder caused by solitons, as discussed in the presentation area. Consequently, solitons can be converted from an attractive state to a state of separation, leading to the decluttering process. This has the potential to alleviate the Internet bottleneck, which poses a growing challenge to the modern telecommunications industry, where the Internet is an essential daily requirement. In the corona virus disease of 2019 pandemic, where business operations primarily occur in the digital realm, ensuring a seamless and continuous transmission of signals is crucial for maintaining uninterrupted internet connectivity. Likewise, in the presence of a background wave, dark solitons can facilitate the transmission of solitons. These visual depictions offer insights into the unique forms and attributes of diverse soliton solutions. Examining these solutions enables researchers to enhance their comprehension of the dynamics and traits exhibited by solitons in nonlinear models.

6 Bifurcation phenomena

We employed bifurcation phenomena [40,41] to reveal all potential phase portraits of the studied equation. To achieve this, we applied the Galilean transformation to Eq. (14) and obtained the subsequent unperturbed planar system:

(35) d ϒ d ξ = M , d M d ξ = φ 1 ϒ − φ 2 ϒ 2 + φ 3 ϒ 3 ,

where φ 1 = μ 2 p 4 ( δ + λ 2 ) , φ 2 = 3 μ 2 p 3 ( δ + λ 2 ) , and φ 3 = 1 2 p 2 ( δ + λ 2 ) . The Hamiltonian of (35), is expressed as follows:

(36) N ( ϒ , M ) = M 2 2 − φ 1 ϒ 2 2 + φ 2 ϒ 3 3 − φ 3 ϒ 4 4 ,

which satisfies

∂ N ∂ ϒ = − M ′ and ∂ N ∂ M = ϒ ′ .

In reality, system (35) forms a planar system, and the paths within its phase space are regulated by the vector field linked to (35). Therefore, it is essential to explore various phase plots of system (35) by examining various parameter setups. Initially, we will identify the critical points C j ( ϒ * , M * ) for system (35). These equilibrium points arise when d ϒ d ξ = 0 and d M d ξ = 0 , resulting in the determination of three critical points based on certain parameters:

C 1 ( ϒ * , M * ) = C 1 ( 0 , 0 ) , C 2 ( ϒ * , M * ) = C 2 φ 2 + φ 2 2 − 4 φ 1 φ 3 2 φ 3 , 0 , C 3 ( ϒ * , M * ) = C 3 φ 2 − φ 2 2 − 4 φ 1 φ 3 2 φ 3 , 0 .

Furthermore, the Jacobian of system (35) will be

(37) J ( ϒ * , M * ) = 0 1 φ 1 − 2 φ 2 ϒ + 3 φ 3 ϒ 2 0 .

The determinant and trace of (37) at the critical points C j ( ϒ * , M * ) are indicated by F and T , respectively, and these values are provided as follows:

T = trace ( J ) ∣ C j = 0 , F = det ( J ) ∣ C j = − φ 1 + 2 φ 2 ϒ − 3 φ 3 ϒ 2 .

The critical point is classified as a saddle when F < 0 , a central point when F > 0 and T = 0 , a cusp when F = 0 , and a node if F > 0 and T 2 − 4 F > 0 . In essence, the behavior of phase plots is dictated by the characteristics of roots. Consequently, the analysis of phase plots for the dynamical system (35) relies on the physical parameters φ 1 , φ 2 , and φ 3 .

• Case 1: Let φ 1 > 0 , φ 2 > 0 , and φ 3 > 0 .

For μ = p = δ = λ = 1 , φ 1 = 0.5 , φ 2 = 0.75 , and φ 3 = 0.25 , system (35) has three critical points C 1 = ( 0 , 0 ) , C 2 = ( 1 , 0 ) , and C 3 = ( 2 , 0 ) . In this instance, critical points C 1 and C 3 serve as saddle points for system (35) due to the negativity of F at these points. Similarly, C 2 serves as a center for system (35) as F is positive at this particular critical point. The phase plot analysis of system (35) at these points is illustrated in Figure 6.

$Figure 6 Phase plots for system (35) when φ 1 > 0 {\varphi }_{1}\gt 0 , φ 2 > 0 {\varphi }_{2}\gt 0 , and φ 3 > 0 {\varphi }_{3}\gt 0 .$

Figure 6

Phase plots for system (35) when φ 1 > 0 , φ 2 > 0 , and φ 3 > 0 .

• Case 2: Let φ 1 < 0 , φ 2 < 0 , and φ 3 < 0 .

For μ = p = λ = 1 , δ = − 2 , φ 1 = − 1 , φ 2 = − 1.5 , and φ 3 = − 0.5 , system (35) has three critical points C 1 = ( 0 , 0 ) , C 2 = ( 1 , 0 ) , and C 3 = ( 2 , 0 ) . In this instance, critical points C 1 and C 3 serve as central points for system (35) due to the positivity of F at these points. Similarly, C 2 serves as a saddle point for system (35) as F is negative at this particular critical point. The phase plot analysis of the system (35) at these points is illustrated in Figure 7.

$Figure 7 Phase plots for system (35) when φ 1 < 0 {\varphi }_{1}\lt 0 , φ 2 < 0 {\varphi }_{2}\lt 0 , and φ 3 < 0 {\varphi }_{3}\lt 0 .$

Figure 7

Phase plots for system (35) when φ 1 < 0 , φ 2 < 0 , and φ 3 < 0 .

7 Chaotic phenomena

In this segment, we introduce an outward periodic force into system (35) to examine the dynamics of both quasi-periodic and chaotic phenomena [40,41]. The modified system can be expressed in the following manner:

(38) d ϒ d ξ = M , d M d ξ = φ 1 ϒ − φ 2 ϒ 2 + φ 3 ϒ 3 + ζ 0 cos ( χ ) , d χ d ξ = ϕ .

The disturbed phase of the system mentioned earlier is defined by parameters ζ 0 and ϕ , representing the intensity and frequency of an outward force applied to the system (35). The introduction of an outward force to a system can lead to unpredictable and apparently chaotic responses. In our study, we have observed this phenomenon in a specific system (38) that exhibits erratic motion. Over the course of time, the trajectories of the system deviate from conventional patterns, underscoring the unpredictability and complexity associated with chaotic behaviors. To investigate this phenomenon, a variety of methods were utilized to identify chaos, including visualization of 2D phase portraits, examination of time series, and computation of Lyapunov exponents (LEs). Collectively, these approaches allowed us to identify the existence of chaotic dynamics within the analyzed system. To further explore this matter, we will investigate the impact of both the intensity, denoted as ζ 0 , and the frequency, represented by ϕ . Throughout our analysis, we will maintain consistency by keeping all other parameters fixed at φ 1 = − 0.5 , φ 2 = − 0.75 , and φ 3 = − 0.25 . In Figure 8, temporal representations are depicted, while Figure 9 illustrates two-dimensional graphs. Various combinations of intensity and frequency values were employed, leading to occurrences of periodic, quasi-periodic, and chaotic tendencies.

$Figure 8 Detection of chaotic phenomena in the perturbed system (38) via time plots. (a) ζ 0 = 0.01 {\zeta }_{0}=0.01 , ϕ = 0.01 \phi =0.01 , (b) ζ 0 = 0.5 {\zeta }_{0}=0.5 , ϕ = 0.1 \phi =0.1 , (c) ζ 0 = 1.6 {\zeta }_{0}=1.6 , ϕ = π \phi =\pi , (d) ζ 0 = 2.8 {\zeta }_{0}=2.8 , ϕ = 1.2 \phi =1.2 , (e) ζ 0 = 2.8 {\zeta }_{0}=2.8 , ϕ = π \phi =\pi , (f) ζ 0 = 3.5 {\zeta }_{0}=3.5 , ϕ = 2 π \phi =2\pi .$

Figure 8

Detection of chaotic phenomena in the perturbed system (38) via time plots. (a) ζ 0 = 0.01 , ϕ = 0.01 , (b) ζ 0 = 0.5 , ϕ = 0.1 , (c) ζ 0 = 1.6 , ϕ = π , (d) ζ 0 = 2.8 , ϕ = 1.2 , (e) ζ 0 = 2.8 , ϕ = π , (f) ζ 0 = 3.5 , ϕ = 2 π .

$Figure 9 Detection of chaotic phenomena in the perturbed system (38) via two-dimensional plots: (a) ζ 0 = 0.01 {\zeta }_{0}=0.01 , ϕ = 0.01 \phi =0.01 , (b) ζ 0 = 0.5 {\zeta }_{0}=0.5 , ϕ = 0.1 \phi =0.1 , (c) ζ 0 = 1.6 {\zeta }_{0}=1.6 , ϕ = π \phi =\pi , (d) ζ 0 = 2.8 {\zeta }_{0}=2.8 , ϕ = 1.2 \phi =1.2 , (e) ζ 0 = 2.8 {\zeta }_{0}=2.8 , ϕ = π \phi =\pi , and (f) ζ 0 = 3.5 {\zeta }_{0}=3.5 , ϕ = 2 π \phi =2\pi .$

Figure 9

Detection of chaotic phenomena in the perturbed system (38) via two-dimensional plots: (a) ζ 0 = 0.01 , ϕ = 0.01 , (b) ζ 0 = 0.5 , ϕ = 0.1 , (c) ζ 0 = 1.6 , ϕ = π , (d) ζ 0 = 2.8 , ϕ = 1.2 , (e) ζ 0 = 2.8 , ϕ = π , and (f) ζ 0 = 3.5 , ϕ = 2 π .

The LE quantifies the divergence of proximate trajectories within a dynamic system, signifying the extent of disorder. A positive LE denotes the chaotic behavior, while a negative one suggests stability. The magnitude influences the pace of divergence. In Figure 10, the LE derived from time evolution reveals chaos within the modified dynamic system (38).

$Figure 10 Detection of chaotic phenomena in the perturbed system (38) via LE with the initial condition (0.2, 0.2, 0.2). (a) φ 1 = ‒ 0.5 , φ 2 = ‒ 0.75 , φ 3 = ‒ 0.25 , ζ = 3.5 {\varphi }_{1}=‒0.5,{\varphi }_{2}=‒0.75,{\varphi }_{3}=‒0.25,\zeta =3.5 , and ϕ = 2 π \phi =2\pi .$

Figure 10

Detection of chaotic phenomena in the perturbed system (38) via LE with the initial condition (0.2, 0.2, 0.2). (a) φ 1 = ‒ 0.5 , φ 2 = ‒ 0.75 , φ 3 = ‒ 0.25 , ζ = 3.5 , and ϕ = 2 π .

8 Sensitivity analysis

This section delves into the sensitivity of the proposed model, examining how mathematical models respond to uncertainties in their parameters across various fields. The quantification of the impact of uncertainties on the input parameters of the model and its subsequent effect on the model’s output is achieved through sensitivity analysis. Across various academic domains, addressing the challenge of sensitivity concerning parameter uncertainty in mathematical equations is a common endeavor. The main objective of this examination is to quantitatively assess the output disturbance arising from input perturbation. This inquiry focuses on the allocation and classification of uncertainty in both the effectiveness of a mathematical equation and the framework’s response to uncertainties introduced by its inputs. In this section, outcomes are showcased using relevant parameter values, highlighting the significance of small input adjustments on the results. A minimal effect suggests low sensitivity, while a significant change indicates high sensitivity. In this segment, we explore the sensitivity characteristics of system (35) illustrated in Figure 11 using specific initial values. A thorough examination of the proposed system is outlined in the following.

$Figure 11 Sensitivity analysis across various initial values: (a) sensitivity analysis of system (35) for initial values ( ϒ , M ) = ( 0.01 , 0 ) \left(\Upsilon ,M)=\left(0.01,0) in red and ( ϒ , M ) = ( 0.02 , 0 ) \left(\Upsilon ,M)=\left(0.02,0) in blue; (b) sensitivity analysis of system (35) for initial values ( ϒ , M ) = ( 0.01 , 0 ) \left(\Upsilon ,M)=\left(0.01,0) in red and ( ϒ , M ) = ( 0.05 , 0 ) \left(\Upsilon ,M)=\left(0.05,0) in green; (c) sensitivity analysis of system (35) for initial values ( ϒ , M ) = ( 0.02 , 0 ) \left(\Upsilon ,M)=\left(0.02,0) in blue and ( ϒ , M ) = ( 0.05 , 0 ) \left(\Upsilon ,M)=\left(0.05,0) in green; and (d) sensitivity analysis of system (35) for initial values ( ϒ , M ) = ( 0.01 , 0 ) \left(\Upsilon ,M)=\left(0.01,0) in red, ( ϒ , M ) = ( 0.02 , 0 ) \left(\Upsilon ,M)=\left(0.02,0) in blue and ( ϒ , M ) = ( 0.05 , 0 ) \left(\Upsilon ,M)=\left(0.05,0) in green.$

Figure 11

Sensitivity analysis across various initial values: (a) sensitivity analysis of system (35) for initial values ( ϒ , M ) = ( 0.01 , 0 ) in red and ( ϒ , M ) = ( 0.02 , 0 ) in blue; (b) sensitivity analysis of system (35) for initial values ( ϒ , M ) = ( 0.01 , 0 ) in red and ( ϒ , M ) = ( 0.05 , 0 ) in green; (c) sensitivity analysis of system (35) for initial values ( ϒ , M ) = ( 0.02 , 0 ) in blue and ( ϒ , M ) = ( 0.05 , 0 ) in green; and (d) sensitivity analysis of system (35) for initial values ( ϒ , M ) = ( 0.01 , 0 ) in red, ( ϒ , M ) = ( 0.02 , 0 ) in blue and ( ϒ , M ) = ( 0.05 , 0 ) in green.

9 Conclusion

In conclusion, this study delves into the exploration of the nonlinear dynamics inherent in the fractional-coupled WBK equation. Employing two distinctive analytical methods, namely, the GLE approach and the unified approach, we constructed various solitons, including bright, singular, kink, and dark solitons, for the proposed equation. These solutions hold significant importance in various applied sciences, particularly in nonlinear fiber optics, as they possess stability during wave propagation, allowing for the efficient transmission of large amounts of data at high speeds. Bright solitons, which have been extensively researched in optical communications, remain a preferred choice in communication network systems due to their high intensity peaks, despite the recent discovery of dark soliton transmission in optical fibers. Although dark solitons offer advantages such as fewer fiber losses and reduced sensitivity to noise, their utilization in communication systems is less prevalent compared to bright solitons. On the other hand, kink solitons find applications in diverse fields such as condensed matter physics and nonlinear optics, serving as stable, localized energy configurations with potential uses in information transmission and materials science. The visual representation of these findings through three-dimensional, two-dimensional, and density plots provides comprehensive insight into the complex behavior of the system. Additionally, we assessed the qualitative nature of the unperturbed dynamical system, facilitated by bifurcation theory, as detailed in Figures 6 and 7. Subsequently, the introduction of an outward periodic force to induce disruption in the system reveals the emergence of a perturbed dynamical system. The detection of chaotic trajectories in the perturbed system is achieved through the implementation of various chaos detection tools, including time series plots, two-dimensional plots, and LEs, as illustrated in Figures 8–10. The sensitivity of the governing model was assessed using the RK method, as presented in Figure 11. The results obtained from this investigation are captivating and make a substantial contribution to the understanding of soliton dynamics, as well as the practical applications of dynamical systems across diverse areas of nonlinear science. Additionally, these findings can aid in future investigations aimed at developing generalized fractional derivatives. Further exploration of the model with alternative exact analytical methods is also anticipated. Consequently, significant research efforts remain necessary to fully grasp and explore the potential of this model. These advancements may lead to the development of more effective mathematical models and numerical methods designed for solving this equation.

10 Future directions

Extensive research efforts are still required to comprehensively understand and unlock the full potential of the WBK model. Additionally, there is anticipation for further exploration using alternative precise analytical methods. Looking ahead, there is a promising avenue for extracting multi-wave soliton solutions, including 1-soliton, 2-soliton, 3-solitons, and higher-dimensional solitons, within the scope of the studied model [42,43].

Acknowledgement

This article has been produced with the financial support of the European Union under the REFRESH – Research Excellence For Region Sustainability and High-tech Industries project number CZ.10.03.01/00/22_003/0000048 via the Operational Programme Just Transition.

Funding information: The authors state no funding involved.
Author contributions: Conceptualization: M.B.R. and A.J.; methodology: M.B.R. and A.J.; software: S.S.K., J.M. and M.B.R.; validation: M.B.R., and A.J.; formal analysis: M.B.R. and A.J.; investigation: M.B.R. and A.J.; data curation: S.S.K.; writing – original draft preparation: S.S.K.; writing – review and editing: M.B.R., J.M. and A.J.; visualization: M.B.R. and A.J.; supervision: A.J.; project administration: J.M. All authors have read and agreed to the published version of the manuscript.
Conflict of interest: The authors declare that they have no conflicts of interest.

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Received: 2024-03-18

Revised: 2024-05-13

Accepted: 2024-06-21

Published Online: 2024-11-27

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/nleng-2024-0024

Schlagwörter für diesen Artikel

coupled fractional Whitham–Broer–Kaup model; the generalized logistic equation method; unified method; Soliton structures; phenomena of bifurcation and chaos; sensitivity

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