Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives

Mohammed Aldandani; Syed T. R. Rizvi; Abdulmohsen Alruwaili; Aly R. Seadawy

doi:10.1515/phys-2024-0080

Article Open Access

Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives

Mohammed Aldandani , Syed T. R. Rizvi , Abdulmohsen Alruwaili and Aly R. Seadawy

Published/Copyright: September 16, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Open Physics Volume 22 Issue 1

Abstract

This research explores new soliton solutions to the Atangana–Baleanu derivative (ABD) fractional system of equations for ion sound and Langmuir waves (FISLW). We utilize the fractional ABD operator to convert our system into an ordinary differential equations. In recent years, machine learning (ML) evolves significantly in the context of data analysis and computing different solutions, which typically enables systems to operate wisely. Now, we are going to use numerous ML tools including matplotlib. pyplot as plt, scipy.integrate, mpl toolkits.mplot3d, and Axes3D to generate various types of optical solutions by using complete discriminant of the polynomial method. We will also analyze solutions for the hyperbolic function, trigonometric function, Jacobian elliptic function (JEF), and other solitary wave solutions. Solitons have extensive uses in pure and applied mathematics, including nonlinear partial differential equations: the Boussinesq equation, the nonlinear Schrödinger equation, and the sine-Gordon equation, Lie groups, Lie algebras, and differential and algebraic geometry. In addition, we study the chaotic behaviour, i.e., 2D, 3D, time series, Poincarè maps, and sensitivity analysis of our governing model. Sensitivity analysis explores how changes in a system’s variables affects its behaviour.

Keywords: Atangana–Baleanu derivative; qualitative analysis; complete discriminant of the polynomial method; solitary waves

1 Introduction

Over the past 25 years, investigations on soliton solutions have been one of the most significant research areas. Solitons are solitary waves (SWs) that preserve their form and speed while traveling with a constant velocity. In a number of applied scientific fields, the idea of soliton has been used. By using a technique known as the inverse scattering transform (IST), which is essentially a nonlinear analogue of the Fourier transform for linear partial differential equations, it has been possible to determine that many dynamical systems that are known to be “integrable” have soliton solutions. Currently, integer-order (nonlinear partial differential equations (NLPDEs)) have been used to formulate a variety of real phenomena. Solitons are a useful tool for modelling natural events in almost all engineering problems. The fundamental reason for this is that NLPDEs and nonlinear evolution equations (NLEEs) may be used for modelling each and every natural event that occurs in our environment. These supermodels are investigated in many scientific fields, including physics, optics, engineering, chemistry, and biology. Due to its importance for research and potential applications, nonlinear equations has received a lot of attention in recent years [1–4]. The nonlocal nonlinear Schrodinger equation (NLSE) governs the propagation of nonlocal nonlinear media in optics. On the other hand, as laser technology has advanced, it has been discovered that angular momentum applies a torque to the beams, there is a lot of interest in the propagation of beams and solitons in strongly nonlocal nonlinear media [5–8]. Numerous studies have been conducted on the propagation of beams and solitons in strongly nonlocal nonlinear media. Sun et al. investigated the coherent interaction between solitons and airy beams in nonlocal nonlinear media [9]. Shen et al. specifically determined a soliton solution of the nonlocal NLSE in nonlocal nonlinear media employing complex-valued astigmatic cosine-Gaussian in transmission of light intensity patterns and second-order moment beam widths [10]. In Laguerre–Gaussian and Hermite–Gaussian solitons solution, the propagation expression was also produced by Song et al., along with the evaluation of the propagation parameters, which included the axial light intensity, phase change, and intensity distribution [11]. By using nonzero boundary conditions, Zou and Guo methodically provided the IST for the higher-order G-I problem. By resolving the matrix Riemann–Hilbert problem, they have derived the expression for the N-soliton solution with the reflectionless potentials [12]. Shen et al. investigated the propagation properties of complex-valued hyperbolic-cosine-Gaussian beams in strongly nonlocal nonlinear media using the nonlocal nonlinear Schrödinger equation [13–15]. Li and Guo looked into semi-rational solutions on periodic backdrops for the coupled Lakshmanan–Porsezian–Daniel equations as well as nonlinear waves such as solitons, breathers, rogue waves, and others. In addition, the dynamic behaviours of various nonlinear waves were investigated along with their interactions [16].

Different methods have been systematized in fractional NLPDEs to identify this type of similarity because it is insufficient to employ integer order when the nonlocal attribute does not show in these forms. Certain derivatives can effectively explain the relationship among the several definitions of fractional derivatives in the local and nonlocal categories. A notable benefit of fractional derivatives is their ability to adjust the memories, inheritance characteristics in various materials, and processes. One of the shortcomings of the well-known Riemann–Liouville and Caputo derivatives is the singularity of the kernel in these derivatives [17]. Jeelani et al. choose the Mittag–Leffler nonsingular function as the fractional derivative kernel [18]. This function is important in the theory of fractional calculus [19]. Using various integration strategies like ( ϕ 6 )-model expansion [20], ( G ′ G )-expansion [21], tan( Φ ( ρ ) 2 )-expansion [22], Kudryashove scheme [23], exp( ( − Ψ ′ Ψ ) η )-expansion [24], extended auxiliary equation technique [25], and so on, these operators have been used to estimate the precise and numerical solutions of fractional order NLPDEs. Li et al. used the Riemann–Hilbert method to solve the cauchy problem of the general N-soliton solutions. They utilized parameter modulation to examine their multisoliton solutions in depth like parallel propagation, elastic collision, soliton reflection, time-periodic propagation, and (space, time)-periodic propagation. Several hypotheses regarding the dynamic characteristics of N-soliton solutions were observed. Several extremely amazing characteristics of integral NLEE included traveling wave solutions, which typically took the shape of SWs, precise N-soliton solutions, bilinear forms, and bi-Hamiltonian structures [26]. Li et al. discussed the Cauchy problem of the Wadati–Konno–Ichikawa (WKI) equation for asymptotic stability of N-soliton and their resolution with finite density initial data [27]. They also studied the WKI and complex short equations with ∂ -steepest descent method having initial conditions in weighted Sobolev space H ( R ) [28,29]. In this article, we consider fractional system of equations for ion sound and Langmuir waves (FISLW) model as follows [30]:

(1) i D t β AB a + 1 2 a x x − b a = 0 , D 2 AB β t b − b x x − 2 ( ∣ a ∣ 2 ) x x = 0 , t > 0 , 0 < β ≤ 1 ,

where a e i α q t and b , respectively, represent the normalized electric field of the intensity disturbance and the wavelength of the Langmuir oscillation. Normalized variables are x and t , and D t β AB is the AB fractional operator in the t direction. The definition of an Atangana–Baleanu derivative (ABD) operator is as follows:

(2) D c + β ABD P ( t ) = Z ( β ) 1 − β d d t ∫ c t P ( x ) F β − β ( t − β ) β 1 − β d x ,

F β is the Mittag–Leffler function, defined as follows:

(3) F β − β ( t − β ) β 1 − β = ∑ n = 0 ∞ − β 1 − β g ( t − x ) β g Γ ( β g + 1 ) ,

Z ( β ) is a normalized function, which satisfies Z ( 1 ) = Z ( 0 ) = 1 . Since, we have

(4) D c + β ABD P ( t ) = Z ( β ) 1 − β ∑ n = 0 ∞ − β 1 − β g ( I β β g RL P ( t ) ) .

For further information on this operator’s properties, this leads to the following form:

(5) a ( x , t ) = h ( u ) e i γ , b ( x , t ) = j ( u ) ,

where γ = c x + α ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) , u = l x + ζ ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) , and β , α are arbitrary constants.

(6) 1 2 l 2 h ″ + i ( ζ + c l ) h ′ − 1 2 ( c 2 + 2 α ) h − h j = 0 , ( ζ 2 − l 2 ) h ″ − 4 l 2 ( h ′ 2 + h h ″ ) = 0 ,

where h and j are the function of u . Now by solving imaginary part of above first equation we have

(7) ζ + c l = 0 , ζ = − c l ,

and finally, by integrating the second part of Eq. (6) twice with respect to u , we obtain

(8) j = 2 l 2 h 2 − l 2 + ζ 2 = 2 h 2 c 2 − 1 ,

Eqs (7) and (8) alter Eq. (6) into the following form:

(9) h ″ − 4 h 3 l 2 ( c 2 − 1 ) − ( c 2 + 2 α ) h l 2 = 0 ,

The objective of our current work is to apply the complete discriminant of the polynomial method (CDSPM) approach via machine learning (ML) technologies to find exact solutions of Eq. (1), sensitivity analysis and Quasi periodic behaviour through the usage of several Python packages [31,32]. A subfield of computer science is called ML. Nowadays, everything is linked to a data source, and our entire life is digitally recorded because we are living in the era of data. The fundamentals of various ML approaches in a wide range of real-world application areas, including e-commerce, health care, agriculture, smart cities, cyber security, and among many others. The two primary goals of ML are to streamline model-related data and generate future outcomes that are predicted by these models. Mainly, the quality and attributes of the data as well as the functionality of the learning algorithms determine how successful and efficient a ML solution is. To improve the organization and depiction of our model, we offer a thorough overview of the many kinds of ML techniques in this work.

The remaining part of this article is arranged as follows: Section 3 examines the quasi-periodic behaviour of the governing model; sensitivity analysis is presented in Section 4; exact and solitary wave solutions are offered in Sections 6 and 7; a detailed explanation of the results is provided in Section 8; and a summary of the study is given in Section 9.

2 Analysis of the method

To determine the soliton solutions of NLPDEs, we give a brief overview of CDSPM, that has been utilized to solve numerous nonlinear models [33,34]. first, consider NLPDE:

(10) M ( Θ , Θ t , Θ x x , … ) = 0 .

Step 1: By using the transformation Θ ( x , t ) = Θ ( ε ) exp i κ , and ε = x − ω t . We have the following ordinary differential equation (ODE):

(11) M ( Θ , Θ ′ , Θ ″ , … ) = 0 ,

Step 2: After various modifications, the previously mentioned nonlinear ODE is reduced to the following ODE form:

(12) ( Θ ′ ) 2 = Θ 4 + q 2 Θ 2 + q 1 Θ + q 0 ,

The integral form is as follows:

(13) ± ( ε − ε 0 ) = ∫ d Θ Θ 4 + q 2 Θ 2 + q 1 Θ + q 0 .

Step 3: Let, P ( Θ ) = Θ 4 + q 2 Θ 2 + q 1 Θ + q 0 . By using CDSPM of fourth-order polynomial, we have

(14) Y 1 = 4 , Y 2 = − q 2 , Y 3 = − 2 q 2 3 + 8 q 2 q 0 − 9 q 1 2 , Y 4 = − q 2 3 q 1 2 + 4 q 2 2 q 0 + 36 q 2 q 1 2 q 0 − 32 q 2 2 q 0 − − 27 4 q 1 4 + 63 q 0 3 , Z 2 = 9 q 2 2 − 32 q 2 q 0 .

It will be possible to categorize the equation’s solutions.

3 Mathematical description

In this section, the first step is integrate after multiply Eq. (6) by h ′ is as follows [35–39]:

(15) ( h ′ ) 2 2 − h 4 l 2 ( c 2 − 1 ) − ( c 2 + 2 α ) l 2 h 2 2 + G = 0 ,

Now multiply above Eq. (15) by 2 then we have,

(16) ( h ′ ) 2 = 2 h 4 l 2 ( c 2 − 1 ) + ( c 2 + 2 α ) h 2 l 2 − 2 G

where G is integration constant. Let a 1 = 2 l 2 ( c 2 − 1 ) , a 2 = ( c 2 + 2 α ) l 2 , a 4 = 2 G , and a 3 = 0 to determine the fourth degree polynomial qualitative behaviour. Then Eq. (16) becomes:

(17) ( h ′ ) 2 = a 1 h 4 − a 2 a 1 h 2 + a 3 a 1 h − a 4 a 1 .

Since, above equation transformed is as follows:

(18) ( h ′ ) 2 = a 1 ( h 4 + ξ 2 h 2 + ξ 1 h − ξ 0 ) ,

where ξ 2 = a 2 a 1 , ξ 1 = a 3 a 1 , and ξ 0 = a 4 a 1 . The a 1 < 0 and a 1 > 0 cases are now addressed in Eq. (18). Eq. (18)’s integral form is:

(19) ∫ d h h 4 + ξ 2 h 2 + ξ 1 h − ξ 0 = ± a 1 ( u − u 0 ) ,

It contains an integration constant, i.e., u 0 . In this instance, ϒ ( h ) = h 4 + ξ 2 h 2 + ξ 1 h − ξ 0 and CDSPM as follows:

(20) E 1 = 4 , E 2 = − ξ 2 , E 3 = − 2 ξ 2 3 + 8 ξ 2 ξ 0 − 9 ξ 1 2 , E 4 = − ξ 2 3 ξ 1 2 + 4 ξ 2 2 ξ 0 + 36 ξ 2 ξ 1 2 ξ 0 − 32 ξ 2 2 ξ 0 − − 27 4 ξ 1 4 + 63 ξ 0 3 , F 2 = 9 ξ 2 2 − 32 ξ 2 ξ 0 .

Here is a representation of the dynamical system of Eq. (18) (Table 1).

(21) h ′ = q ( u ) , q ′ = a 1 ( 4 h 3 − 2 ξ 2 h + ξ 1 ) ,

Table 1

Regression learner python libraries for qualitative analysis, exact, and SW solutions

Library	Usage
Pandas and numpy	Data analysis
matplotlib.pyplot as plt	Graphical and plotting interface
mpl toolkits.mplot3d	Access to further abilities for creating and modifying 3D plots
Axes3D	To create and manipulate 3D axes
solve_ivp	Designed for resolving first order ode problems easily also for dynamical system
scipy.integrate	Offers a platform for conducting multiple approaches of numerical integration

4 Phase patterns for quasi-periodic behaviour

Quasi-periodic behaviour is made up of a number of chaotic and predictable patterns. It occurs when several wave motions are joined mistakenly. For this reason, as compared to a fundamental pattern, the pattern is nonrepeating and intricate. It is not particularly organized, but it’s not as chaotic as chaos either. Through analyzing particular elements of these patterns and identifying a structure within the complexity, we may be able explore more deeply. Sophisticated phase patterns associated with quasi-periodic behaviour provide a unique perspective on dynamic systems [40]. We now explore how the system exhibits quasi-periodic and chaotic behaviour, which is as follows:

(22) h ′ = q ( u ) , q ′ = a 1 ( 4 h 3 − 2 ξ 2 h + ξ 1 ) + ϖ 0 cos ( ζ ψ ) ,

where a 1 = 2 l 2 ( c 2 − 1 ) , ξ 2 = a 2 a 1 , ξ 1 = a 3 a 1 , and a 3 = 0 .

Here, ζ indicates the frequency of the disruption and ϖ 0 indicates its strength in the system (22) that was already presented. In system (21), the external periodic force of ϖ 0 cos ( ζ ψ ) , which is present in system (22) is absent and required to examine the periodic and chaotic patterns of Eq. (1). Our approaches for addressing this issue include phase portrait approach, time series profile, and Poincarè maps. To study the problem from various perspectives, we will analyze the influence of the parameters ξ 1 , ξ 2 , a 1 , ϖ 0 , and ζ using two different situations. We keep ξ 1 , ξ 2 , a 1 , and ζ unchanged for the first condition and examine the impact of ϖ 0 . In the other situation, we will examine the effects of changing additional elements: ξ 1 , ξ 2 , and a 1 while maintaining ϖ 0 and ζ constants.

For the following values: a 1 = − 0.96 , ξ 2 = 4.28 , ξ 1 = 0 , ϖ 0 = 3.5 , and ζ = 2 π , Figure 1 shows Poincarè maps, 3D, 2D phase plots, and time analysis graphs. System (22) shows up to be functioning quasi-periodically at ϖ 0 = 3.5 . With the same components as shown in Figures 1 and 2 for ϖ 0 = 2.5 shows 2D, 3D plots, time analysis graphs, and Poincarè maps. System (22), for ϖ 0 = 2.5 , seems to exhibit quasi-periodic characteristics. With the following values, a 1 = − 3.3 , ξ 2 = − 0.28 , ξ 1 = 0 , ϖ 0 = 1.5 , and ζ = 2 π , Figure 3 displays time series profiles, Poincarè maps, and 3D, 2D plots. System (22) appears to demonstrate quasi-periodic features, whereas the Poincarè section includes several disordered points indicating chaotic motion for the parameters that have been specified. While keeping the same value of ϖ 0 = 1.5 , Figure 4 displays time series profiles, 3D and 2D plots, and Poincarè maps for a 1 = − 1.96 , ξ 2 = − 3.28 , ξ 1 = 0 , and ζ = 2 π . The quasi-periodic pattern of System (22) combines with the chaotic behaviour and significant variety of discontinuous points of the Poincarè map. For a very long period, the chaos has been researched. Poincarè is usually recognized to be the first person to study chaos.

$Figure 1 Poincarè maps, 3D, 2D phase plots, and time analysis graphs with a 1 = ‒ 0.96 {a}_{1}=‒0.96 , ξ 2 = 4.28 {\xi }_{2}=4.28 , ξ 1 = 0 {\xi }_{1}=0 , ϖ 0 = 3.5 {\varpi }_{0}=3.5 , and ζ = 2 π \zeta =2\pi .$

Figure 1

Poincarè maps, 3D, 2D phase plots, and time analysis graphs with a 1 = ‒ 0.96 , ξ 2 = 4.28 , ξ 1 = 0 , ϖ 0 = 3.5 , and ζ = 2 π .

$Figure 2 The use of several chaos-detecting techniques for the initial condition (0.08, 1.08) indicates the nonlinear dynamical system (22) for the following values: a 1 = ‒ 0.96 {a}_{1}=‒0.96 , ξ 2 = 4.28 {\xi }_{2}=4.28 , ξ 1 = 0 {\xi }_{1}=0 , ϖ 0 = 3.5 {\varpi }_{0}=3.5 , and ζ = 2 π \zeta =2\pi .$

Figure 2

The use of several chaos-detecting techniques for the initial condition (0.08, 1.08) indicates the nonlinear dynamical system (22) for the following values: a 1 = ‒ 0.96 , ξ 2 = 4.28 , ξ 1 = 0 , ϖ 0 = 3.5 , and ζ = 2 π .

$Figure 3 Time series profiles, Poincarè maps, and 3D, 2D plots With the following values, a 1 = ‒ 3.3 {a}_{1}=‒3.3 , ξ 2 = ‒ 0.28 {\xi }_{2}=‒0.28 , ξ 1 = 0 {\xi }_{1}=0 , ϖ 0 = 1.5 {\varpi }_{0}=1.5 , and ζ = 2 π \zeta =2\pi .$

Figure 3

Time series profiles, Poincarè maps, and 3D, 2D plots With the following values, a 1 = ‒ 3.3 , ξ 2 = ‒ 0.28 , ξ 1 = 0 , ϖ 0 = 1.5 , and ζ = 2 π .

$Figure 4 Considering the nonlinear dynamical system (22) with the initial condition (1.08, 1.08) determined through numerous chaos-detecting computations for the following values: a 1 = ‒ 0.96 {a}_{1}=‒0.96 , ξ 2 = 4.28 {\xi }_{2}=4.28 , ξ 1 = 0 {\xi }_{1}=0 , ϖ 0 = 2.5 {\varpi }_{0}=2.5 , and ζ = 2 π \zeta =2\pi .$

Figure 4

Considering the nonlinear dynamical system (22) with the initial condition (1.08, 1.08) determined through numerous chaos-detecting computations for the following values: a 1 = ‒ 0.96 , ξ 2 = 4.28 , ξ 1 = 0 , ϖ 0 = 2.5 , and ζ = 2 π .

5 Sensitivity analysis

A mathematical model or system’s output variation or uncertainty can be attributed to modifications or uncertainties in its input variables by using the sensitivity analysis approach. It presents important insights into the functioning of the model and helps decision-makers fully understand the impact of various scenarios and uncertainties [40]. The objective was initially approached from three distinct beginning points: the black curve plots ( h , q ) = ( 0.18 , 0 ) , the blue curve plots ( h , q ) = ( 0.17 , 0 ) , and the red solid line indicates ( h , q ) = ( 0.16 , 0 ) . Maintaining the identical beginning conditions under examination, two distinct situations are investigated.

Now, with a small modification to the beginning values shown in Figures 5 and 6, the outcomes are obtained with the same properties as in the preceding system. This indicates that the outcome is unaffected by even slight changes to the starting conditions. As a result, we observe that the proposed explanation is not very sensitive.

Figure 5

Time series, Poincarè maps, and 2D and 3D phase patterns.

$Figure 6 The approximate values of the nonlinear dynamical system (76) are a 1 = ‒ 3.3 {a}_{1}=‒3.3 , ξ 2 = ‒ 0.28 {\xi }_{2}=‒0.28 , ξ 1 = 0 {\xi }_{1}=0 , ϖ 0 = 1.5 {\varpi }_{0}=1.5 , and ζ = 2 π \zeta =2\pi , whereas the initial condition (1.08, 1.08) demonstrates a variety of chaos-detecting strategies.$

Figure 6

The approximate values of the nonlinear dynamical system (76) are a 1 = ‒ 3.3 , ξ 2 = ‒ 0.28 , ξ 1 = 0 , ϖ 0 = 1.5 , and ζ = 2 π , whereas the initial condition (1.08, 1.08) demonstrates a variety of chaos-detecting strategies.

6 Exact solutions

It has been demonstrated by earlier research that periodic and soliton solutions are authentic. Using CDSPM to ascertain the precise solution is the next step. The subsequent nine options are examined in the manner described as follows:

Case 1: If E 2 < 0 , E 3 = 0 , and E 4 = 0 ,

(23) ϒ ( h ) = ( ( h − M ) 2 + N 2 ) 2 ,

where a 1 > 0 For, N > 0 putting Eq. (23) into Eq. (19). Then, we obtain

(24) h = M + N tan ( N a 1 ( u − u 0 ) ) .

In this way, we evaluate the following solutions

(25) a ( x , t ) = ( M + N tan ( N a 1 ( u − u 0 ) ) ) e i γ ,

(26) b ( x , t ) = 2 l 2 − l 2 + ζ 2 ( M + N tan ( N a 1 ( u − u 0 ) ) ) 2 e i γ ,

where γ = c x + α ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) , u = l x + ζ ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) . Eqs (25) and (26) give triangular periodic solution.

Case 2: E 2 > 0 , E 3 = 0 , E 4 = 0 , and F 2 > 0 . Then,

(27) ϒ ( h ) = ( h − R ) 2 ( h − S ) 2 ,

where h < S , h > R , a 1 > 0 , and R < S . Since, we obtain

(28) h = S − R 2 coth ( S − R ) a 1 ( u − u 0 ) 2 − 1 + S ,

when S < h < R , thus, we have

(29) h = S − R 2 tanh ( S − R ) a 1 ( u − u 0 ) 2 − 1 + S ,

In addition, we reach at the following conclusions:

(30) a ( x , t ) = S − R 2 coth ( S − R ) a 1 ( u − u 0 ) 2 − 1 + S e i γ .

(31) a ( x , t ) = S − R 2 tanh ( S − R ) a 1 ( u − u 0 ) 2 − 1 + S e i γ .

(32) b ( x , t ) = 2 l 2 − l 2 + ζ 2 S − R 2 coth ( S − R ) a 1 ( u − u 0 ) 2 − 1 ) + S ) 2 e i γ .

(33) b ( x , t ) = 2 l 2 − l 2 + ζ 2 S − R 2 tanh ( S − R ) a 1 ( u − u 0 ) 2 − 1 ) + S ) 2 e i γ .

Here γ = c x + α ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) , u = l x + ζ ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) .

Eqs (30)–(33) represents SW solitons.

Case 3: E 2 = 0 , E 3 = 0 , and E 4 = 0 . Since,

(34) ϒ ( h ) = h 4 .

Then, we have

(35) h = − 1 a 1 ( u 0 − u ) − 1 ,

where a 1 > 0 . Hence, the following solutions are obtained

(36) a ( x , t ) = − 1 a 1 ( u 0 − u ) − 1 e i γ ,

(37) b ( x , t ) = 2 l 2 − l 2 + ζ 2 − 1 a 1 ( u 0 − u ) − 1 2 e i γ ,

where γ = c x + α ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) , u = l x + ζ ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) .

Eqs (36) and (37) are a rational function solution.

Case 4: E 2 > 0 , E 3 = 0 , E 4 = 0 , and F 2 = 0 . Thus,

(38) ϒ ( h ) = ( h − χ 1 ) 3 ( h − χ 2 ) ,

where a 1 < 0 , h > χ 1 , h > χ 2 , or h < χ 1 , h < χ 2 . Hence, we have

(39) h = 4 a 1 ( χ 1 − χ 2 ) ( χ 2 − χ 1 ) 2 ( u − u 0 ) 2 − 4 + χ 1 ,

(40) h = 4 a 1 ( χ 1 − χ 2 ) − ( χ 2 − χ 1 ) 2 ( u − u 0 ) 2 − 4 + χ 1 .

In addition, we have the following solutions:

(41) a ( x , t ) = 4 a 1 ( χ 1 − χ 2 ) ( χ 2 − χ 1 ) 2 ( u − u 0 ) 2 − 4 + χ 1 e i γ .

(42) a ( x , t ) = 4 a 1 ( χ 1 − χ 2 ) − ( χ 2 − χ 1 ) 2 ( u − u 0 ) 2 − 4 + χ 1 e i γ .

(43) b ( x , t ) = 2 l 2 − l 2 + ζ 2 4 a 1 ( χ 1 − χ 2 ) ( χ 2 − χ 1 ) 2 ( u − u 0 ) 2 − 4 + χ 1 2 e i γ .

(44) b ( x , t ) = 2 l 2 − l 2 + ζ 2 4 a 1 ( χ 1 − χ 2 ) − ( χ 2 − χ 1 ) 2 ( u − u 0 ) 2 − 4 + χ 1 2 e i γ .

Here γ = c x + α ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) , u = l x + ζ ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) .

Eqs (41)–(44) gives rational function solutions.

Case 5: E 2 E 3 < 0 , and E 4 = 0 . Then,

(45) ϒ ( h ) = ( h − β 1 ) 2 ( ( h − β 2 ) 2 + β 3 2 ) ,

where β 1 , β 2 , and β 3 are real coefficients. Since, we have

(46) h = ( e ± ( β 1 − β 2 ) 2 + β 3 2 a 1 ( u − u 0 ) − ν ) + ( β 1 − β 2 ) 2 + β 3 2 ( 2 − ν ) ( e ± ( β 1 − β 2 ) 2 + β 3 2 a 1 ( u − u 0 ) − ν ) 2 − 1 ,

where a 1 > 0 and ν = β 1 − 2 β 2 ( β 1 − β 2 ) 2 + β 3 2 .

(47) a ( x , t ) = ( e ± ( β 1 − β 2 ) 2 + β 3 2 a 1 ( u − u 0 ) − ν ) + ( β 1 − β 2 ) 2 + β 3 2 ( 2 − ν ) ( e ± ( β 1 − β 2 ) 2 + β 3 2 a 1 ( u − u 0 ) − ν ) 2 − 1 e i γ ,

(48) b ( x , t ) = 2 l 2 − l 2 + ζ 2 × ( e ± ( β 1 − β 2 ) 2 + β 3 2 a 1 ( u − u 0 ) − ν ) + ( β 1 − β 2 ) 2 + β 3 2 ( 2 − ν ) ( e ± ( β 1 − β 2 ) 2 + β 3 2 a 1 ( u − u 0 ) − ν ) 2 − 1 2 e i γ ,

where γ = c x + α ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) , u = l x + ζ ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) .

Eqs (47) and (48) represents SW solution.

Case 6: E 2 > 0 , E 3 > 0 , and E 4 > 0 . Since,

(49) ϒ ( h ) = ( h − p 1 ) ( h − p 2 ) ( h − p 3 ) ( h − p 4 ) ,

where p 1 > p 2 > p 3 > p 4 , and p 1 , p 2 , p 3 , and p 4 are real numbers. If h > p 1 or h < p 4 , then we obtain

(50) h = p 2 ( p 1 − p 4 ) sn 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν − p 1 ( p 2 − p 4 ) ( p 1 − p 4 ) sn 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν − ( p 2 − p 4 ) .

On the other hand, if p 3 < h < p 2 , then we obtain

(51) h = p 4 ( p 2 − p 3 ) sn 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν − p 3 ( p 2 − p 4 ) ( p 2 − p 3 ) sn 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν − ( p 2 − p 4 ) ,

where a 1 > 0 and ν = ( p 1 − p 4 ) ( p 2 − p 3 ) ( p 1 − p 3 ) ( p 2 − p 4 ) . Furthermore, we have the following circumstances:

(52) a ( x , t ) = p 2 ( p 1 − p 4 ) sn 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν − p 1 ( p 2 − p 4 ) ( p 1 − p 4 ) sn 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν − ( p 2 − p 4 ) e i γ ,

(53) a ( x , t ) = p 4 ( p 2 − p 3 ) sn 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν − p 3 ( p 2 − p 4 ) ( p 2 − p 3 ) sn 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν − ( p 2 − p 4 ) e i γ ,

(54) b ( x , t ) = 2 l 2 − l 2 + ζ 2 p 2 ( p 1 − p 4 ) sn 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν − p 1 ( p 2 − p 4 ) ( p 1 − p 4 ) sn 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν − ( p 2 − p 4 ) 2 e i γ ,

(55) b ( x , t ) = 2 l 2 − l 2 + ζ 2 p 4 ( p 2 − p 3 ) sn 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν − p 3 ( p 2 − p 4 ) ( p 2 − p 3 ) sn 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν − ( p 2 − p 4 ) 2 e i γ ,

where γ = c x + α ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) , u = l x + ζ ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) .

Eqs (52)–(55) solutions represents jacobian elliptic double periodic soliton.

Case 7: E 2 E 3 ≥ 0 , and E 4 < 0 . Since,

(56) ϒ ( h ) = ( h − A ) ( h − B ) ( ( h − C ) 2 + D 2 ) ,

where A , B , C , and D are all real numbers. If A > B , D > 0 , then

(57) c 1 = 1 2 ( A + B ) d 1 − 1 2 ( A − B ) d 2 , c 2 = 1 2 ( A + B ) d 2 − 1 2 ( A − B ) d 1 , d 1 = A − C − D s 2 , d 2 = A − C − D s 2 , s 1 = D 2 + ( A − C ) ( B − C ) D ( A − B ) , s 2 = s 1 ± s 1 2 + 1 .

Since,

(58) h = c 1 cn ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) , ν + c 2 d 1 cn ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) , ν + d 2 ,

where a 1 > 0 , ν = 2 1 + s 2 2 and s 2 > 0 . As a result, we obtain the following solutions:

(59) a ( x , t ) = c 1 cn ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) , ν + c 2 d 1 cn ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) , ν + d 2 e i γ ,

(60) b ( x , t ) = 2 l 2 − l 2 + ζ 2 c 1 cn ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) , ν + c 2 d 1 cn ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) , ν + d 2 2 e i γ ,

where γ = c x + α ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) , u = l x + ζ ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) .

Eqs (59) and (60) are jacobian elliptic double periodic soliton.

Case 8: E 2 E 3 ≤ 0 , and E 4 > 0 . Then,

(61) ϒ ( h ) = ( ( h − w 1 ) 2 + w 2 2 ) ( ( h − w 3 ) 2 + w 4 2 ) ,

where w 2 ≥ w 4 > 0 and w 1 , w 2 , w 3 , and w 3 are real coefficients. Since,

(62) c 1 = w 1 d 1 + w 2 d 2 c 2 = w 1 d 1 − w 2 d 2 , d 1 = − w 1 − w 2 s 2 , d 2 = w 1 − w 3 , s 1 = ( w 1 − w 3 ) 2 + w 2 2 + w 4 2 2 w 2 w 4 , s 2 = s 1 + s 1 2 − 1 .

Hence, we obtain

(63) h = c 1 sn ( η a 1 ( u − u 0 ) , ν ) + c 2 cn ( η a 1 ( u − u 0 ) , ν ) c 2 sn ( η a 1 ( u − u 0 ) , ν ) + d 2 cn ( η a 1 ( u − u 0 ) , ν ) ,

where a 1 > 0 , ν 2 = s 2 2 − 1 s 2 2 , and η = w 4 ( d 1 2 + d 2 2 ) ( s 2 2 d 1 2 + d 2 2 ) d 1 2 + d 2 2 . Subsequently, we acquire

(64) a ( x , t ) = c 1 sn ( η a 1 ( u − u 0 ) , ν ) + c 2 cn ( η a 1 ( u − u 0 ) , ν ) c 2 sn ( η a 1 ( u − u 0 ) , ν ) + d 2 cn ( η a 1 ( u − u 0 ) , ν ) e i γ ,

(65) b ( x , t ) = 2 l 2 − l 2 + ζ 2 c 1 sn ( η a 1 ( u − u 0 ) , ν ) + c 2 cn ( η a 1 ( u − u 0 ) , ν ) c 2 sn ( η a 1 ( u − u 0 ) , ν ) + d 2 cn ( η a 1 ( u − u 0 ) , ν ) 2 e i γ ,

where γ = c x + α ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) , u = l x + ζ ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) .

Eqs (64) and (65) are double periodic elliptic function solution.

Case 9: E 3 > 0 , E 4 = 0 , and E 2 > 0 . Since,

(66) ϒ ( h ) = ( h − s ) 2 ( h − a ) ( h − n ) ,

where s = − a + n 2 , p = ( s − a ) ( s − n ) and s , a , and n are real numbers.

If ϒ > a and a > h > n , then

(67) h = 2 p 2 a + ( a − n ) sinh ( − p a 1 ( u − u 0 ) ) ,

and if a 1 > 0 , s > a and s < n , then

(68) h = 2 p 2 a + ( a − n ) cosh ( p a 1 ( u − u 0 ) ) ,

Consequently, we obtain

(69) a ( x , t ) = 2 p 2 a + ( a − n ) sinh ( − p a 1 ( u − u 0 ) ) e i γ ,

(70) a ( x , t ) = 2 p 2 a + ( a − n ) cosh ( p a 1 ( u − u 0 ) ) e i γ ,

(71) b ( x , t ) = 2 l 2 − l 2 + ζ 2 2 p 2 a + ( a − n ) sinh ( − p a 1 ( u − u 0 ) ) 2 e i γ ,

(72) b ( x , t ) = 2 l 2 − l 2 + ζ 2 2 p 2 a + ( a − n ) cosh ( p a 1 ( u − u 0 ) ) 2 e i γ ,

where γ = c x + α ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) , u = l x + ζ ( 1 − β ) t − g Z ( β ) ∑ n = 0 ∞ − β 1 − β g Γ ( 1 − β g ) .

Eqs (69)–(72) give SW solutions.

Conclusions indicate that the CDSPM allows us to immediately offer a variety of solutions in explicit form and characterize the crucial area.

7 The SW solutions

Analyzing multiple exact and precise localized solutions scattered throughout a fiber optic system is an interesting problem. To obtain the SW solution in the aforementioned instance, we convert JEF to hyperbolic and trigonometric functions. In the long-wave limit, the JEF transforms into trigonometric and hyperbolic functions, representing ν → 0 and ν → 1 , accordingly.

7.1 Double hyperbolic type-I

By using the limit ν → 1 in Eqs (52)–(55), it becomes

( sn ) 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν → tanh 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) .

The SW solution is provided by [41]:

(73) a ( x , t ) = p 2 ( p 1 − p 4 ) tanh 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − p 1 ( p 2 − p 4 ) ( p 1 − p 4 ) tanh 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − ( p 2 − p 4 ) e i γ ,

(74) a ( x , t ) = p 4 ( p 2 − p 3 ) tanh 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − p 3 ( p 2 − p 4 ) ( p 2 − p 3 ) tanh 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − ( p 2 − p 4 ) e i γ ,

(75) b ( x , t ) = 2 l 2 − l 2 + ζ 2 p 2 ( p 1 − p 4 ) tanh 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − p 1 ( p 2 − p 4 ) ( p 1 − p 4 ) tanh 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − ( p 2 − p 4 ) 2 e i γ ,

(76) b ( x , t ) = 2 l 2 − l 2 + ζ 2 p 4 ( p 2 − p 3 ) tanh 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − p 3 ( p 2 − p 4 ) ( p 2 − p 3 ) tanh 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − ( p 2 − p 4 ) 2 e i γ .

7.2 Double periodic type-I

Containing the limit ν → 0 in Eqs. (52)–(55), it turns into

( sn ) 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) , ν → sin 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 )

that offer the SW solution [41]:

(77) a ( x , t ) = p 2 ( p 1 − p 4 ) sin 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − p 1 ( p 2 − p 4 ) ( p 1 − p 4 ) sin 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − ( p 2 − p 4 ) e i γ ,

(78) a ( x , t ) = p 4 ( p 2 − p 3 ) sin 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − p 3 ( p 2 − p 4 ) ( p 2 − p 3 ) sin 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − ( p 2 − p 4 ) e i γ ,

(79) b ( x , t ) = 2 l 2 − l 2 + ζ 2 p 2 ( p 1 − p 4 ) tanh 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − p 1 ( p 2 − p 4 ) ( p 1 − p 4 ) sin 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − ( p 2 − p 4 ) 2 e i γ ,

(80) b ( x , t ) = 2 l 2 − l 2 + ζ 2 p 4 ( p 2 − p 3 ) tanh 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − p 3 ( p 2 − p 4 ) ( p 2 − p 3 ) sin 2 ( p 1 − p 3 ) ( p 2 − p 4 ) 2 a 1 ( u − u 0 ) − ( p 2 − p 4 ) 2 e i γ .

7.3 Double hyperbolic type-II

By using the limit ν → 1 in Eqs. (59) and (60), the function transforms into

cn ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) , ν → sech ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) , ν ,

which demonstrates the SW resolution [41]:

(81) a ( x , t ) = c 1 sech ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) + c 2 d 1 sech ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) + d 2 e i γ

(82) b ( x , t ) = 2 l 2 − l 2 + ζ 2 c 1 sech ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) + c 2 d 1 sech ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) + d 2 2 e i γ .

7.4 Double periodic type-II SW

By taking limit ν → 0 in Eqs (59) and (60), then the function changes into

cn ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) , ν → cos ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) , ν ,

which exhibits the SW accuracy [41]:

(83) a ( x , t ) = c 1 cos ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) + c 2 d 1 cos ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) + d 2 e i γ ,

(84) b ( x , t ) = 2 l 2 − l 2 + ζ 2 c 1 cos ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) + c 2 d 1 cos ± 2 D s 2 ( A − B ) 2 s 2 ν a 1 ( u − u 0 ) + d 2 2 e i γ .

7.5 Double hyperbolic type-III

Considering the limit ν → 1 in Eqs (64) and (65), the functions turns into

sn ( η a 1 ( u − u 0 ) , ν ) → tanh ( η a 1 ( u − u 0 ) )

and

cn ( η a 1 ( u − u 0 ) , ν ) → sech ( η a 1 ( u − u 0 ) , ν ) ,

and this demonstrates the SW resolution [41]:

(85) a ( x , t ) = c 1 tanh ( η a 1 ( u − u 0 ) ) + c 2 sech ( η a 1 ( u − u 0 ) ) c 2 tanh ( η a 1 ( u − u 0 ) ) + d 2 sech ( η a 1 ( u − u 0 ) ) e i γ ,

(86) b ( x , t ) = 2 l 2 − l 2 + ζ 2 × c 1 tanh ( η a 1 ( u − u 0 ) ) + c 2 sech ( η a 1 ( u − u 0 ) ) c 2 tanh ( η a 1 ( u − u 0 ) ) + d 2 sech ( η a 1 ( u − u 0 ) ) 2 e i γ ,

where η = w 4 ( d 1 2 + d 2 2 ) ( s 2 2 d 1 2 + d 2 2 ) d 1 2 + d 2 2 .

7.6 Double periodic type-III

By using the limit μ → 0 in Eqs (64) and (65), the functions turns into

sn ( η a 1 ( u − u 0 ) , ν ) → sin ( η a 1 ( u − u 0 ) )

and

cn ( η a 1 ( u − u 0 ) , ν ) → cos ( η a 1 ( u − u 0 ) , ν ) ,

which indicates the SW solution [41]:

(87) a ( x , t ) = c 1 sin ( η a 1 ( u − u 0 ) ) + c 2 cos ( η a 1 ( u − u 0 ) ) c 2 sin ( η a 1 ( u − u 0 ) ) + d 2 cos ( η a 1 ( u − u 0 ) ) e i γ

and

(88) b ( x , t ) = 2 l 2 − l 2 + ζ 2 × c 1 sin ( η a 1 ( u − u 0 ) ) + c 2 cos ( η a 1 ( u − u 0 ) ) c 2 sin ( η a 1 ( u − u 0 ) ) + d 2 cos ( η a 1 ( u − u 0 ) ) 2 e i γ ,

where η = w 4 ( d 1 2 + d 2 2 ) ( s 2 2 d 1 2 + d 2 2 ) d 1 2 + d 2 2 .

8 Results and discussion

In this section, our findings and the present phase of research on our suggested approach will be compared. Younas et al. investigated the construction of analytical wave solutions to the conformable fractional dynamical system of ion sound and Langmuir waves [35]. Atangana and Koca examined the chaos in a simple nonlinear system with ABD fractional order [36]. Algahtani studied the comparison of ABD and Caputo Fabrizio derivative with fractional order [37]. Alkahtani obtained Chua’s circuit model with ABD with fractional order [38]. Seadawy et al. described the application of mathematical methods on the system of dynamical equations for the ion sound and Langmuir waves [39]. The CDSPM technique is one of the most effective methods for solving NLPDEs. Moreover, we convert our model into a dynamic system and look at the qualitative analysis of the model under discussion displayed in Figures 1–4 to achieve a quasi-periodic behaviour. When a system’s trajectory is plotted in time space, a quasi- periodic cycle can be seen as a dense layer covering a toroidal surface (a surface of revolution with a hole in the middle). Toroidal shapes demonstrate quasi periodicity, as their motions cover their surfaces extensively without precisely repeating. Also when an external periodic strength is applied in dynamical system, quasi-periodic behaviours are observed. A number of methods are discussed to identify chaos including: time series, Poincarè maps, and 2D and 3D phase patterns. In addition, sensitivity analysis is conducted under different initial conditions as illustrated in Figures 5, 6, 7, 8. The sensitivity analysis for the three initial conditions shown in red, black, and green can be seen in these figures. Sensitivity analysis along with periodic structure indicate a nonlinear dynamical system. We made a small modification to the starting condition, and with the same parameters, we saw that the solution was unaffected by the change, suggesting that the model we were given was not extremely sensitive.

Figure 7

Time series, Poincarè maps, and 2D and 3D phase patterns.

$Figure 8 The nonlinear dynamical system (22) is described using several chaos-detecting techniques and initial conditions (1.08, 1.08), assuming the following assumptions: a 1 = ‒ 1.96 {a}_{1}=‒1.96 , ξ 2 = ‒ 3.28 {\xi }_{2}=‒3.28 , ξ 1 = 0 {\xi }_{1}=0 , ϖ 0 = 1.5 {\varpi }_{0}=1.5 , and ζ = 2 π \zeta =2\pi .$

Figure 8

The nonlinear dynamical system (22) is described using several chaos-detecting techniques and initial conditions (1.08, 1.08), assuming the following assumptions: a 1 = ‒ 1.96 , ξ 2 = ‒ 3.28 , ξ 1 = 0 , ϖ 0 = 1.5 , and ζ = 2 π .

We are now conducting a detailed analysis of the findings. The periodic solutions for triangle functions obtained from Eqs (25) and (26) are displayed in Figures 9 and 10. In the fields of science and engineering, periodic solutions which exhibit recurring behaviours are crucial. The shapes of these figures remain constant and repeat at regular intervals; notable peaks indicate the point at which the function value is significantly greater than the surroundings. The color shift indicates the function’s magnitude, the color transition depicts the traveling structure, and the abrupt change close to the peak indicates a steep gradient. The SW solutions for Eqs (30)–(33), (47), (48), (69)–(72) are shown in Figures 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20. Because of their unique properties, SW solutions find application in many fields of science, including physics, biology, nonlinear optics, and plasma physics. Furthermore, the surrounding oscillatory pattern suggests that periodic solutions or multi-soliton interactions may exist, these figures represents both the stability and the complex dynamics of the system also the property of troughs and crests represents the stable localized solitons. Figures 21, 22, 23, 24, 25, and 26 show the rational solutions for Eqs (36), (37), and (41)–(44). When considering fields like physics, mathematics, and engineering, geometry plays a crucial role in comprehending the behaviour of rational solutions. The shape of rational soliton is stable when they propagate from one medium to another, and the prominent peak indicates that the shape of the soliton is conserved because it is a component of nonlinear waves. The Jacobi elliptic functions (JEF) are a set of basic elliptic functions in mathematics. Different approaches for JEF double periodic are found in Eqs (52)–(55), (59), (60), (64), and (65) are illustrated in Figures 27, 28, 29, 30, 31, 32, 33, and 34. It is a soliton that is a generalization of trigonometry and hyperbolic functions. It has multiple localized peaks, and the amplitude of these peaks is consistent, indicating a repeating and stable structure. This also suggests that the elliptic function is periodic. Eqs (52)–(55) transformed into SW double hyperbolic type-I and SW double periodic type-I in Eqs (73)–(76) and Eqs (77)–(80), respectively, by applying ν → 1 and ν → 0 , while the variables and characteristics of the solutions demonstrate in Figures 35, 36, 37, 38, 39, 40, 41, 42. Meanwhile, JEF double periodic solutions in Eqs (59) and (60) becomes a double hyperbolic type-II SW in Eqs (81) and (82) and a double periodic type-II SW in Eqs (83) and (46), by utilizing the limit ν → 1 and ν → 0 ; however, Figures 43, 44, 45, and 46 provide a graphical representation of these solutions. Although Eqs (64) and (65) are also the JEF double periodic solution, they become the double hyperbolic type-III in Eqs (85) and (86) and the double periodic type-III in Eqs (87) and (88) upon examination of the limits ν → 1 and ν → 0 . These equations can be seen graphically in Figures 47, 48, 49, 50. A number of soliton solutions provide important insights into the functions performed by wave phenomena in many physical systems. They are useful in quantum mechanics, plasma physics, and telecommunications. They are also crucial to our understanding of the dispersion of stable, confined waveforms. Moreover, solitons have been employed to investigate a broad variety of significant real-world problems in domains like fluid dynamics, astronomy, molecular biology, plasma, and nonlinear optics (Figures 51, 52, 53, 54).

$Figure 9 The variables in the system Eq. (22) sensitivity profile are a 1 = 0.7 {a}_{1}=0.7 , ξ 2 = ‒ 2.3 {\xi }_{2}=‒2.3 , ξ 1 = 0 {\xi }_{1}=0 , ϖ 0 = 3.5 {\varpi }_{0}=3.5 , and ζ = 2 π \zeta =2\pi , with initial values of (0.18, 0) in black, (0.17, 0) in blue, and (0.16, 0) in red.$

Figure 9

The variables in the system Eq. (22) sensitivity profile are a 1 = 0.7 , ξ 2 = ‒ 2.3 , ξ 1 = 0 , ϖ 0 = 3.5 , and ζ = 2 π , with initial values of (0.18, 0) in black, (0.17, 0) in blue, and (0.16, 0) in red.

$Figure 10 With the parameters as follows: a 1 = 1.5 {a}_{1}=1.5 , ξ 2 = ‒ 4.2 {\xi }_{2}=‒4.2 , ξ 1 = 0 {\xi }_{1}=0 , ϖ 0 = 2.5 {\varpi }_{0}=2.5 , and ζ = 2 π \zeta =2\pi , the sensitivity profile of Eq. (2)) is displayed. The beginning points are (0.15, 0) in red, (0.16, 0) in black, and (0.27, 0) in blue.$

Figure 10

With the parameters as follows: a 1 = 1.5 , ξ 2 = ‒ 4.2 , ξ 1 = 0 , ϖ 0 = 2.5 , and ζ = 2 π , the sensitivity profile of Eq. (2)) is displayed. The beginning points are (0.15, 0) in red, (0.16, 0) in black, and (0.27, 0) in blue.

$Figure 11 Sensitivity profile for System (22), with initial values in blue (0.10, 0), red (0.19, 0), and black (0.40, 0), with the following parameters: a 1 = 0.7 {a}_{1}=0.7 , ξ 2 = ‒ 2.3 {\xi }_{2}=‒2.3 , ξ 1 = 0 {\xi }_{1}=0 , ϖ 0 = 3.5 {\varpi }_{0}=3.5 , and ζ = 2 π \zeta =2\pi .$

Figure 11

Sensitivity profile for System (22), with initial values in blue (0.10, 0), red (0.19, 0), and black (0.40, 0), with the following parameters: a 1 = 0.7 , ξ 2 = ‒ 2.3 , ξ 1 = 0 , ϖ 0 = 3.5 , and ζ = 2 π .

$Figure 12 Using starting values of (0.30, 0) in red, (0.46,0) in black, and (0.57, 0) in blue, the variables that constitute the sensitivity of system equation (23) are: a 1 = 1.5 {a}_{1}=1.5 , ξ 2 = ‒ 4.2 {\xi }_{2}=‒4.2 , ξ 1 = 0 {\xi }_{1}=0 , ϖ 0 = 2.5 {\varpi }_{0}=2.5 , and ζ = 2 π \zeta =2\pi .$

Figure 12

Using starting values of (0.30, 0) in red, (0.46,0) in black, and (0.57, 0) in blue, the variables that constitute the sensitivity of system equation (23) are: a 1 = 1.5 , ξ 2 = ‒ 4.2 , ξ 1 = 0 , ϖ 0 = 2.5 , and ζ = 2 π .

$Figure 13 The graphical illustration of a ( x , t ) a\left(x,t) in Eq. (25) can be generated by setting: M = 1.9 M=1.9 , N = ‒ 1.009 N=‒1.009 , and β = ‒ 2.5 \beta =‒2.5 .$

Figure 13

The graphical illustration of a ( x , t ) in Eq. (25) can be generated by setting: M = 1.9 , N = ‒ 1.009 , and β = ‒ 2.5 .

$Figure 14 The graphical representation of b ( x , t ) b\left(x,t) , as defined in Eq. (26) can be developed by modifying: M = 0.2 M=0.2 , N = ‒ 3.009 N=‒3.009 , l = ‒ 4.5 l=‒4.5 , g = 4.008 g=4.008 .$

Figure 14

The graphical representation of b ( x , t ) , as defined in Eq. (26) can be developed by modifying: M = 0.2 , N = ‒ 3.009 , l = ‒ 4.5 , g = 4.008 .

$Figure 15 Eq. (41) illustrates the behaviour of a ( x , t ) a\left(x,t) for: χ 2 = 0.8 {\chi }_{2}=0.8 , g = 1.5 g=1.5 , β = ‒ 5.3 \beta =‒5.3 , and c = 0.6 c=0.6 .$

Figure 15

Eq. (41) illustrates the behaviour of a ( x , t ) for: χ 2 = 0.8 , g = 1.5 , β = ‒ 5.3 , and c = 0.6 .

$Figure 16 Eq. (42) represents the graph of a ( x , t ) a\left(x,t) for: χ 1 = 0.8 {\chi }_{1}=0.8 , g = ‒ 0.5 g=‒0.5 , χ 2 = ‒ 4.3 {\chi }_{2}=‒4.3 , and β = 3.6 \beta =3.6 .$

Figure 16

Eq. (42) represents the graph of a ( x , t ) for: χ 1 = 0.8 , g = ‒ 0.5 , χ 2 = ‒ 4.3 , and β = 3.6 .

$Figure 17 The configuration of dynamical plot of a ( x , t ) a\left(x,t) in Eq. (59) can be observed by using: A = 0.2 , u 0 = 0.3 , B = 0.9 , C = 0.1 A=0.2,{u}_{0}=0.3,B=0.9,C=0.1 , and g = 0.09 g=0.09 .$

Figure 17

The configuration of dynamical plot of a ( x , t ) in Eq. (59) can be observed by using: A = 0.2 , u 0 = 0.3 , B = 0.9 , C = 0.1 , and g = 0.09 .

$Figure 18 The graphical behaviour of b ( x , t ) b\left(x,t) in Eq. (60) can be determined by setting: D = 0.2 D=0.2 , g = 0.3 g=0.3 , A = 0.9 A=0.9 , l = 0.1 l=0.1 , and u 0 = 0.09 {u}_{0}=0.09 .$

Figure 18

The graphical behaviour of b ( x , t ) in Eq. (60) can be determined by setting: D = 0.2 , g = 0.3 , A = 0.9 , l = 0.1 , and u 0 = 0.09 .

$Figure 19 The fluctuating trajectory of a ( x , t ) a\left(x,t) in Eq. (64) can be produced by using: w 1 = 1.8 {w}_{1}=1.8 , g = 2.3 i g=2.3i , c = 1.8 c=1.8 , w 4 = 0.63 {w}_{4}=0.63 , and w 2 = 0.01 {w}_{2}=0.01 .$

Figure 19

The fluctuating trajectory of a ( x , t ) in Eq. (64) can be produced by using: w 1 = 1.8 , g = 2.3 i , c = 1.8 , w 4 = 0.63 , and w 2 = 0.01 .

$Figure 20 The configuration of b ( x , t ) b\left(x,t) in Eq. (65) can be represented by using: u 0 = 4.8 {u}_{0}=4.8 , l = 9.3 i l=9.3i , g = 9.8 g=9.8 , w 1 = 0.85 {w}_{1}=0.85 , w 4 = ‒ 2.84 {w}_{4}=‒2.84 , and β = 1.01 \beta =1.01 .$

Figure 20

The configuration of b ( x , t ) in Eq. (65) can be represented by using: u 0 = 4.8 , l = 9.3 i , g = 9.8 , w 1 = 0.85 , w 4 = ‒ 2.84 , and β = 1.01 .

$Figure 21 To obtained the graphical plot of a ( x , t ) a\left(x,t) in Eq. (30) set: S = 0.8 S=0.8 , R = 0.9 R=0.9 , u 0 = 9.8 {u}_{0}=9.8 , and c = 9.008 c=9.008 .$

Figure 21

To obtained the graphical plot of a ( x , t ) in Eq. (30) set: S = 0.8 , R = 0.9 , u 0 = 9.8 , and c = 9.008 .

$Figure 22 The configuration of a ( x , t ) a\left(x,t) according to Eq. (31) is visualized when: S = 0.9 S=0.9 , u 0 = 0.3 , R = 2.4 , β = 3.006 {u}_{0}=0.3,R=2.4,\beta =3.006 , and g = 6.008 g=6.008 .$

Figure 22

The configuration of a ( x , t ) according to Eq. (31) is visualized when: S = 0.9 , u 0 = 0.3 , R = 2.4 , β = 3.006 , and g = 6.008 .

$Figure 23 The graphical representation of b ( x , t ) b\left(x,t) in Eq. (32) for: S = 0.8 S=0.8 , g = 0.9 g=0.9 , R = 0.4 R=0.4 , and u 0 = 9.8 {u}_{0}=9.8 .$

Figure 23

The graphical representation of b ( x , t ) in Eq. (32) for: S = 0.8 , g = 0.9 , R = 0.4 , and u 0 = 9.8 .

$Figure 24 The visual illustration of b ( x , t ) b\left(x,t) in Eq. (33) for: S = 0.76 S=0.76 , g = 0.5 g=0.5 , R = 0.5 R=0.5 , and u 0 = 8.9 {u}_{0}=8.9 .$

Figure 24

The visual illustration of b ( x , t ) in Eq. (33) for: S = 0.76 , g = 0.5 , R = 0.5 , and u 0 = 8.9 .

$Figure 25 Eq. (36) illustrates the fluctuating trajectory of a ( x , t ) a\left(x,t) for: g = 6.4 g=6.4 , c = 4.05 c=4.05 , β = 2.008 \beta =2.008 , and u 0 = 0.8 {u}_{0}=0.8 .$

Figure 25

Eq. (36) illustrates the fluctuating trajectory of a ( x , t ) for: g = 6.4 , c = 4.05 , β = 2.008 , and u 0 = 0.8 .

$Figure 26 Eq. (37) shows the graphical behaviour of b ( x , t ) b\left(x,t) for: l = 0.4 l=0.4 , c = ‒ 9.05 c=‒9.05 , g = 6.008 g=6.008 , and u 0 = ‒ 9.8 {u}_{0}=‒9.8 .$

Figure 26

Eq. (37) shows the graphical behaviour of b ( x , t ) for: l = 0.4 , c = ‒ 9.05 , g = 6.008 , and u 0 = ‒ 9.8 .

$Figure 27 Eq. (43) gives the graphical behaviour of b ( x , t ) b\left(x,t) for: l = 0.8 l=0.8 , g = ‒ 4.3 g=‒4.3 , u 0 = 3.6 {u}_{0}=3.6 , and c = ‒ 0.5 c=‒0.5 .$

Figure 27

Eq. (43) gives the graphical behaviour of b ( x , t ) for: l = 0.8 , g = ‒ 4.3 , u 0 = 3.6 , and c = ‒ 0.5 .

$Figure 28 Eq. (44) represents the variation of b ( x , t ) b\left(x,t) for: χ 2 = 1.8 {\chi }_{2}=1.8 , χ 1 = ‒ 0.4 {\chi }_{1}=‒0.4 , g = 4.3 g=4.3 , and c = 3.6 c=3.6 .$

Figure 28

Eq. (44) represents the variation of b ( x , t ) for: χ 2 = 1.8 , χ 1 = ‒ 0.4 , g = 4.3 , and c = 3.6 .

$Figure 29 The configuration of b ( x , t ) b\left(x,t) in Eq. (47) shows graphical behaviour for: β = 0.8 \beta =0.8 , β 1 = ‒ 0.5 {\beta }_{1}=‒0.5 , β 2 = ‒ 4.3 {\beta }_{2}=‒4.3 , and β 3 = 3.6 {\beta }_{3}=3.6 .$

Figure 29

The configuration of b ( x , t ) in Eq. (47) shows graphical behaviour for: β = 0.8 , β 1 = ‒ 0.5 , β 2 = ‒ 4.3 , and β 3 = 3.6 .

$Figure 30 The configuration of b ( x , t ) b\left(x,t) in Eq. (48) is visual illustrated by setting: β 3 = 0.8 {\beta }_{3}=0.8 , β 2 = ‒ 0.5 {\beta }_{2}=‒0.5 , l = ‒ 4.3 l=‒4.3 , and c = 3.6 c=3.6 .$

Figure 30

The configuration of b ( x , t ) in Eq. (48) is visual illustrated by setting: β 3 = 0.8 , β 2 = ‒ 0.5 , l = ‒ 4.3 , and c = 3.6 .

$Figure 31 The graphical depiction of a ( x , t ) a\left(x,t) as given by Eq. (52) is presented under the following parameter values: p 1 = 0.5 {p}_{1}=0.5 , p 2 = 4.03 {p}_{2}=4.03 , p 3 = ‒ 0.6 {p}_{3}=‒0.6 , c = ‒ 0.3 c=‒0.3 , g = ‒ 0.4 g=‒0.4 , and β = 0.8 \beta =0.8 .$

Figure 31

The graphical depiction of a ( x , t ) as given by Eq. (52) is presented under the following parameter values: p 1 = 0.5 , p 2 = 4.03 , p 3 = ‒ 0.6 , c = ‒ 0.3 , g = ‒ 0.4 , and β = 0.8 .

$Figure 32 The graphical illustration of a ( x , t ) a\left(x,t) in Eq. (53) for: p 1 = 2.5 {p}_{1}=2.5 , p 2 = 6.03 {p}_{2}=6.03 , p 3 = 1.4 {p}_{3}=1.4 , c = 0.48 c=0.48 , g = ‒ 0.086 g=‒0.086 , and u 0 = 1.8 {u}_{0}=1.8 .$

Figure 32

The graphical illustration of a ( x , t ) in Eq. (53) for: p 1 = 2.5 , p 2 = 6.03 , p 3 = 1.4 , c = 0.48 , g = ‒ 0.086 , and u 0 = 1.8 .

$Figure 33 The visual representation of b ( x , t ) b\left(x,t) in Eq. (54) can be obtained by setting following parameter: p 1 = 0.5 {p}_{1}=0.5 , g = 4.03 g=4.03 , l = ‒ 0.6 l=‒0.6 , p 2 = ‒ 0.3 {p}_{2}=‒0.3 , p 3 = ‒ 0.4 {p}_{3}=‒0.4 , and β = 0.8 \beta =0.8 .$

Figure 33

The visual representation of b ( x , t ) in Eq. (54) can be obtained by setting following parameter: p 1 = 0.5 , g = 4.03 , l = ‒ 0.6 , p 2 = ‒ 0.3 , p 3 = ‒ 0.4 , and β = 0.8 .

$Figure 34 Eq. (55) describes the graphical illustration of b ( x , t ) b\left(x,t) for: p 1 = 0.5 {p}_{1}=0.5 , p 2 = 1.3 {p}_{2}=1.3 , p 4 = 1.4 {p}_{4}=1.4 , β = 4.03 \beta =4.03 , c = 1.6 c=1.6 , and g = 0.4 g=0.4 .$

Figure 34

Eq. (55) describes the graphical illustration of b ( x , t ) for: p 1 = 0.5 , p 2 = 1.3 , p 4 = 1.4 , β = 4.03 , c = 1.6 , and g = 0.4 .

$Figure 35 The graph of a ( x , t ) a\left(x,t) in Eq. (69) are observed under the following conditions: a = 0.8 a=0.8 , g = 0.3 g=0.3 , n = 0.25 n=0.25 , and β = 5.01 \beta =5.01 .$

Figure 35

The graph of a ( x , t ) in Eq. (69) are observed under the following conditions: a = 0.8 , g = 0.3 , n = 0.25 , and β = 5.01 .

$Figure 36 The plot for a ( x , t ) a\left(x,t) in Eq. (70) can be represented by setting: u 0 = 5.8 {u}_{0}=5.8 , g = 8.3 g=8.3 , l = 3.8 l=3.8 , and β = 3.1 \beta =3.1 .$

Figure 36

The plot for a ( x , t ) in Eq. (70) can be represented by setting: u 0 = 5.8 , g = 8.3 , l = 3.8 , and β = 3.1 .

$Figure 37 For the visual representation of b ( x , t ) b\left(x,t) in Eq. (71) by using: β = 2.8 \beta =2.8 , s = 0.6 s=0.6 , g = 0.75 g=0.75 , n = 1.8 n=1.8 , and l = 0.01 l=0.01 .$

Figure 37

For the visual representation of b ( x , t ) in Eq. (71) by using: β = 2.8 , s = 0.6 , g = 0.75 , n = 1.8 , and l = 0.01 .

$Figure 38 Graph of b ( x , t ) b\left(x,t) in Eq. (72) is determined by setting: β = 8.8 \beta =8.8 , a = 0.5 a=0.5 , n = 0.2 n=0.2 , g = 0.854 g=0.854 , and l = 3.68 l=3.68 .$

Figure 38

Graph of b ( x , t ) in Eq. (72) is determined by setting: β = 8.8 , a = 0.5 , n = 0.2 , g = 0.854 , and l = 3.68 .

$Figure 39 Eq. (73) gives the graphical behaviour of a ( x , t ) a\left(x,t) by using following parameters: p 1 = ‒ 0.6 {p}_{1}=‒0.6 , p 2 = 0.08 {p}_{2}=0.08 , p 3 = 0.5 {p}_{3}=0.5 , p 4 = ‒ 0.4 {p}_{4}=‒0.4 , and u 0 = 4.03 {u}_{0}=4.03 .$

Figure 39

Eq. (73) gives the graphical behaviour of a ( x , t ) by using following parameters: p 1 = ‒ 0.6 , p 2 = 0.08 , p 3 = 0.5 , p 4 = ‒ 0.4 , and u 0 = 4.03 .

$Figure 40 Eq. (74) determined the visual illustration of a ( x , t ) a\left(x,t) by using: p 1 = 9.36 {p}_{1}=9.36 , p 2 = 2.1 {p}_{2}=2.1 , p 3 = 1.9 {p}_{3}=1.9 , p 4 = 5.2 {p}_{4}=5.2 , and u = 0.9 u=0.9 .$

Figure 40

Eq. (74) determined the visual illustration of a ( x , t ) by using: p 1 = 9.36 , p 2 = 2.1 , p 3 = 1.9 , p 4 = 5.2 , and u = 0.9 .

$Figure 41 The configuration of b ( x , t ) b\left(x,t) in Eq. (75) can be determined by setting: l = 4.1 l=4.1 , g = 4.238 g=4.238 , p 1 = 3.5 {p}_{1}=3.5 , p 3 = ‒ 4.325 {p}_{3}=‒4.325 , and p 4 = 0.368 {p}_{4}=0.368 .$

Figure 41

The configuration of b ( x , t ) in Eq. (75) can be determined by setting: l = 4.1 , g = 4.238 , p 1 = 3.5 , p 3 = ‒ 4.325 , and p 4 = 0.368 .

$Figure 42 The visual illustration of b ( x , t ) b\left(x,t) in Eq. (76) can be determined by setting: l = 0.3 l=0.3 , g = 2.8 g=2.8 , ζ = ‒ 4.24 \zeta =‒4.24 , p 3 = ‒ 9.325 {p}_{3}=‒9.325 , and p 4 = ‒ 3.368 {p}_{4}=‒3.368 .$

Figure 42

The visual illustration of b ( x , t ) in Eq. (76) can be determined by setting: l = 0.3 , g = 2.8 , ζ = ‒ 4.24 , p 3 = ‒ 9.325 , and p 4 = ‒ 3.368 .

$Figure 43 Eq. (77) shows the graphical behaviour of a ( x , t ) a\left(x,t) by using: p 1 = 0.43 {p}_{1}=0.43 , p 2 = 9.6 {p}_{2}=9.6 , p 3 = ‒ 3.6 {p}_{3}=‒3.6 , p 4 = ‒ 4.3 {p}_{4}=‒4.3 , and u 0 = 0.5 {u}_{0}=0.5 .$

Figure 43

Eq. (77) shows the graphical behaviour of a ( x , t ) by using: p 1 = 0.43 , p 2 = 9.6 , p 3 = ‒ 3.6 , p 4 = ‒ 4.3 , and u 0 = 0.5 .

$Figure 44 Eq. (78) shows the visual illustration of a ( x , t ) a\left(x,t) by setting: p 1 = 2.3 {p}_{1}=2.3 , p 2 = ‒ 0.6 {p}_{2}=‒0.6 , p 3 = 4.05 {p}_{3}=4.05 , and u = 0 u=0 .$

Figure 44

Eq. (78) shows the visual illustration of a ( x , t ) by setting: p 1 = 2.3 , p 2 = ‒ 0.6 , p 3 = 4.05 , and u = 0 .

$Figure 45 The graphical depiction of b ( x , t ) b\left(x,t) as given by Eq. (79) is presented under the following parameter values: l = 0.2 l=0.2 , g = 4.3 g=4.3 , p 1 = 2.6 {p}_{1}=2.6 , and p 4 = ‒ 9.5 {p}_{4}=‒9.5 .$

Figure 45

The graphical depiction of b ( x , t ) as given by Eq. (79) is presented under the following parameter values: l = 0.2 , g = 4.3 , p 1 = 2.6 , and p 4 = ‒ 9.5 .

$Figure 46 For the following parameters: p 1 = 4.6 {p}_{1}=4.6 , p 2 = 1.6 {p}_{2}=1.6 , p 3 = 9.1 {p}_{3}=9.1 , and u 0 = 0.008 {u}_{0}=0.008 , a graph of b ( x , t ) b\left(x,t) is shown in Eq. (80).$

Figure 46

For the following parameters: p 1 = 4.6 , p 2 = 1.6 , p 3 = 9.1 , and u 0 = 0.008 , a graph of b ( x , t ) is shown in Eq. (80).

$Figure 47 Eq. (81) represents the dynamic illustration of a ( x , t ) a\left(x,t) occurs when: A = 0.1 A=0.1 , B = 1.03 B=1.03 , D = ‒ 2.4 D=‒2.4 , and u 0 = 1.09 {u}_{0}=1.09 .$

Figure 47

Eq. (81) represents the dynamic illustration of a ( x , t ) occurs when: A = 0.1 , B = 1.03 , D = ‒ 2.4 , and u 0 = 1.09 .

$Figure 48 For Eq. (82), a dynamical representation of b ( x , t ) b\left(x,t) occurs when: l = 3.6 l=3.6 , g = 4.3 g=4.3 , A = ‒ 3.9 A=‒3.9 , D = 3.9 , D=3.9, and B = 1.9 B=1.9 .$

Figure 48

For Eq. (82), a dynamical representation of b ( x , t ) occurs when: l = 3.6 , g = 4.3 , A = ‒ 3.9 , D = 3.9 , and B = 1.9 .

$Figure 49 Eq. (83) display the graph of a ( x , t ) a\left(x,t) under the following conditions: A = ‒ 3.3 A=‒3.3 , B = 1.4 B=1.4 , D = ‒ 4.2 D=‒4.2 , u 0 = 0.09 {u}_{0}=0.09 , and a 1 = 3.7 {a}_{1}=3.7 .$

Figure 49

Eq. (83) display the graph of a ( x , t ) under the following conditions: A = ‒ 3.3 , B = 1.4 , D = ‒ 4.2 , u 0 = 0.09 , and a 1 = 3.7 .

$Figure 50 Eq. (84) shows the graphical behaviour of b ( x , t ) b\left(x,t) under the following conditions: A = 0.3 A=0.3 , B = 4.4 B=4.4 , l = ‒ 1.2 l=‒1.2 , g = 9.34 g=9.34 , and a 1 = 2.86 {a}_{1}=2.86 .$

Figure 50

Eq. (84) shows the graphical behaviour of b ( x , t ) under the following conditions: A = 0.3 , B = 4.4 , l = ‒ 1.2 , g = 9.34 , and a 1 = 2.86 .

$Figure 51 A graphical depiction of a ( x , t ) a\left(x,t) can be identified in Eq. (85) under the following circumstances: d 1 = ‒ 0.008 {d}_{1}=‒0.008 , d 2 = 0.8 {d}_{2}=0.8 , a 1 = ‒ 0.46 {a}_{1}=‒0.46 , s 2 = ‒ 0.9 {s}_{2}=‒0.9 , and u 0 = 0.8 {u}_{0}=0.8 .$

Figure 51

A graphical depiction of a ( x , t ) can be identified in Eq. (85) under the following circumstances: d 1 = ‒ 0.008 , d 2 = 0.8 , a 1 = ‒ 0.46 , s 2 = ‒ 0.9 , and u 0 = 0.8 .

$Figure 52 A graphical behaviour of b ( x , t ) b\left(x,t) for Eq. (79) presented by using: l = ‒ 2.4 l=‒2.4 , g = 2.56 g=2.56 , d 1 = 1.4 {d}_{1}=1.4 , and u 0 = 0 {u}_{0}=0 .$

Figure 52

A graphical behaviour of b ( x , t ) for Eq. (79) presented by using: l = ‒ 2.4 , g = 2.56 , d 1 = 1.4 , and u 0 = 0 .

$Figure 53 When a 1 = ‒ 0.9 {a}_{1}=‒0.9 , u 0 = 0 {u}_{0}=0 , s 2 = ‒ 0.74 {s}_{2}=‒0.74 , c 1 = 2.7 {c}_{1}=2.7 , and c 2 = 1.5 {c}_{2}=1.5 , the dynamics of a ( x , t ) a\left(x,t) is represented in Eq. (87).$

Figure 53

When a 1 = ‒ 0.9 , u 0 = 0 , s 2 = ‒ 0.74 , c 1 = 2.7 , and c 2 = 1.5 , the dynamics of a ( x , t ) is represented in Eq. (87).

$Figure 54 The visual representation of b ( x , t ) b\left(x,t) in Eq. (88) can be determined by using: s 2 = ‒ 1.8 {s}_{2}=‒1.8 , g = 2.3 g=2.3 , ζ = ‒ 0.43 \zeta =‒0.43 , d 1 = 0.35 {d}_{1}=0.35 , l = 5.9 l=5.9 , and u 0 = 0 {u}_{0}=0 .$

Figure 54

The visual representation of b ( x , t ) in Eq. (88) can be determined by using: s 2 = ‒ 1.8 , g = 2.3 , ζ = ‒ 0.43 , d 1 = 0.35 , l = 5.9 , and u 0 = 0 .

9 Conclusion

With the use of the ABD, we were able to effectively derive a few novel analytical solutions for FISLWS in this research. These exact solutions are obtained in the form of JEF, hyperbolic, trigonometric, and exponential functions. Consequently, rational, periodic, SW, bright and dark solitons, and other new travelling wave solutions are obtained and visually presented in 2D and 3D plots. Also we used ML tools like: solve_ivp, Pandas and numpy, and scipy.integrate in this work. Furthermore, after transforming the model into a dynamic system, we conducted a quasi-periodic behaviour analysis and sensitivity analysis on it. A variety of methods, including time analysis, Poincarè maps, and 2D and 3D plots, were utilized to identify chaos. In addition, sensitivity analysis under different initial conditions was carried out, and the outcome demonstrated that the governing model is not very sensitive. These solutions have significant applications in various fields of engineering, physics, and other disciplines. Finally, by comparing our solutions with earlier studies, we found that they are distinct and more precise.

aabdelalim@taibahu.edu.sa

Funding information: This work was funded by the Deanship of Graduate Studies and Scientific Research at Jouf University under grant No. DGSSR-2023-02-02535.
Author contribution: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

References

[1] Seadawy AR. Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput Math Appl. 2014;67:172–80. 10.1016/j.camwa.2013.11.001Search in Google Scholar

[2] Seadawy AR. Stability analysis solutions for nonlinear three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation in a magnetized electron-positron plasma. Stat Mechanics Appl. 2016;455:44–51. 10.1016/j.physa.2016.02.061Search in Google Scholar

[3] Iqbal M, Seadawy AR, Lu D. Applications of nonlinear longitudinal wave equation in a magneto-electro-elastic circular rod and new solitary wave solutions. Modern Phys Lett B. 2019;33(18):1950210. 10.1142/S0217984919502105Search in Google Scholar

[4] Jhangeer A, Rezazadeh H, Seadawy A. A study of travelling, periodic, quasiperiodic and chaotic structures of perturbed Fokas-Lenells model. Pramana. 2021;95:41. 10.1007/s12043-020-02067-9Search in Google Scholar

[5] Seadawy AR, Rizvi STR, Ali I, Younis M, Ali K, Makhlouf MM, et al. Conservation laws, optical molecules, modulation instability and Painlevé analysis for the Chen-Lee-Liu model. Optical Quantum Electron. 2021;53:172. 10.1007/s11082-021-02823-0Search in Google Scholar

[6] Wang J, Shehzad K, Seadawy AR, Arshad M, Asmat F. Dynamic study of multi-peak solitons and other wave solutions of new coupled KdV and new coupled Zakharov-Kuznetsov systems with their stability. J Taibah Univ Sci. 2023;17(1):2163872. 10.1080/16583655.2022.2163872Search in Google Scholar

[7] Seadawy AR, Ali KK, Nuruddeen RI. A variety of soliton solutions for the fractional Wazwaz-Benjamin-Bona-Mahony Equations. Results Phys. 2019;12:2234–41. 10.1016/j.rinp.2019.02.064Search in Google Scholar

[8] Seadawy AR, Alsaedi B. Contraction of variational principle and optical soliton solutions for two models of nonlinear Schrodinger equation with polynomial law nonlinearity. AIMS Math. 2024;9(3):6336–67. 10.3934/math.2024309Search in Google Scholar

[9] Sun ZY, Deng D, Pang ZG, Yang ZJ. Nonlinear transmission dynamics of mutual transformation between array modes and hollow modes in elliptical sine-Gaussian cross-phase beams. Chaos, Solitons and Fractals 2024;178:114398. 10.1016/j.chaos.2023.114398Search in Google Scholar

[10] Shen S, Yang ZJ, Pang ZG, Ge YR. The complex-valued astigmatic cosine-Gaussian soliton solution of the nonlocal nonlinear Schrödinger equation and its transmission characteristics. Appl Math Lett. 2022;125:107755. 10.1016/j.aml.2021.107755Search in Google Scholar

[11] Song LM, Yang ZJ, Li XL, Zhang SM. Coherent superposition propagation of Laguerre-Gaussian and Hermite-Gaussian solitons. Appl Math Lett. 2020;102:106114. 10.1016/j.aml.2019.106114Search in Google Scholar

[12] Zou Z, Guo R. The Riemann–Hilbert approach for the higher-order Gerdjikov-Ivanov equation, soliton interactions and position shift. Commun Nonl Sci Numer Simulat. 2023;124:107316. 10.1016/j.cnsns.2023.107316Search in Google Scholar

[13] Shen S, Yang Z, Li XL, Zhang S. Periodic propagation of complex-valued hyperbolic-cosine-Gaussian solitons and breathers with complicated light field structure in strongly nonlocal nonlinear media. Commun Nonl Sci Numer Simulat. 2021;103:106005. 10.1016/j.cnsns.2021.106005Search in Google Scholar

[14] Seadawy AR, Alsaedi BA. Soliton solutions of nonlinear Schrödinger dynamical equation with exotic law nonlinearity by variational principle method. Opt Quantum Electron. 2024;56:700. 10.1007/s11082-024-06367-xSearch in Google Scholar

[15] Seadawy AR, Arshad M, Lu D. The weakly nonlinear wave propagation theory for the Kelvin-Helmholtz instability in magnetohydrodynamics flows. Chaos Solitons Fractals. 2020;139:110141. 10.1016/j.chaos.2020.110141Search in Google Scholar

[16] Li XL, Guo R, Interactions of localized wave structures on periodic backgrounds for the coupled Lakshmanan-Porsezian-Daniel equations in birefringent optical fibers. Annalen der Physik. 2022;535:2200472. 10.1002/andp.202200472Search in Google Scholar

[17] Abdeljawad T. On Riemann and Caputo fractional differences. Comput Math Appl. 2011;62(3):1602–11. 10.1016/j.camwa.2011.03.036Search in Google Scholar

[18] Jeelani MB, Alnahdi AS, Almalahi MA, Abdo MS, Wahash HA, Abdelkawy MA. Study of the Atangana-Baleanu-Caputo type fractional system with a generalized Mittag–Leffler kernel. AIMS Mathematics. 2022;7(2):2001–18. 10.3934/math.2022115Search in Google Scholar

[19] Batiha IM, Alshorm S, Jebril IH, Hammad MA. A brief review about fractional calculus. Int J Open Problems Compt Math. 2022;15(4):39–56. Search in Google Scholar

[20] Seadawy AR, Bilal M, Younis M, Rizvi STR, Althobaiti S, Makhlouf MM. Analytical mathematical approaches for the double chain model of DNA by a novel computational technique. Chaos Solitons Fractals. 2021;144:110669. 10.1016/j.chaos.2021.110669Search in Google Scholar

[21] Farah N, Seadawy AR, Ahmad S, Rizvi STR, Younis M. Interaction properties of soliton molecules and Painleve analysis for nano bioelectronics transmission model. Opt Quantum Electron. 2020;52(7):1–15. 10.1007/s11082-020-02443-0Search in Google Scholar

[22] Raza N, Jhangeer A, Rezazadeh H, Bekir A. Explicit solutions of the (2+1)-dimensinal Hirota Maccari system arising in nonlinear optics. Int J Modern Phys B. 2020;33:1950360. 10.1142/S0217979219503600Search in Google Scholar

[23] Gaber AA, Aljohani AF, Ebaid A, Machado JT. The generalized Kudryashov method for nonlinear space-time fractional partial differential equations of burgers type. Nonl Dyn. 2019;95(1):361–8. 10.1007/s11071-018-4568-4Search in Google Scholar

[24] Ghaffar A, Ali A, Ahmed S, Akram S, Baleanu D, Nisar KS. A novel analytical technique to obtain the solitary solutions for nonlinear evolution equation of fractional order. Adv Differ Equ. 2020;1:1–15. 10.1186/s13662-020-02751-5Search in Google Scholar

[25] Rizvi STR, Ali K, Ahmad M. Optical solitons for Biswas-Milovic equation by new extended auxiliary equation method. Optik. 2020;204:164181. 10.1016/j.ijleo.2020.164181Search in Google Scholar

[26] Li Y, Tian SF, Yang JJ. Riemann–Hilbert problem and interactions of solitons in the component nonlinear Schrödinger equations. Stud Appl Math. 2022;148(2):577–605. 10.1111/sapm.12450Search in Google Scholar

[27] Li ZQ, Tian SF, Yang JJ. On the soliton resolution and the asymptotic stability of N-soliton solution for the Wadati-Konno-Ichikawa equation with finite density initial data in space-time solitonic regions. Adv Math. 2022;409:108639. 10.1016/j.aim.2022.108639Search in Google Scholar

[28] Li ZQ, Tian SF, Yang JJ. Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data. Ann Henri Poincare. 2022;23:2611–55. 10.1007/s00023-021-01143-zSearch in Google Scholar

[29] Li ZQ, Tian SF, Yang JJ, Fan E. Soliton resolution for the complex short pulse equation with weighted Sobolev initial data in space-time solitonic regions. J Differ Equ. 2022;329:31–88. 10.1016/j.jde.2022.05.003Search in Google Scholar

[30] Seadawy AR. Approximation solutions of derivative nonlinear Schrodinger equation with computational applications by variational method. Europ Phys J Plus. 2015;130:182:1–10. 10.1140/epjp/i2015-15182-5Search in Google Scholar

[31] Li Z, Huang C. Bifurcation, phase portrait, chaotic pattern and optical soliton solutions of the conformable Fokas-Lenells model in optical fibers. Chaos Solitons Fractals. 2023;169:113237. 10.1016/j.chaos.2023.113237Search in Google Scholar

[32] Rafiq MH, Raza N, Jhangeer A. Dynamic study of bifurcation, chaotic behaviour and multi-soliton profiles for the system of shallow water wave equations with their stability. Chaos Solitons Fractals. 2023;171:113436. 10.1016/j.chaos.2023.113436Search in Google Scholar

[33] Rizvi STR, Shabbir S. Optical soliton solution via complete discrimination system approach along with bifurcation and sensitivity analyses for the Gerjikov-Ivanov equation. Optik. 2023;294:171456. 10.1016/j.ijleo.2023.171456Search in Google Scholar

[34] Kai Y, Chen S, Zheng B, Zhang K, Yang N, Xu W. Qualitative and quantitative analysis of nonlinear dynamics by the complete discrimination system for polynomial method. Chaos Solitons Fractals. 2020;141:110314. 10.1016/j.chaos.2020.110314Search in Google Scholar

[35] Younas U, Seadawy AR, Younis M, Rizvi STR. Construction of analytical wave solutions to the conformable fractional dynamical system of ion sound and Langmuir waves. Waves Random Complex Media. 2020;32(6):2587–605. 10.1080/17455030.2020.1857463Search in Google Scholar

[36] Atangana A, Koca A. Chaos in a simple nonlinear system with Atangana Baleanu derivatives with fractional order. Chaos Solitons Fractals. 2016;89:447–54. 10.1016/j.chaos.2016.02.012Search in Google Scholar

[37] Algahtani OJJ. Comparing the Atangana Baleanu and Caputo Fabrizio derivative with fractional order: Allen Cahn model. Chaos Solitons Fractals. 2016;89:552–9. 10.1016/j.chaos.2016.03.026Search in Google Scholar

[38] Alkahtani BST. Chuaas circuit model with Atangana Baleanu derivative with fractional order. Chaos Solitons Fractals. 2016;89:547–51. 10.1016/j.chaos.2016.03.020Search in Google Scholar

[39] Seadawy AR, Lu D, Iqbal M. Application of mathematical methods on the system of dynamical equations for the ion sound and Langmuir waves. Pramana. 2019;93:10. 10.1007/s12043-019-1771-xSearch in Google Scholar

[40] Kazmi SS, Jhangeer A, Raza N, Alrebdi HI, Aty AHA, Eleuch H. The analysis of bifurcation, quasi periodic and solitons patterns to the new form of the generalized q-deformed Sinh-Gordon equation. Symmetry. 2023;15:13–24. 10.3390/sym15071324Search in Google Scholar

[41] Filiz A, Ekici M, Sonmezoglu A. F-Expansion method and new exact solutions of the Schrödinger-KdV equation. Scientif World J. 2014;2014:534063. 10.1155/2014/534063Search in Google Scholar PubMed PubMed Central

Received: 2024-06-03

Revised: 2024-08-04

Accepted: 2024-08-06

Published Online: 2024-09-16

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

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0080

Keywords for this article

Atangana–Baleanu derivative; qualitative analysis; complete discriminant of the polynomial method; solitary waves

Creative Commons

BY 4.0