Exploring dynamical features like bifurcation assessment, sensitivity visualization, and solitary wave solutions of the integrable Akbota equation

Dean Chou; Azad Ali Sagher; Muhammad Imran Asjad; Yasser Salah Hamed

doi:10.1515/nleng-2024-0040

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Exploring dynamical features like bifurcation assessment, sensitivity visualization, and solitary wave solutions of the integrable Akbota equation

Dean Chou , Azad Ali Sagher , Muhammad Imran Asjad und Yasser Salah Hamed

Veröffentlicht/Copyright: 25. Februar 2025

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Abstract

The Akbota equation (AE), as a Heisenberg ferromagnetic-type equation, can be extremely valuable in the study of curve and surface geometry. In this study, we employ the well-known two analytical techniques, the modified Khater method and the new sub-equation approach, to construct the solitary wave solution of AE. Transform the partial differential equation into an ordinary differential equation using the wave transformation. The graphical visualization of select wave solutions is carried out using Wolfram Mathematica software. By utilizing appropriate parametric values across various wave velocities, this process unveils the intricate internal structures and provides a comprehensive understanding of wave behavior. The visual representations are rendered in 3D, 2D, and contour surfaces, capturing a range of solitonic phenomena. These include multiple kink solitons, flat kink, kink-peakon, kink solitons, and singular kink solitons, offering detailed insights into the complex dynamics of the system under study. Newly obtained soliton solutions are compared with available soliton solutions in the literature. The new results indicate that these obtained solutions can be a part of completing the family of solutions, and the considered methods are effective, simple, and easy to use. For qualitative assessment, convert the ordinary differential into a dynamical system by using the Galilean transformation to conduct the sensitivity visualization and bifurcation assessment along with phase portraits and chaos analysis of the considered model. Bifurcation analysis is crucial in soliton dynamics, as it influences the behavior and characteristics of solitons in various systems, with the results presented through phase portraits. Sensitivity visualization illustrates how parametric values affect the system’s behavior. The solutions obtained have broad applications in surface geometry and electromagnetism theory. The aim of this study is to enhance the understanding of complex nonlinear dynamics and their relevance in curve and surface geometry.

Keywords: Akbota equation; modified Khater method; new sub-equation method; soliton; bifurcation; sensitivity visualization

1 Introduction

A different name used for the mathematical model known as nonlinear partial differential equations (NLPDEs) is nonlinear mathematical physics equations or nonlinear evolution. In many relevant scientific and mathematical domains, such as physical chemistry, biology, and atmospheric and space sciences, it exhibits nonlinear phenomena [1–15]. In recent years, partial differential equations (PDEs) have many applications in the fields of physics, applied mathematics, astronomy, and many other branches of science. PDEs are used as effective tools for modeling in diverse fields of the natural sciences and engineering.

The integrable Akbota system equation in the study of Mathanaranjan and Myrzakulov [16] has many applications in the study of the curve and surface geometry.

(1) ι Ω t + α Ω x x + β Ω x t + γ ℛ Ω = 0 , ℛ x − 2 ε ( α ∣ Ω ∣ x 2 + β ∣ Ω ∣ t 2 ) = 0 ,

where Ω = Ω ( x , t ) is represented as a complex function, ℛ = ℛ ( x , t ) is considered a real function, while α , β , and γ are the arbitrary constants, and ε = ± 1 . The Akbota equation (AE), a Heisenberg ferromagnetic-type equation, is a valuable model for studying nonlinear phenomena in magnets, optics, and differential geometry. When α = 0 , the AE becomes the Kuralay equation. When β = 0 , the AE becomes the well-known Schrodinger equation. The AE is considered in this study due to its critical role in modeling nonlinear wave phenomena across various physical systems. This equation is particularly relevant for describing the evolution and interaction of wave structures in contexts such as fluid dynamics, optical fibers, and plasma physics. The focus on the AE is driven by its ability to capture complex, nonlinear behaviors that are essential for understanding and predicting wave dynamics in these areas. Investigating this equation aims to uncover new exact solutions and solitonic structures, contributing to the broader theoretical framework of nonlinear wave propagation.The findings from this research could also have significant implications for experimental and applied sciences, offering deeper insights into the mechanisms governing wave phenomena. Soliton theory is a key topic in mathematical physics and applied mathematics, both of which have grown quickly since the 1960s. Solitons, also known as solitary waves, are found in solutions to a variety of NLPDEs. They have various noteworthy qualities and can be used to explain a wide range of significant scientific events. Solitons are commonly seen in nature and display both particle and wave properties. There are numerous significant challenges with solitons theory in study domains such as fluid dynamics, plasma physics, nonlinear optics, classical and quantum field theory, and so on. In recent years, the idea of soliton has been studied in a broader context. Solitons, for example, are static solutions with certain features [17].

Handling and evaluating nonlinear equations employing solitary waves have become common. Finding closed-form, exact solutions has attracted a lot of attention from researchers recently. These solutions are vital for understanding the characteristics and stability of physical systems. Kink and bell-shaped solitons are extensively utilized to approximate nonlinear oscillatory phenomena in a variety of fields, including hydrodynamics, optical fibers, and physics. Solitary wave solutions are obtained by utilizing the different methods and techniques in the NLPDE. In the previous study, the variety of solitary wave solutions was obtained by Khater using the Khater II method [18], unified auxiliary equation method [19], simple extended [20], extended tanh method, and modified extended tanh method [21], ( 1 ⁄ G ′ ) -expansion method [22], variational iteration techniques [23], extended exponential function technique [24], power series methodology [25], Hirota bilinear method [26], and numerous others [27–33]. Kumar and Kukkar explored new exact soliton solutions for the (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain (HFSC) equation by the new Inverse G ′ ⁄ G -Expansion Method[34]. Kumar and Kukkar studied the dynamics of solitary waves to a (3+1)-dimensional nonlinear Schrödinger equation with parabolic law using the extended generalized Riccati equation mapping method [35]. Mann et al. discussed the dynamical behavior of the governing (1+1)-dimensional time-space Salerno equation, which describes the discrete electrical lattice with nonlinear dispersion. The different kinds of solutions in the form of periodic, Jacobi elliptic, and exponential functions were obtained by the two analytical approaches, namely, the modified generalized exponential rational function method and the extended sine–Gordon equation expansion method [36]. Niwas et al. utilized the multivariate generalized exponential rational integral function approach for solving the Hirota bilinear problem in (2+1)-dimensions [37]. Kumar et al. investigated the coupled breaking soliton (cBS) model using two distinct analytical methods, the Lie symmetry approach and the unified method, to construct breather solitons, cone-shaped solitons, lump solitons, patterns of flower petals, periodic solitons, and solitary waves [38]. Kumar and Mohan [39] formulated a generalized nonlinear fifth-order Korteweg–De Vries-type equation with multiple soliton solutions with Painlevé analysis and the Hirota bilinear technique. Kumar and Hamid investigated the (2+1)-dimensional nonlinear electrical transmission line model and proposed a modified generalized Riccati equation mapping approach for obtaining analytical soliton solutions [40]. Uddin et al. [41] explored the bifurcation and phase portraits of prevalent nonlinear pseudo-parabolic physical Oskolkov models for Kevin–Voigt fluids for different parametric conditions via a dynamical system approach. They also derived smooth waves of the bright bell and dark bell, periodic waves, and singular waves of dark and bright cusps, in correspondence to homoclinic, periodic, and open orbits with cusp, respectively. In a previous study, Sagidullayeva et al. [42], explored the lax pair representation and gauge counterparts of AE. Kong and Guo [43], using the Darbox transformation method acquired some solitary wave solutions of AE. Conservation laws, soliton solutions, and stability analysis for the AE were used by Mathanaranjan and Myrzakulov [16]. This study was inspired by the need to fill a substantial research vacuum in the existing literature on the AE. Specifically, there has been an absence of comprehensive research into sensitivity visualization, bifurcation analysis, chaos analysis, and solitary wave solutions using the New Sub-Equation (NSE) and Modified Khater (MK) methods. In order to fill this gap, we intend to perform an in-depth investigation of these features within the context of the AE. To the best of our knowledge, no previous research has developed or investigated the sensitivity visualization, bifurcation behavior, and solitary wave solutions to the AE making use of the NSE and MK approaches. By concentrating on these unknown regions, we want to add new insights and techniques to the discipline. Results of this work are intended to improve understanding of nonlinear dynamics associated with the AE and provide an adequate foundation for future research in this field.

The NSE and MK techniques are used during this study to discover optical soliton solutions. In comparison with previous methods, our current technique offers a more efficient and generalized solution with several additional advantages. These solutions address a variety of waves: multiple kink solitons, flat kink, kink-peakon, kink soliton, and singular kink soliton. We visually evaluate the solutions to the integrable system equations by utilizing Wolfram Mathematica software. By adjusting the parameter values, we illustrate the solutions diagrammatically, providing an in-depth understanding of their characteristics and behaviors of wave pattern. The structure of the current study is as follows: Discuss and explore the mathematical analysis of the considered model in Section 2. Our suggested techniques and their implementation are listed in Section 3. Graphical illustrations of some selected solutions in comparison with previous studies are explored in Section 4. Section 5 presents the dynamic assessment with sensitivity visualization, bifurcation assessment, and chaos analysis along phase portraits. Finally, Section 6 gives the concluding remarks.

2 Mathematical analysis

Consider the integrable AE as follows:

(2) ι Ω t + α Ω x x + β Ω x t + γ ℛ Ω = 0 , ℛ x − 2 ε ( α ∣ Ω ∣ x 2 + β ∣ Ω ∣ t 2 ) = 0 .

Apply the wave transformation in the ways as listed below:

(3) Ω ( x , t ) = ℳ ( ξ ) e ι ( − k x + ω t ) , ℛ ( x , t ) = Ψ ( ξ ) , ξ = x − ν t .

Applying the wave transformation in Eq. (2) and separating the real and imaginary parts, we obtain

(4) ( − k 2 α ω + k β ω + γ Ψ ( ξ ) ) ℳ + ( α − β ν ) ℳ ″ ( ξ ) = 0 ,

(5) ( − 2 k α ν + k β ν + β ω ) ℳ ( ξ ) = 0 ,

(6) − 4 ε ( α − β ν ) ℳ ( ξ ) ℳ ′ ( ξ ) + Ψ ′ ( ξ ) = 0 .

Eq. (5) implies that

(7) ω = 2 k α + ν − k β ν β .

Integrating (6) with respect to ξ and the constant of integration is supposed to be zero, we obtain

(8) Ψ ( ξ ) = 2 ε ( α β ν ) ℳ ( ξ ) 2 .

The following equation is obtained by substituting Eqs (8) and (7) in Eq. (4)

(9) ( − ν + k ( − 2 + k β ) ( α − β ν ) ) ℳ + 2 β γ ε ( α − β ν ) ℳ 3 + β ( α − β ν ) ℳ ″ = 0 ,

where ( − β ν + α ) ≠ 0 .

3 The proposed techniques

In this section, we explore and implement the proposed methods on the integral AE to construct the soliton solution.

3.1 Basic stages of the proposed approaches

The algorithm of the proposed methods contains the following steps:

Let us assume that the NLPDE is

(10) N ( f , f t , f x , f x x , f x t , f t t , f t x , … ) = 0 .

Stage 1: Utilize the wave transformation such that

(11) ℛ = r ( ξ ) and ξ = x + y + m z − c t

are used in Eq. (10), which leads to the ordinary differential equation

(12) T ( f , f ′ , f ″ , … ) = 0 .

Stage 2: Let the solution form of Eq. (9) can be presented as reported by Tripathy et al. [44],

NSE method:

(13) ℳ ( ξ ) = ∑ i = 0 k d i Z i h ( ξ ) .

MK method:

(14) ℳ ( ξ ) = ∑ l = 0 k d l Z l h ( ξ ) + ∑ l = 0 k c l Z − l h ( ξ ) + r o .

As c l , d i , d l , and r o are necessary to compute and z h ( ξ ) satisfy the following:

(15) h ′ ( ξ ) = λ 1 Z − h ( ξ ) + λ 2 + λ 3 Z h ( ξ ) l n ( Z ) ,

we obtained the positive number N , by utilizing the balancing principle as in Eqs (13) and (14).

Stage 3: The solutions by NSE and MK methods possesses the following cases given as Eq. (15).

Case 1: If ( D = λ 2 2 − 4 λ 3 λ 1 < 0 , λ 3 ≠ 0 ) ,

(16) Z 1 h ( ξ ) = − λ 2 2 λ 3 + − D 2 λ 3 tan − D 2 ( ξ ) ,

(17) Z 2 h ( ξ ) = − λ 2 2 λ 3 + − D 2 λ 3 cot − D 2 ( ξ ) .

Case 2: If ( D = λ 2 2 − 4 λ 3 λ 1 > 0 , λ 3 ≠ 0 ) ,

(18) Z 3 h ( ξ ) = − λ 2 2 λ 3 + D 2 λ 3 tan D 2 ( ξ ) ,

(19) Z 4 h ( ξ ) = − λ 2 2 λ 3 + D 2 λ 3 cot D 2 ( ξ ) .

Case 3: If ( E = λ 2 2 + 4 λ 1 2 > 0 , λ 3 ≠ 0 , λ 3 = − λ 1 ) ,

(20) Z 5 h ( ξ ) = λ 2 2 λ 1 + E 2 λ 1 tanh E 2 ( ξ ) ,

(21) Z 6 h ( ξ ) = λ 2 2 λ 1 + E 2 λ 1 coth E 2 ( ξ ) .

Case 4: If ( E = λ 2 2 + 4 λ 1 2 < 0 , λ 3 ≠ 0 , λ 3 = − λ 1 ) ,

(22) Z 7 h ( ξ ) = λ 2 2 λ 1 + − E 2 λ 1 tan − E 2 ( ξ ) ,

(23) Z 8 h ( ξ ) = λ 2 2 λ 1 + − E 2 λ 1 cot − E 2 ( ξ ) .

Case 5: If ( F = λ 2 2 − 4 λ 1 2 < 0 , λ 3 = λ 1 ) ,

(24) Z 9 h ( ξ ) = − λ 2 2 λ 1 + − F 2 λ 1 tan − F 2 ( ξ ) ,

(25) Z 10 h ( ξ ) = − λ 2 2 λ 1 − − F 2 λ 1 cot − F 2 ( ξ ) .

Case 6: If ( F = λ 2 2 − 4 λ 1 2 > 0 , λ 3 = λ 1 ) ,

(26) Z 11 h ( ξ ) = − λ 2 2 λ 1 − F 2 λ 1 tanh F 2 ( ξ ) ,

(27) Z 12 h ( ξ ) = − λ 2 2 λ 1 − F 2 λ 1 coth F 2 ( ξ ) .

Case 7: If ( λ 3 λ 1 < 0 , λ 3 ≠ 0 , λ 2 = 0 ) ,

(28) Z 13 h ( ξ ) = − − λ 1 λ 3 tanh ( − λ 3 λ 1 ( ξ ) ) ,

(29) Z 14 h ( ξ ) = − − λ 1 λ 3 coth ( − λ 3 λ 1 ( ξ ) ) .

Case 8: If ( λ 3 λ 1 > 0 , λ 3 ≠ 0 , λ 2 = 0 ) ,

(30) Z 15 h ( ξ ) = λ 1 λ 3 tan ( λ 3 λ 1 ( ξ ) ) ,

(31) Z 16 h ( ξ ) = − λ 1 λ 3 cot ( λ 3 λ 1 ( ξ ) ) .

Case 9: If ( λ 2 = 0 , λ 1 = − λ 3 ) ,

(32) Z 17 h ( ξ ) = − tanh ( λ 1 ξ ) ,

(33) Z 18 h ( ξ ) = − coth ( λ 1 ξ ) .

Case 10: If ( λ 2 2 = 4 λ 3 λ 1 ) ,

(34) Z 19 h ( ξ ) = − 2 λ 1 ( λ 2 ξ + 2 ) λ 2 2 ξ .

Case 11: If ( λ 2 = ν , λ 3 = 0 , λ 1 = 2 ν ) ,

(35) Z 20 h ( ξ ) = ( e ν ξ − 2 ) .

Case 12: If ( λ 2 = ν , λ 1 = 0 , λ 3 = 2 ν ) ,

(36) Z 21 h ( ξ ) = e ν ξ 1 − 2 e ν ξ .

Case 13: If ( λ 2 = λ 3 = 0 ) ,

(37) Z 22 h ( ξ ) = [ λ 1 ξ ] .

Case 14: ( λ 2 = λ 1 = 0 ) ,

(38) Z 23 h ( ξ ) = − 1 λ 3 ξ .

Case 15: ( λ 2 = 0 , λ 1 = λ 3 ) ,

(39) Z 24 h ( ξ ) = tan ( λ 1 ξ ) ,

(40) Z 25 h ( ξ ) = − cot ( λ 1 ξ ) .

Case 16: ( λ 3 = 0 ) :

(41) Z 26 h ( ξ ) = e λ 2 ξ − λ 1 λ 2 .

Stage 4: By substituting Eqs (13)–(15) in Eq. (9), we obtain a system of equations and gather all of the terms that are equal to zero and have the same power of z h ( ξ ) . The unknown values of c l , d i , and r o will be determined by symbolic calculations.

Stage 5: By substituting the values of unknowns in (13)–(15), we obtain the required solution.

3.2 Implementation of the suggested approach

To construct the solitary wave solution of the AE, we utilize the NSE and MK methods in this section.

3.2.1 Traveling wave solution via NSE method

Using the balancing principal algorithm from Eq. (6), we can obtain N = 1 , and the general solution of Eq. (13) as follows:

(42) ℳ = d o + d 1 Z h ( ξ ) .

Now, by substituting Eqs (15) and (42) in Eq. (9) and following the stages of the suggested method mentioned above, we obtain the following sets of solutions.

(43) d 0 = ± λ 2 2 γ ε , d 1 = ± λ 3 γ ε , λ 1 = λ 2 2 β + 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β λ 3 .

Family 1: If ( D = λ 2 2 − 4 λ 3 λ 1 < 0 , λ 3 ≠ 0 ) ,

(44) ℳ 1 ( ξ ) = ± λ 2 2 γ ε + λ 3 γ ε − λ 2 2 λ 3 + − D 2 λ 3 tan − D 2 ( ξ ) × e i ( − k x + ω t ) ,

(45) ℳ 2 ( ξ ) = ± λ 2 2 γ ε + λ 3 γ ε − λ 2 2 λ 3 + − D 2 λ 3 cot − D 2 ( ξ ) × e i ( − k x + ω t ) .

Family 2: If ( D = λ 2 2 − 4 λ 3 λ 1 > 0 , λ 3 ≠ 0 ) ,

(46) ℳ 3 ( ξ ) = ± λ 2 2 γ ε + λ 3 γ ε − λ 2 2 λ 3 + D 2 λ 3 tan D 2 ( ξ ) × e i ( − k x + ω t ) ,

(47) ℳ 4 ( ξ ) = ± λ 2 2 γ ε + λ 3 γ ε − λ 2 2 λ 3 + D 2 λ 3 cot D 2 ( ξ ) × e i ( − k x + ω t ) .

Family 3: If ( E = λ 2 2 + 4 λ 1 2 > 0 , λ 3 ≠ 0 , λ 3 = − λ 1 ) ,

(48) ℳ 5 ( ξ ) = ± λ 2 2 γ ε − λ 1 γ ε λ 2 2 λ 1 + E 2 λ 1 tanh E 2 ( ξ ) × e i ( − k x + ω t ) ,

(49) ℳ 6 ( ξ ) = ± λ 2 2 γ ε − λ 1 γ ε λ 2 2 λ 1 + E 2 λ 1 coth E 2 ( ξ ) × e i ( − k x + ω t ) .

Family 4: If ( E = λ 2 2 + 4 λ 1 2 < 0 , λ 3 ≠ 0 , λ 3 = − λ 1 ) ,

(50) ℳ 7 ( ξ ) = ± λ 2 2 γ ε − λ 1 γ ε λ 2 2 λ 1 + − E 2 λ 1 tan − E 2 ( ξ ) × e i ( − k x + ω t ) ,

(51) ℳ 8 ( ξ ) = ± λ 2 2 γ ε − λ 1 γ ε λ 2 2 λ 1 + − E 2 λ 1 cot − E 2 ( ξ ) × e i ( − k x + ω t ) .

Family 5: If ( F = λ 2 2 − 4 λ 1 2 < 0 , λ 3 = λ 1 ) ,

(52) ℳ 9 ( ξ ) = ± λ 2 2 γ ε + λ 1 γ ε − λ 2 2 λ 1 + − F 2 λ 1 tan − F 2 ( ξ ) × e i ( − k x + ω t ) ,

(53) ℳ 10 ( ξ ) = ± λ 2 2 γ ε + λ 1 γ ε − λ 2 2 λ 1 − − F 2 λ 1 cot − F 2 ( ξ ) × e i ( − k x + ω t ) .

Family 6: If ( F = λ 2 2 − 4 λ 1 2 > 0 , λ 3 = λ 1 ) ,

(54) ℳ 11 ( ξ ) = ± λ 2 2 γ ε + λ 1 γ ε − λ 2 2 λ 1 − F 2 λ 1 tanh ( F 2 ( ξ ) ) × e i ( − k x + ω t ) ,

(55) ℳ 12 ( ξ ) = ± λ 2 2 γ ε + λ 1 γ ε − λ 2 2 λ 1 − F 2 λ 1 coth F 2 ( ξ ) × e i ( − k x + ω t ) .

Family 7: If ( λ 3 λ 1 < 0 , λ 3 ≠ 0 , λ 2 = 0 ) ,

(56) ℳ 13 ( ξ ) = ± λ 3 γ ε − 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β λ 3 2 tanh ( − 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β ( ξ ) ) e i ( − k x + ω t ) ,

(57) ℳ 14 ( ξ ) = ± λ 3 γ ε − 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β λ 3 2 coth ( − 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β ( ξ ) ) e i ( − k x + ω t ) .

Family 8: If ( λ 3 λ 1 > 0 , λ 3 ≠ 0 , λ 2 = 0 ) ,

(58) ℳ 15 ( ξ ) = ± λ 3 γ ε λ 2 2 β + 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β λ 3 2 tan λ 2 2 β + 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β ( ξ ) e i ( − k x + ω t ) ,

(59) ℳ 16 ( ξ ) = ± λ 3 γ ε λ 2 2 β + 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β λ 3 2 cot λ 2 2 β + 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β ( ξ ) e i ( − k x + ω t ) .

Family 9: If ( λ 2 = 0 , λ 1 = − λ 3 ) ,

(60) ℳ 17 ( ξ ) = ± λ 3 γ ε ( − 1 tan ( − λ 3 ( ξ ) ) ) × e i ( − k x + ω t ) ,

(61) ℳ 18 ( ξ ) = ± λ 3 γ ε ( − 1 cot ( − λ 3 ( ξ ) ) ) × e i ( − k x + ω t ) .

Family 10: If ( λ 2 2 = 4 λ 3 λ 1 ) ,

(62) ℳ 19 ( ξ ) = ± 2 λ 3 λ 1 2 γ ε + λ 3 γ ε − 4 λ 3 λ 1 β + 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 8 β λ 3 2 λ 1 ξ × ( 2 λ 3 λ 1 ( ξ ) + 2 ) e i ( − k x + ω t ) .

Family 11: If ( λ 2 = ν , λ 3 = 0 , λ 1 = 2 ν ) ,

(63) ℳ 20 ( ξ ) = ± ν 2 γ ε ( e ν ξ − 2 ) × e i ( − k x + ω t ) .

Family 12: If ( λ 2 = ν , λ 1 = 0 , λ 3 = 2 ν ) ,

(64) ℳ 21 ( ξ ) = ± ν 2 γ ε + 2 ν γ ε e ν ξ 1 − 2 e ν ξ × e i ( − k x + ω t ) .

Family (13) has a constant solution.

Family 14: If ( λ 2 = λ 1 = 0 ) ,

(65) ℳ 23 ( ξ ) = ± λ 3 γ ε − 1 λ 3 ξ × e i ( − k x + ω t ) .

Family 15: If ( λ 2 = 0 , λ 1 = λ 3 ) ,

(66) ℳ 24 ( ξ ) = ± λ 3 γ ε ( tan ( λ 3 ξ ) ) × e i ( − k x + ω t ) ,

(67) ℳ 25 ( ξ ) = ± λ 2 2 γ ε + λ 3 γ ε ( − cot ( λ 3 ξ ) ) × e i ( − k x + ω t ) .

Family 16: If ( λ 3 = 0 ) ,

(68) ℳ 26 ( ξ ) = ± λ 2 2 γ ε e λ 2 ξ − λ 1 λ 2 × e i ( − k x + ω t ) .

3.2.2 Visual presentation of exact solutions via NSE method

Figures 1, 2, 3 exhibit graphical representations of some selected solutions employing the NSE method.

$Figure 1 The parameters λ 1 = 0.4 {\lambda }_{1}=0.4 , λ 2 = 0.25 {\lambda }_{2}=0.25 , λ 3 = 1.0025 {\lambda }_{3}=1.0025 , γ = 1.75 \gamma =1.75 , and ε = 3.0075 \varepsilon =3.0075 depict the physical structure of the ℳ 1 ( x , t ) {{\mathcal{ {\mathcal M} }}}_{1}\left(x,t) . (a) 3D visualization of traveling wave at ν = 0.9 \nu =0.9 , (b) contour visualization of traveling wave at ν = 0.9 \nu =0.9 , (c) 2D visualization of traveling wave at ν = 0.9 \nu =0.9 , (d) 3D visualization of traveling wave at ν = 1.9 \nu =1.9 , (e) contour visualization of traveling wave at ν = 1.9 \nu =1.9 , (f) 2D visualization of traveling wave at ν = 1.9 \nu =1.9 , (g) 3D of traveling wave at ν = 2.9 \nu =2.9 , (h) contour visualization of traveling wave at ν = 2.9 \nu =2.9 , and (i) 2D of traveling wave at ν = 2.9 \nu =2.9 .$

Figure 1

The parameters λ 1 = 0.4 , λ 2 = 0.25 , λ 3 = 1.0025 , γ = 1.75 , and ε = 3.0075 depict the physical structure of the ℳ 1 ( x , t ) . (a) 3D visualization of traveling wave at ν = 0.9 , (b) contour visualization of traveling wave at ν = 0.9 , (c) 2D visualization of traveling wave at ν = 0.9 , (d) 3D visualization of traveling wave at ν = 1.9 , (e) contour visualization of traveling wave at ν = 1.9 , (f) 2D visualization of traveling wave at ν = 1.9 , (g) 3D of traveling wave at ν = 2.9 , (h) contour visualization of traveling wave at ν = 2.9 , and (i) 2D of traveling wave at ν = 2.9 .

$Figure 2 The parameters, λ 1 = 0.4 {\lambda }_{1}=0.4 , λ 2 = 0.25 {\lambda }_{2}=0.25 , λ 3 = 1.0025 {\lambda }_{3}=1.0025 , γ = 1.75 \gamma =1.75 , and ε = 3.0075 \varepsilon =3.0075 depict the physical structure of ℳ 5 ( x , t ) {{\mathcal{ {\mathcal M} }}}_{5}\left(x,t) . (a) 3D visualization of traveling wave at ν = 0.9 \nu =0.9 , (b) contour visualization of traveling wave at ν = 0.9 \nu =0.9 , (c) 2D visualization of traveling wave at ν = 0.9 \nu =0.9 , (d) 3D visualization of traveling wave at ν = 1.9 \nu =1.9 , (e) contour visualization of traveling wave at ν = 1.9 \nu =1.9 , (f) 2D visualization of traveling wave at ν = 1.9 \nu =1.9 , (g) 3D visualization of traveling wave at ν = 2.9 \nu =2.9 , (h) contour visualization of traveling wave at ν = 2.9 \nu =2.9 , and (i) 2D visualization of traveling wave at ν = 2.9 \nu =2.9 .$

Figure 2

The parameters, λ 1 = 0.4 , λ 2 = 0.25 , λ 3 = 1.0025 , γ = 1.75 , and ε = 3.0075 depict the physical structure of ℳ 5 ( x , t ) . (a) 3D visualization of traveling wave at ν = 0.9 , (b) contour visualization of traveling wave at ν = 0.9 , (c) 2D visualization of traveling wave at ν = 0.9 , (d) 3D visualization of traveling wave at ν = 1.9 , (e) contour visualization of traveling wave at ν = 1.9 , (f) 2D visualization of traveling wave at ν = 1.9 , (g) 3D visualization of traveling wave at ν = 2.9 , (h) contour visualization of traveling wave at ν = 2.9 , and (i) 2D visualization of traveling wave at ν = 2.9 .

$Figure 3 The parameters, λ 1 = 0.4 {\lambda }_{1}=0.4 , λ 2 = 0.25 {\lambda }_{2}=0.25 , λ 3 = 1.0025 {\lambda }_{3}=1.0025 , γ = 1.75 \gamma =1.75 , and ε = 3.0075 \varepsilon =3.0075 depict the physical structure of the ℳ 6 ( x , t ) {{\mathcal{ {\mathcal M} }}}_{6}\left(x,t) . (a) 3D visualization of traveling wave at ν = 0.9 \nu =0.9 , (b) contour visualization of traveling wave at ν = 0.9 \nu =0.9 , (c) 2D visualization of traveling wave at ν = 0.9 \nu =0.9 , (d) 3D visualization of traveling wave at ν = 1.9 \nu =1.9 , (e) contour visualization of traveling wave at ν = 1.9 \nu =1.9 , (f) 2D visualization of traveling wave at c = 1.9 c=1.9 , (g) 3D visualization of traveling wave at ν = 2.9 \nu =2.9 , (h) contour visualization of traveling wave at ν = 2.9 \nu =2.9 , and (i) 2D visualization of traveling wave at ν = 2.9 \nu =2.9 .$

Figure 3

The parameters, λ 1 = 0.4 , λ 2 = 0.25 , λ 3 = 1.0025 , γ = 1.75 , and ε = 3.0075 depict the physical structure of the ℳ 6 ( x , t ) . (a) 3D visualization of traveling wave at ν = 0.9 , (b) contour visualization of traveling wave at ν = 0.9 , (c) 2D visualization of traveling wave at ν = 0.9 , (d) 3D visualization of traveling wave at ν = 1.9 , (e) contour visualization of traveling wave at ν = 1.9 , (f) 2D visualization of traveling wave at c = 1.9 , (g) 3D visualization of traveling wave at ν = 2.9 , (h) contour visualization of traveling wave at ν = 2.9 , and (i) 2D visualization of traveling wave at ν = 2.9 .

3.3 Traveling wave solution via MK method

Using the balancing principal algorithm from Eq. (9), we can obtain N = 1 , and the general solution of Eq. (14) as follows:

(69) ℳ ( ξ ) = d 0 + d 1 Z h ( ξ ) + c 1 Z − h ( ξ ) .

After inserting Eqs (15) and (69) in Eq. (9) and following the stages of the suggested method mentioned above, we obtain the following sets of solutions:

(70) d o = ± λ 2 γ ε , c 1 = ± 1 γ ε , λ 3 = λ 2 2 β + 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β λ 1 .

Family 1: If ( D = λ 2 2 − 4 λ 3 λ 1 < 0 , λ 3 ≠ 0 ) ,

(71) ℳ 27 ( ξ ) = ± λ 2 γ ε + 1 γ ε − λ 2 2 λ 3 + − D 2 λ 3 tan − D 2 ( ξ ) × e i ( − k x + ω t ) ,

(72) ℳ 28 ( ξ ) = ± λ 2 γ ε + 1 γ ε − λ 2 2 λ 3 + − D 2 λ 3 cot − D 2 ( ξ ) × e i ( − k x + ω t ) .

Family 2: If ( D = λ 2 2 − 4 λ 3 λ 1 > 0 , λ 3 ≠ 0 ) ,

(73) ℳ 29 ( ξ ) = ± λ 2 γ ε + 1 γ ε − λ 2 2 λ 3 + D 2 λ 3 tan D 2 ( ξ ) × e i ( − k x + ω t ) ,

(74) ℳ 30 ( ξ ) = ± λ 2 γ ε + 1 γ ε − λ 2 2 λ 3 + D 2 λ 3 cot D 2 ( ξ ) × e i ( − k x + ω t ) .

Family 3: If ( E = λ 2 2 + 4 λ 1 2 > 0 , λ 3 ≠ 0 , λ 3 = − λ 1 ) ,

(75) ℳ 31 ( ξ ) = ± λ 2 γ ε − λ 1 γ ε λ 2 2 λ 1 + E 2 λ 1 tanh E 2 ( ξ ) × e i ( − k x + ω t ) ,

(76) ℳ 32 ( ξ ) = ± λ 2 γ ε − λ 1 γ ε λ 2 2 λ 1 + E 2 λ 1 coth E 2 ( ξ ) × e i ( − k x + ω t ) .

Family 4: If ( E = λ 2 2 + 4 λ 1 2 < 0 , λ 3 ≠ 0 , λ 3 = − λ 1 ) ,

(77) ℳ 33 ( ξ ) = ± λ 2 γ ε − λ 1 γ ε λ 2 2 λ 1 + − E 2 λ 1 tan − E 2 ( ξ ) × e i ( − k x + ω t ) ,

(78) ℳ 34 ( ξ ) = ± λ 2 γ ε − λ 1 γ ε λ 2 2 λ 1 + − E 2 λ 1 cot − E 2 ( ξ ) × e i ( − k x + ω t ) .

Family 5: If ( F = λ 2 2 − 4 λ 1 2 < 0 , λ 3 = λ 1 ) ,

(79) ℳ 35 ( ξ ) = ± λ 2 γ ε + 1 γ ε − λ 2 2 λ 1 + − F 2 λ 1 tan − F 2 ( ξ ) × e i ( − k x + ω t ) ,

(80) ℳ 36 ( ξ ) = ± λ 2 γ ε + 1 γ ε − λ 2 2 λ 1 − − F 2 λ 1 cot − F 2 ( ξ ) × e i ( − k x + ω t ) .

Family 6: If ( F = λ 2 2 − 4 λ 1 2 > 0 , λ 3 = λ 1 ) ,

(81) ℳ 37 ( ξ ) = ± λ 2 γ ε + 1 γ ε − λ 2 2 λ 1 − F 2 λ 1 tanh ( F 2 ( ξ ) ) × e i ( − k x + ω t ) ,

(82) ℳ 38 ( ξ ) = ± λ 2 γ ε + 1 γ ε − λ 2 2 λ 1 − F 2 λ 1 coth F 2 ( ξ ) × e i ( − k x + ω t ) .

Family 7: If ( λ 3 λ 1 < 0 , λ 3 ≠ 0 , λ 2 = 0 ) ,

(83) ℳ 39 ( ξ ) = ± 1 γ ε − 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β λ 3 2 × tanh − 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β ( ξ ) e i ( − k x + ω t ) ,

(84) ℳ 40 ( ξ ) = ± 1 γ ε − 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β λ 3 2 × coth ( − 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β ( ξ ) ) e i ( − k x + ω t ) .

Family 8: If ( λ 3 λ 1 > 0 , λ 3 ≠ 0 , λ 2 = 0 ) ,

(85) ℳ 41 ( ξ ) = ± 1 γ ε λ 2 2 β + 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β λ 3 2 tan × λ 2 2 β + 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β ( ξ ) e i ( − k x + ω t ) ,

(86) ℳ 42 ( ξ ) = ± 1 γ ε λ 2 2 β + 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β λ 3 2 cot × λ 2 2 β + 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 4 β ( ξ ) e i ( − k x + ω t ) .

Family 9: If ( λ 2 = 0 , λ 1 = − λ 3 ) ,

(87) ℳ 43 ( ξ ) = ± 1 γ ε ( − 1 tan ( − λ 3 ( ξ ) ) ) × e i ( − k x + ω t ) ,

(88) ℳ 44 ( ξ ) = ± 1 γ ε ( − 1 cot ( − λ 3 ( ξ ) ) ) × e i ( − k x + ω t ) .

Family 10: If ( λ 2 2 = 4 λ 3 λ 1 ) ,

(89) ℳ 45 ( ξ ) = ± 2 λ 3 λ 1 γ ε + 1 γ ε − 4 λ 3 λ 1 β + 4 κ + 2 ν − 2 κ β ( β ν + α ) 2 8 β λ 3 2 λ 1 ξ × ( 2 λ 3 λ 1 ( ξ ) + 2 ) e i ( − k x + ω t ) .

Family 11: If ( λ 2 = ν , λ 3 = 0 , λ 1 = 2 ν ) ,

(90) ℳ 46 ( ξ ) = ± ν γ ε + 1 γ ε ( e ν ξ − 2 ) e i ( − k x + ω t ) .

Family 12: If ( λ 2 = ν , λ 1 = 0 , λ 3 = 2 ν ) ,

(91) ℳ 47 ( ξ ) = ± ν γ ε + 1 γ ε e ν ξ 1 − 2 e ν ξ × e i ( − k x + ω t ) .

Family 13: If ( λ 2 = λ 3 = 0 ) ,

(92) ℳ 48 ( ξ ) = ± 1 γ ε ( λ 1 ξ ) × e i ( − k x + ω t ) .

Family 14: If ( λ 2 = λ 1 = 0 )

(93) ℳ 49 ( ξ ) = ± 1 γ ε − 1 λ 3 ξ × e i ( − k x + ω t ) .

Family 15: If ( λ 2 = 0 , λ 1 = λ 3 ) ,

(94) ℳ 50 ( ξ ) = ± 1 γ ε ( tan ( λ 3 ξ ) ) × e i ( − k x + ω t ) ,

(95) ℳ 51 ( ξ ) = ± 1 γ ε ( − cot ( λ 3 ξ ) ) × e i ( − k x + ω t ) .

Family 16: If ( λ 3 = 0 ) ,

(96) ℳ 52 ( ξ ) = ± λ 2 γ ε + 1 γ ε e λ 2 ξ − λ 1 λ 2 × e i ( − k x + ω t ) .

3.3.1 Visual presentation of exact solutions via the MK method

Figures 4, 5, 6 show the graphical illustration of some selected wave solutions via the MK method.

$Figure 4 The parameters λ 1 = 2.4 {\lambda }_{1}=2.4 , λ 2 = 2.25 {\lambda }_{2}=2.25 , λ 3 = 2.0025 {\lambda }_{3}=2.0025 , γ = 2.75 \gamma =2.75 , and ε = 3.0075 \varepsilon =3.0075 depict the physical structure of the ℳ 27 ( x , t ) {{\mathcal{ {\mathcal M} }}}_{27}\left(x,t) . (a) 3D visualization of traveling wave at ν = 0.9 \nu =0.9 , (b) contour visualization of traveling wave at ν = 0.9 \nu =0.9 , (c) 2D visualization of traveling wave at ν = 0.9 \nu =0.9 , (d) 3D visualization of traveling wave at ν = 1.9 \nu =1.9 , (e) contour visualization of traveling wave at ν = 1.9 \nu =1.9 , (f) 2D visualization of traveling wave at ν = 1.9 \nu =1.9 , (g) 3D visualization of traveling wave at ν = 2.9 \nu =2.9 , (h) contour visualization of traveling wave at ν = 2.9 \nu =2.9 , and (i) 2D visualization of traveling wave at ν = 2.9 \nu =2.9 .$

Figure 4

The parameters λ 1 = 2.4 , λ 2 = 2.25 , λ 3 = 2.0025 , γ = 2.75 , and ε = 3.0075 depict the physical structure of the ℳ 27 ( x , t ) . (a) 3D visualization of traveling wave at ν = 0.9 , (b) contour visualization of traveling wave at ν = 0.9 , (c) 2D visualization of traveling wave at ν = 0.9 , (d) 3D visualization of traveling wave at ν = 1.9 , (e) contour visualization of traveling wave at ν = 1.9 , (f) 2D visualization of traveling wave at ν = 1.9 , (g) 3D visualization of traveling wave at ν = 2.9 , (h) contour visualization of traveling wave at ν = 2.9 , and (i) 2D visualization of traveling wave at ν = 2.9 .

$Figure 5 The parameters λ 1 = 2.4 {\lambda }_{1}=2.4 , λ 2 = 2.25 {\lambda }_{2}=2.25 , λ 3 = 2.0025 {\lambda }_{3}=2.0025 , γ = 2.75 \gamma =2.75 , and ε = 3.0075 \varepsilon =3.0075 depict the physical structure of the ℳ 31 ( x , t ) {{\mathcal{ {\mathcal M} }}}_{31}\left(x,t) . (a) 3D visualization of traveling wave at ν = 0.9 \nu =0.9 , (b) contour visualization of traveling wave at ν = 0.9 \nu =0.9 , (c) 2D visualization of traveling wave at ν = 0.9 \nu =0.9 , (d) 3D visualization of traveling wave at ν = 1.9 \nu =1.9 , (e) contour visualization of traveling wave at ν = 1.9 \nu =1.9 , (f) 2D visualization of traveling wave at ν = 1.9 \nu =1.9 , (g) 3D visualization of traveling wave at ν = 2.9 \nu =2.9 , (h) contour visualization of traveling wave at ν = 2.9 \nu =2.9 , and (i) 2D visualization of traveling wave at ν = 2.9 \nu =2.9 .$

Figure 5

The parameters λ 1 = 2.4 , λ 2 = 2.25 , λ 3 = 2.0025 , γ = 2.75 , and ε = 3.0075 depict the physical structure of the ℳ 31 ( x , t ) . (a) 3D visualization of traveling wave at ν = 0.9 , (b) contour visualization of traveling wave at ν = 0.9 , (c) 2D visualization of traveling wave at ν = 0.9 , (d) 3D visualization of traveling wave at ν = 1.9 , (e) contour visualization of traveling wave at ν = 1.9 , (f) 2D visualization of traveling wave at ν = 1.9 , (g) 3D visualization of traveling wave at ν = 2.9 , (h) contour visualization of traveling wave at ν = 2.9 , and (i) 2D visualization of traveling wave at ν = 2.9 .

$Figure 6 The parameters λ 1 = 2.4 {\lambda }_{1}=2.4 , λ 2 = 0.25 {\lambda }_{2}=0.25 , λ 3 = 3.0025 {\lambda }_{3}=3.0025 , γ = 1.75 \gamma =1.75 , and ε = 1.0075 \varepsilon =1.0075 depict the physical structure of the ℳ 32 ( x , t ) {{\mathcal{ {\mathcal M} }}}_{32}\left(x,t) . (a) 3D visualization of traveling wave at ν = 0.9 \nu =0.9 , (b) contour visualization of traveling wave at ν = 0.9 \nu =0.9 , (c) 2D visualization of traveling wave at c = 0.9 c=0.9 , (d) 3D visualization of traveling wave at ν = 1.9 \nu =1.9 , (e) contour visualization of traveling wave at ν = 1.9 \nu =1.9 , (f) 2D visualization of traveling wave at ν = 1.9 \nu =1.9 , (g) 3D visualization of traveling wave at ν = 2.9 \nu =2.9 , (h) contour of traveling wave at ν = 2.9 \nu =2.9 , and (i) 2D visualization of traveling wave at ν = 2.9 \nu =2.9 .$

Figure 6

The parameters λ 1 = 2.4 , λ 2 = 0.25 , λ 3 = 3.0025 , γ = 1.75 , and ε = 1.0075 depict the physical structure of the ℳ 32 ( x , t ) . (a) 3D visualization of traveling wave at ν = 0.9 , (b) contour visualization of traveling wave at ν = 0.9 , (c) 2D visualization of traveling wave at c = 0.9 , (d) 3D visualization of traveling wave at ν = 1.9 , (e) contour visualization of traveling wave at ν = 1.9 , (f) 2D visualization of traveling wave at ν = 1.9 , (g) 3D visualization of traveling wave at ν = 2.9 , (h) contour of traveling wave at ν = 2.9 , and (i) 2D visualization of traveling wave at ν = 2.9 .

4 Visual representation

In this section, we present the visual representation of some selected traveling wave solutions of the considered model. For visualization of the graphs, we utilized the modern software Wolfram mathematica in the shapes of 3D, 2D, and contour surfaces, which gives a better understanding of the wave pattern. By selecting suitable values of the parameters at different wave velocities, we check the behavior of the graph, which shows the impact of the wave velocities on the traveling wave solution. These graphical analyses reveal the characteristics and internal structure of the wave pattern.

Figures 1–3 exhibit graphical representations of some selected solutions employing the NSE method. Figures 4–6 show the graphical illustration of some selected wave solutions via the MK method. These graphs clearly show the propagation of waves and make the results easy to understand.

In Figure 1, by utilizing the parameters λ 1 = 0.4 , λ 2 = 0.25 , λ 3 = 1.0025 , γ = 1.75 , and ε = 3.0075 at wave velocity ν = 0.9 , to wave solution ℳ 1 ( x , t ) , the 3D graph reflects the pattern of multiple kink. By increasing the wave velocity, the behavior of the graph remains the same. 2D and contour surfaces are also plotted by utilizing the same parameters and velocities.
In Figure 2, we utilized the soliton solution ℳ 5 ( x , t ) with the same parameters to 3D predict the flat kink pattern. At wave velocities ν = 1.9 and ν = 2.9 , the pattern of the graph remains the same as before, like flat kink. Additionally, 2D and contour are also depicted at the same values of the parameters, which give the internal structure of the wave.
In Figure 3, which utilized the soliton solution ℳ 6 ( x , t ) with the same values of parameters, the graph reflected a kink-peakon shape. Due to the variation in wave velocity at ν = 1.9 and ν = 2.9 , the behavior of the graph changes to a kink pattern, showing the impact of the wave velocity.
Select the ℳ 27 ( x , t ) soliton solution in Figure 4 with the values of parameters λ 1 = 2.4 , λ 2 = 2.25 , λ 3 = 2.0025 , γ = 2.75 , and ε = 3.0075 at ν = 0.9 . The graph illustrates the multiple-kink wave structure. By increasing the wave velocity, the behavior of the graph pattern remains the same.
Figure 5 represents the ℳ 31 ( x , t ) soliton solution with suitable parameter values λ 1 = 2.4 , λ 2 = 2.25 , λ 3 = 2.0025 , γ = 2.75 , and ε = 3.0075 at ν = 0.9 , ν = 1.9 , and ν = 2.9 wave velocities, behaving as kink shape.
In Figure 6, 3D predicts of the singular kink shape are carried out by using the parameter values λ 1 = 2.4 , λ 2 = 0.25 , λ 3 = 3.0025 , γ = 1.75 , and ε = 1.0075 at different wave velocities ν = 1.9 and ν = 2.9 . Additionally, 2D and contour are also depicted at the same values of the parameters, which give the internal structure of the wave.

The results presented illustrate the wide range of soliton structures that build in the dynamical system when specific parameter values and wave velocities are adjusted. Our newly developed results are predicted to make an important contribution to future challenges related to nonlinear evaluation equations.

4.1 Comparison with previous study

In this section, we compare the results of the current study with the existing literature. Mathanaranjan et al. [16] developed dark soliton, singular soliton, singular periodic soliton, and bright soliton. Sagidullayeva et al. [42] carefully investigated the lax pair representation and gauge counterparts of the AE. Kong and Guo [43], using the Darbox transformation method, developed the semi-rational solutions 1-breather, 2-breather, and rogue wave solutions. This comparison demonstrates a variety of soliton solutions investigated in previous studies, as well as the different aspects of our research in this field of study.

By utilizing the NSE method and MK method, we obtained different and novel wave structures, which have many applications in surface and curved geometry. We developed the different types of wave solutions in the form of 3D, 2D, and contour surfaces at different wave velocities. The graphical representation shows multiple-kink shapes, flat kink, kink-peakon, and singular-kink soliton patterns, which show that our work is original and novel. Bifurcation assessment, chaos analysis, and sensitivity visualization were not utilized in AE, to the best of our knowledge. We apply these features to the dynamical system, which explains the qualitative study of the AE. Bifurcation plays an important role in soliton dynamics due to the way it impacts the behavior and characteristics of solitons in various systems. Sensitivity visualization explains how the parametric values impact the system.

5 Dynamical assessment

In the current section of our study, we conduct sensitivity visualization and bifurcation assessment of the dynamical system.

5.1 Sensitivity visualization

Now, by applying the Galilean transformation process to Eq. (9), we obtain the following dynamical system:

(97) d ℳ d ξ = τ , d τ d ξ = − 2 γ ε ℳ 3 − ( − ν + 2 κ ( − 2 + κ β ) ( α − β ν ) ) β ( α − β ν ) ℳ .

The objective is to investigate the changes in the system under different initial conditions through sensitivity analysis. We may find out more about the sensitivity of the system to initial conditions by systematically changing these initial guesses and evaluating how the system’s diverges or converges. The transformation of Eq. (9) into a new dynamical system (97) with parameters ν = 0.45 , κ = 0.20 , β = 1.5 , α = 0.2 , and ε = 1.2 is examined in this section. Choosing suitable initial conditions is important to determine the system’s sensitivity.

This dynamical system, governed by the specified parameters, provides a window into how the system behaves under different initial conditions by employing sensitivity analysis. It allows us to understand the complicated interactions between system dynamics and initial conditions, which opens up new avenues for understanding complex systems and decision-making. Our study shows a wide variety of parameter values, as shown in Figure 7, which illustrate how even little changes in the input can have an important effect on the outcomes. Figure 7 shows how small changes in the initial conditions of the system can have an immense effect on the dynamics of the curve. Implementing this procedure with variation in initial condition at the same level and parametric values gives results as shown in Figure 7(a)–(d). The result suggests that the model exhibits crucial system sensitivity.

Figure 7

Sensitivity demonstration. (a) Sensitivity assessment at (0.02, 0.25) and (0.25, 0.02) for curves 1 and 2, respectively, (b) sensitivity assessment at (0.02, 0.003) and (0.003, 0.02) for curves 1 and 2, respectively, (c) sensitivity assessment at (0.02, 0.04) and (0.04, 0.02) for the curves 1 and 2, respectively, (d) sensitivity assessment at (0.05, 0.003) and (0.003, 0.05) for curves 1 and 2, respectively.

Sensitivity assessment has a variety of applications in risk management, finance, and decision-making.

5.2 Bifurcation assessment

In this section, we examine the bifurcation analysis of the dynamical system Eq. (97), which explains the qualitative behaviors of model. Bifurcation plays an important role in soliton dynamics due to the way it impacts the behavior and characteristics of solitons in various systems. The dynamical system of Eq. (9) can be represented as:

(98) d ℳ d ξ = τ , d τ d ξ = − P 1 ℳ 3 + P 2 ℳ ,

where P 1 = − 2 γ ε , P 2 = − ( − ν + 2 κ ( − 2 + κ β ) ( α − β ν ) ) β ( α − β ν ) .

Define the Hamiltonian function of Eq. (98) as:

ℋ ( ℳ , τ ) = τ 2 2 + 1 4 P 1 ℳ 4 − 1 2 P 2 ℳ 2 = C ,

where C is the Hamiltonian constant. To obtain the equilibrium points of system (97) as follows:

(99) κ = 0 , − P 1 ℳ 3 + P 2 ℳ = 0 ,

For solving the above system, obtain the following equilibrium points:

ℬ 1 = ( 0 , 0 ) , ℬ 2 = S 2 S 1 , 0 , ℬ 3 = − S 2 S 1 , 0 .

Now, by using the definition of the Jacobian matrix, we have developed the Jacobean matrix, and after obtaining the determinant of (99), it is as follows:

(100) D ( ℳ , κ ) = 0 1 − 3 P 1 ℳ 2 + P 2 0 = 3 P 1 ℳ 2 − P 2 .

We know that

( ℳ , τ ) is a saddle point when D ( ℳ , τ ) < 0 .
( ℳ , τ ) is a center when D ( ℳ , τ ) > 0 .
( ℳ , τ ) is a cuspidal point when D ( ℳ , τ ) = 0 .

Variations in the parameters may result in the following outcomes:

Case 1: When S 1 > 0 , S 2 > 0 , by selecting the parameters as γ = − 1 , β = 1 , κ = 1 , α = 1 , and ν = 1 , we determine the three equilibrium points ℬ 1 = ( 0 , 0 ) , ℬ 2 = ( 1 , 0 ) , and ℬ 3 = ( − 1 , 0 ) as shown in Figure 8(a). ℬ 1 = ( 0 , 0 ) is saddle point while ℬ 2 = ( 1 , 0 ) and ℬ 3 = ( − 1 , 0 ) are center.

$Figure 8 Graphical analysis of phase depiction. (a) Phase portraits when S 1 > 0 , S 2 > 0 {{\mathcal{S}}}_{1}\gt 0,{{\mathcal{S}}}_{2}\gt 0 , (b) phase portraits when S 1 < 0 , S 2 > 0 {{\mathcal{S}}}_{1}\lt 0,{{\mathcal{S}}}_{2}\gt 0 , (c) phase portraits when S 1 > 0 , S 2 < 0 {{\mathcal{S}}}_{1}\gt 0,{{\mathcal{S}}}_{2}\lt 0 , and (d) phase portraits when S 1 < 0 , S 2 < 0 {{\mathcal{S}}}_{1}\lt 0,{{\mathcal{S}}}_{2}\lt 0 .$

Figure 8

Graphical analysis of phase depiction. (a) Phase portraits when S 1 > 0 , S 2 > 0 , (b) phase portraits when S 1 < 0 , S 2 > 0 , (c) phase portraits when S 1 > 0 , S 2 < 0 , and (d) phase portraits when S 1 < 0 , S 2 < 0 .

Case 2: When S 1 < 0 , S 2 > 0 , by choosing the following parameters as γ = 1 , β = 1 , κ = 1 , α = 1 , and ν = 2 , only one real point is obtained, which is ℬ 1 = ( 0 , 0 ) as displayed in Figure 8(b).

Case 3: S 1 > 0 , S 2 < 0 , by selecting the parameters as γ = − 1 , β = 1 , κ = 1 , α = 1 , and ν − 3 , we obtain the only real point, i.e., ℬ 1 = ( 0 , 0 ) (the center point) displayed in Figure 8(c).

Case 4: S 1 < 0 , S 2 < 0 , by selecting the parameters as γ = − 1 , β = 1 , κ = 1 , and α = 1 , we obtain the three equilibrium points that are ℬ 1 = ( 0 , 0 ) , ℬ 2 = ( 1 , 0 ) , and ℬ 3 = ( − 1 , 0 ) , as shown in Figure 8(d).

5.3 Chaos visualization

This section delves into the periodic, quasi-periodic chaotic, and quasi-periodic resonant fluctuations of the solitary wave profiles within the perturbed dynamical system (97). To facilitate simulations, we introduce a perturbation factor χ 0 , cos ( ℑ , t ) in Eq. (97). Simulation tools can better understand this system’s dynamics, including 3D phase portrait analysis, Poincare section, and time series analysis in the 2D presence of nonlinear periodic solitary waves.

(101) d ℳ d ξ = τ , d τ d ξ = − 2 γ ε ℳ 3 − ( − ν + 2 κ ( − 2 + κ β ) ( α − β ν ) ) β ( α − β ν ) ℳ + χ 0 cos t ( ℑ t ) .

The dynamical pattern of the system is greatly influenced by perturbation factor [45] and the initial conditions. Therefore, we examine various parameter ranges and systematically adjust the amplitudes ( ℑ ) and angles ( χ ).

The dynamical plane’s perturbed system (101) is shown in Figure 9, where ℑ = 1.5 , χ 0 = 0.7 , and (0,0.25) is the initial condition. The 3D, time series and Poincare graphs show the system’s periodic behavior. Eq. (101) regulates the perturbed system, whose quasi-periodic behavior is depicted by the graphs in Figure 10. ℑ = 2.2 and χ 0 = 1.2 and the initial condition of (0, 0.25) has been used to initialize the system. To understand the system’s behavior, the plot consists of a time series and 2D and 3D graphs that provide specific details about the system’s dynamics. Additionally, Figure 11 shows the perturbed (101) quasi-periodic chaotic behavior with the initial condition set to (0, 0.25) at ℑ = 2.9 and χ 0 = 1.9 .

$Figure 9 Analysis of periodic pattern with the parameters, γ = 0.2 \gamma =0.2 , ν = 0.5 \nu =0.5 , κ = 3.2 \kappa =3.2 , ε = 0.02 \varepsilon =0.02 , β = 0.2 \beta =0.2 , α = 0.03 \alpha =0.03 and the perturbation factor. (a) Phase portrait, (b) Poincare, and (c) ξ \xi and κ \kappa .$

Figure 9

Analysis of periodic pattern with the parameters, γ = 0.2 , ν = 0.5 , κ = 3.2 , ε = 0.02 , β = 0.2 , α = 0.03 and the perturbation factor. (a) Phase portrait, (b) Poincare, and (c) ξ and κ .

$Figure 10 Analysis of quasi-periodic pattern with the parameters, γ = 0.2 \gamma =0.2 , ν = 0.5 \nu =0.5 , κ = 3.2 \kappa =3.2 , ε = 0.02 \varepsilon =0.02 , β = 0.2 \beta =0.2 , α = 0.03 \alpha =0.03 and the perturbation factor. (a) Phase portrait, (b) Poincare, and (c) ξ \xi and κ \kappa .$

Figure 10

Analysis of quasi-periodic pattern with the parameters, γ = 0.2 , ν = 0.5 , κ = 3.2 , ε = 0.02 , β = 0.2 , α = 0.03 and the perturbation factor. (a) Phase portrait, (b) Poincare, and (c) ξ and κ .

$Figure 11 Analysis of quasi-periodic chaotic pattern with the parameters, γ = 0.2 \gamma =0.2 , ν = 0.5 \nu =0.5 , κ = 3.2 \kappa =3.2 , ε = 0.02 \varepsilon =0.02 , β = 0.2 \beta =0.2 , α = 0.03 \alpha =0.03 and the perturbation factor. (a) Phase portrait, (b) Poincare, and (c) ξ \xi and κ \kappa .$

Figure 11

Analysis of quasi-periodic chaotic pattern with the parameters, γ = 0.2 , ν = 0.5 , κ = 3.2 , ε = 0.02 , β = 0.2 , α = 0.03 and the perturbation factor. (a) Phase portrait, (b) Poincare, and (c) ξ and κ .

6 Conclusion

This study successfully developed and analyzed traveling wave solutions for the AE using the NSE and MK methods. Utilizing Wolfram Mathematica, the study provided detailed 3D, 2D, and contour visualizations, which effectively illustrated the behavior of multiple solitonic structures, including multiple kink solitons, flat kink solitons, kink-peakon, kink solitons, and singular kink solitons under various parametric conditions and wave velocities. These visualizations unveiled the complex internal dynamics of the system and deepened the understanding of the solitary wave solutions. The governing equation was transformed into a dynamical system using the Galilean transformation, enabling further qualitative analysis through planar dynamical system theory. Sensitivity visualization and bifurcation analysis were conducted to explore how changes in parameters affect the system’s dynamics. Bifurcation analysis, presented via phase portraits, highlighted critical shifts in soliton behavior across various systems, providing key insights into soliton dynamics. Additionally, the chaos analysis revealed a range of dynamic behaviors, including periodic, quasi-periodic, and chaotic trajectories, as presented in Figures 9–11, illustrating the system’s complex response to minor parameter variations. The findings of this study offer valuable theoretical contributions to understanding the dynamic behavior of the AE and the transmission of solitary wave solutions in nonlinear systems. The original solutions derived are novel and hold potential applications in fields such as nonlinear optical fibers, surface geometry, and curve analysis. These results expand the current understanding of nonlinear wave dynamics and open new pathways for future research in the study of complex soliton systems.

Acknowledgements

The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-47).

Funding information: This research was funded by Taif University, Saudi Arabia, Project No. (TU DSPP-2024-47).
Author contributions: M.I.A. and A.A.S. worked together to define the problem and conduct a formal analysis. M.I.A. and Y.S.H. were responsible for developing the investigation approach and methodology. D.C. and M.I.A. provided guidance and resources throughout the process. A.A.S. and Y.S.H. were responsible for software development, validation, and graphical discussions. M.I.A. was involved in the manuscript review. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

References

[1] Li R, Sinnah ZA, Shatouri ZM, Manafian J, Aghdaei MF, Kadi A. Different forms of optical soliton solutions to the Kudryashov’s quintuple self-phase modulation with dual-form of generalized nonlocal nonlinearity. Results Phys. 2023 Mar 1;46:106293. 10.1016/j.rinp.2023.106293Suche in Google Scholar

[2] Li Z, Liu C. Chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion. Results Phys. 2024 Jan 1;56:107305. 10.1016/j.rinp.2023.107305Suche in Google Scholar

[3] Li Z, Hussain E. Qualitative analysis and optical solitons for the (1+1)-dimensional Biswas-Milovic equation with parabolic law and nonlocal nonlinearity. Results Phys. 2024 Jan 1;56:107304. 10.1016/j.rinp.2023.107304Suche in Google Scholar

[4] Liu X, Abd Alreda B, Manafian J, Eslami B, Aghdaei MF, Abotaleb M, et al. Computational modeling of wave propagation in plasma physics over the Gilson-Pickering equation. Results Phys. 2023 Jul 1;50:106579. 10.1016/j.rinp.2023.106579Suche in Google Scholar

[5] Alshanti WG, Batiha IM, Khalil R. A novel analytical approach for solving partial differential equations via a tensor product theory of Banach spaces. Partial Differ Equ Appl Math. 2023 Dec 1;8:100531. 10.1016/j.padiff.2023.100531Suche in Google Scholar

[6] Li R, Manafian J, Lafta HA, Kareem HA, Uktamov KF, Abotaleb M. The nonlinear vibration and dispersive wave systems with cross-kink and solitary wave solutions. Int J Geometric Methods Modern Phys. 2022 Sep 15;19(10):2250151. 10.1142/S0219887822501511Suche in Google Scholar

[7] Alquran M. Classification of single-wave and bi-wave motion through fourth-order equations generated from the Ito model. Phys Scr. 2023 Jul 5;98(8):085207. 10.1088/1402-4896/ace1afSuche in Google Scholar

[8] Alquran M, Najadat O, Ali M, Qureshi S. New kink-periodic and convex-concave-periodic solutions to the modified regularized long wave equation by means of modified rational trigonometric-hyperbolic functions. Nonl Eng. 2023 Aug 4;12(1):20220307. 10.1515/nleng-2022-0307Suche in Google Scholar

[9] Ghiasi M, Niknam T, Wang Z, Mehrandezh M, Dehghani M, Ghadimi N. A comprehensive review of cyber-attacks and defense mechanisms for improving security in smart grid energy systems: past, present and future. Electric Power Syst Res. 2023 Feb 1;215:108975. 10.1016/j.epsr.2022.108975Suche in Google Scholar

[10] Abdulazeez ST, Modanli M. Analytic solution of fractional order pseudo-hyperbolic telegraph equation using modified double Laplace transform method. Int J Math Comput Eng. 2023;1(1):105–114. 10.2478/ijmce-2023-0008Suche in Google Scholar

[11] Saeedi M, Moradi M, Hosseini M, Emamifar A, Ghadimi N. Robust optimization based optimal chiller loading under cooling demand uncertainty. Appl Therm Eng. 2019 Feb 5;148:1081–91. 10.1016/j.applthermaleng.2018.11.122Suche in Google Scholar

[12] Ghanbari B, Baleanu D. New optical solutions of the fractional Gerdjikov-Ivanov equation with conformable derivative. Front Phys. 2020 May 15;8:167. 10.3389/fphy.2020.00167Suche in Google Scholar

[13] Akram G, Arshed S, Sadaf M, Maqbool M. Comparison of fractional effects for Phi-4 equation using beta and M-truncated derivatives. Opt Quantum Electron. 2023 Mar;55(3):282. 10.1007/s11082-023-04549-7Suche in Google Scholar

[14] Khater M, Ghanbari B. On the solitary wave solutions and physical characterization of gas diffusion in a homogeneous medium via some efficient techniques. Europ Phys J Plus. 2021 Apr;136(4):1–28. 10.1140/epjp/s13360-021-01457-1Suche in Google Scholar

[15] Sagher AA, Majid SZ, Asjad MI, Muhammad T. Analyzing optical solitary waves in Fokas system equation insight mono-mode optical fibres with generalized dynamical evaluation. Opt Quantum Electron. 2024 Apr 5;56(5):873. 10.1007/s11082-024-06697-wSuche in Google Scholar

[16] Mathanaranjan T, Myrzakulov R. Integrable Akbota equation: conservation laws, optical soliton solutions and stability analysis. Opt Quantum Electron. 2024 Jan 30;56(4):564. 10.1007/s11082-023-06227-0Suche in Google Scholar

[17] Manukure S, Booker T. A short overview of solitons and applications. Partial Differ Equ Appl Math. 2021 Dec 1;4:100140. 10.1016/j.padiff.2021.100140Suche in Google Scholar

[18] Khater MM. Physics of crystal lattices and plasma; analytical and numerical simulations of the Gilson-Pickering equation. Results Phys. 2023 Jan 1;44:106193. 10.1016/j.rinp.2022.106193Suche in Google Scholar

[19] Zayed EM, Gepreel KA, Shohib RM, Alngar ME, Yıldırım Y. Optical solitons for the perturbed Biswas-Milovic equation with Kudryashov’s law of refractive index by the unified auxiliary equation method. Optik. 2021 Mar 1;230:166286. 10.1016/j.ijleo.2021.166286Suche in Google Scholar

[20] Khater MM, Jhangeer A, Rezazadeh H, Akinyemi L, Akbar MA, Inc M, et al. New kinds of analytical solitary wave solutions for ionic currents on microtubules equation via two different techniques. Opt Quantum Electron. 2021 Nov;53:1–27. 10.1007/s11082-021-03267-2Suche in Google Scholar

[21] Mannaf MA, Islam ME, Bashar H, Basak US, Akbar MA. Dynamic behavior of optical self-control soliton in a liquid crystal model. Results Phys. 2024 Feb 1;57:107324. 10.1016/j.rinp.2024.107324Suche in Google Scholar

[22] Durur H, Yokuş A, Duran S. Investigation of exact soliton solutions of nematicons in liquid crystals according to nonlinearity conditions. Int J Modern Phys B. 2024 Feb 10;38(4):2450054. 10.1142/S0217979224500541Suche in Google Scholar

[23] Tian SF, Xu MJ, Zhang TT. A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation. Proc R Soc A. 2021 Nov 24;477(2255):20210455. 10.1098/rspa.2021.0455Suche in Google Scholar

[24] Khodadad FS, Mirhosseini-Alizamini SM, Günay B, Akinyemi L, Rezazadeh H, Inc M. Abundant optical solitons to the Sasa-Satsuma higher-order nonlinear Schrödinger equation. Opt Quantum Electron. 2021 Dec;53(12):702. 10.1007/s11082-021-03338-4Suche in Google Scholar

[25] Ali KK, Yokus A, Seadawy AR, Yilmazer R. The ion sound and Langmuir waves dynamical system via computational modified generalized exponential rational function. Chaos Solitons Fractals. 2022 Aug 1;161:112381. 10.1016/j.chaos.2022.112381Suche in Google Scholar

[26] Abdulkadir Sulaiman T, Yusuf A. Dynamics of lump-periodic and breather waves solutions with variable coefficients in liquid with gas bubbles. Waves Random Complex Media. 2023 Jul 4;33(4):1085–98. 10.1080/17455030.2021.1897708Suche in Google Scholar

[27] Khater MM. Soliton propagation under diffusive and nonlinear effects in physical systems; (1+1)-dimensional MNW integrable equation. Phys Lett A. 2023 Aug 28;480:128945. 10.1016/j.physleta.2023.128945Suche in Google Scholar

[28] Khater MM. Horizontal stratification of fluids and the behavior of long waves. Europ Phys J Plus. 2023 Aug 15;138(8):715. 10.1140/epjp/s13360-023-04336-zSuche in Google Scholar

[29] Asjad MI, Faridi WA, Alhazmi SE, Hussanan A. The modulation instability analysis and generalized fractional propagating patterns of the Peyrard-Bishop DNA dynamical equation. Opt Quantum Electron. 2023 Mar;55(3):232. 10.1007/s11082-022-04477-ySuche in Google Scholar

[30] Khater MM. Characterizing shallow water waves in channels with variable width and depth; computational and numerical simulations. Chaos Solitons Fractals. 2023 Aug 1;173:113652. 10.1016/j.chaos.2023.113652Suche in Google Scholar

[31] Majid SZ, Asjad MI, Faridi WA. Solitary traveling wave profiles to the nonlinear generalized Calogero-Bogoyavlenskii-Schiff equation and dynamical assessment. Europ Phys J Plus. 2023 Nov 24;138(11):1040. 10.1140/epjp/s13360-023-04681-zSuche in Google Scholar

[32] Khater MM. Advancements in computational techniques for precise solitary wave solutions in the (1+1)-dimensional Mikhailov-Novikov-Wang equation. Int J Theoret Phys. 2023 Jul 14;62(7):152. 10.1007/s10773-023-05402-zSuche in Google Scholar

[33] Khater MM Long waves with a small amplitude on the surface of the water behave dynamically in nonlinear lattices on a non-dimensional grid. Int J Modern Phys B. 2023 Jul 30;37(19):2350188. 10.1142/S0217979223501886Suche in Google Scholar

[34] Kumar S, Niwas M. Exploring lump soliton solutions and wave interactions using new Inverse (G/G)-expansion approach: applications to the (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation. Nonl Dyn. 2023 Nov;111(21):20257–73. 10.1007/s11071-023-08937-2Suche in Google Scholar

[35] Kumar S, Kukkar A. Dynamics of several optical soliton solutions of a (3+1)-dimensional nonlinear Schrödinger equation with parabolic law in optical fibers. Modern Phys Lett B. 2024 Jun 22;2024:2450453. 10.1142/S0217984924504530Suche in Google Scholar

[36] Mann N, Rani S, Kumar S, Kumar R. Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation. Math Comput Simulat. 2024 May 1;219:473–90. 10.1016/j.matcom.2023.12.031Suche in Google Scholar

[37] Niwas M, Kumar S, Rajput R, Chadha D. Exploring localized waves and different dynamics of solitons in (2+1)-dimensional Hirota bilinear equation: a multivariate generalized exponential rational integral function approach. Nonl Dyn. 2024;2:9431–44.10.1007/s11071-024-09555-2Suche in Google Scholar

[38] Kumar S, Dhiman SK. Exploring cone-shaped solitons, breather, and lump-forms solutions using the lie symmetry method and unified approach to a coupled breaking soliton model. Phys Script. 2024 Jan 24;99(2):025243. 10.1088/1402-4896/ad1d9eSuche in Google Scholar

[39] Kumar S, Mohan B. A generalized nonlinear fifth-order KdV-type equation with multiple soliton solutions: Painlevé analysis and Hirota Bilinear technique. Phys Scripta. 2022 Nov 24;97(12):125214. 10.1088/1402-4896/aca2faSuche in Google Scholar

[40] Kumar S, Hamid I. New interactions between various soliton solutions, including bell, kink, and multiple soliton profiles, for the (2+1)-dimensional nonlinear electrical transmission line equation. Opt Quantum Electron. 2024 Jul;56(7):1–24. 10.1007/s11082-024-06960-0Suche in Google Scholar

[41] Uddin S, Karim S, Alshammari FS, Roshid HO, Noor NF, Hoque F, et al. Bifurcation analysis of travelling waves and multi-rogue wave solutions for a nonlinear pseudo-parabolic model of visco-elastic Kelvin-Voigt fluid. Math Probl Eng. 2022;2022(1):8227124. 10.1155/2022/8227124Suche in Google Scholar

[42] Sagidullayeva Z, Yesmakhanova K, Serikbayev N, Nugmanova G, Yerzhanov K, Myrzakulov R. Integrable generalized Heisenberg ferromagnet equations in 1+1 dimensions: reductions and gauge equivalence. 2022 May 4. arXiv: http://arXiv.org/abs/arXiv:2205.02073. Suche in Google Scholar

[43] Kong HY, Guo R. Dynamic behaviors of novel nonlinear wave solutions for the Akbota equation. Optik. 2023 Jul 1;282:170863. 10.1016/j.ijleo.2023.170863Suche in Google Scholar

[44] Tripathy A, Sahoo S, Rezazadeh H, Izgi ZP, Osman MS. Dynamics of damped and undamped wave natures in ferromagnetic materials. Optik. 2023 Jun 1;281:170817. 10.1016/j.ijleo.2023.170817Suche in Google Scholar

[45] Lakshmanan M, Rajaseekar S. Nonlinear dynamics: integrability, chaos and patterns. Berlin/Heidelberg, Germany: Springer Science & Business Media; 2012 Dec 6. Suche in Google Scholar

Received: 2024-07-25

Revised: 2024-09-10

Accepted: 2024-09-13

Published Online: 2025-02-25

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

Artikel in diesem Heft

https://doi.org/10.1515/nleng-2024-0040

Schlagwörter für diesen Artikel

Akbota equation; modified Khater method; new sub-equation method; soliton; bifurcation; sensitivity visualization

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