The fractional soliton solutions of dynamical system arising in plasma physics: The comparative analysis

Alexandria Engineering Journal 95 (2024) 247–261 Available online 4 April 2024 1110-0168/© 2024 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Contents lists available at ScienceDirect Alexandria Engineering Journal journal homepage: www.elsevier.com/locate/aej Original Article The fractional soliton solutions of dynamical system arising in plasma physics: The comparative analysis Waqas Ali Faridi a, Mujahid Iqbalb, Muhammad Bilal Riaz c,∗, Salman A. AlQahtanid, Abdul-Majid Wazwaz e aDepartment of Mathematics, University of Management and Technology, Lahore, Pakistan bSchool of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, China cIT4Innovations, VSB – Technical University of Ostrava, Ostrava, Czech Republic dComputer Engineering Department, College of Computer and Information Sciences, King Saud University, Riyadh, Saudi Arabia eDepartment of Mathematics, Saint Xavier University, Chicago, IL 60655, USA A R T I C L E I N F O A B S T R A C T Keywords: The ion sound and Langmuir waves Fractional derivatives New auxiliary equation method Analytical exact solutions In light of fractional theory, this paper presents several new effective solitonic formulations for the Langmuir and ion sound wave equations. Prior to this study, no previous research has presented the comparision and obtained the generalized fractional soliton solutions of this kind with power law kernel and Mittag-Leffler kernel. The ion sound and Langmuir wave equations are essential in plasma physics, offering insights into the collective behavior of charged particles in plasmas and enabling diagnostics and control of these complex, ionized gas systems. The two distinct fractional order differential operators are substituted for the traditional order derivative to reshape the examined model. The Atangana-Baleanu non-singular and non-local operator and conformable fractional operator are the fractional-order operators that are used to create the fractional complex system equations for Langmuir waves and ion sound. A constructive approach new auxiliary equation method utilizes to obtain the exact analytical soliton solutions for ion sound and Langmuir wave equation. A wide range of soliton solutions is obtained, including mixed complex solitary shock solutions, singular solutions, mixed shock singular solutions, mixed trigonometric solutions, mixed singular solutions, exact solutions, mixed periodic solutions, and mixed hyperbolic solutions, dark soliton, bright soliton, trigonometric solutions, periodic results, and hyperbolic results. The solitons solution of the ion sound and Langmuir wave equations lies in their ability to maintain wave stability, their role in modeling wave propagation and nonlinear effects, their potential use as diagnostic tools, and their relevance in wave-particle interactions in plasma physics. The solitons provide a valuable framework for understanding the behavior of waves in plasmas and offer insights into the complex dynamics of these charged particle systems. A graphical comparison analysis of a few solutions is also shown here, taking into account appropriate parametric values through the use of the software package. Moreover, the results of this study have important implications for Hamilton’s equations and generalized momentum, where solitons are employed in long-range interactions. 1. Introduction In many branches of science and engineering, the non-linear structure of partial differential equations is important. The applications that use non-linear partial differential equations within maths and quantum physics are quite diverse [1], soil mechanics [2], mechanical statistics [3], civil engineering [4], solid-state physics [5], plasma physics [6], * Corresponding author at: IT4Innovations, VSB – Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic. E-mail addresses: [email protected] (W.A. Faridi), [email protected] (M. Iqbal), [email protected] (M.B. Riaz), [email protected] (S.A. AlQahtani), [email protected] (A.-M. Wazwaz). plasma waves [7], economics [8], biology [9], and population ecology [10]etc. Control issues are governed by the best non-linear partial differential equations, and Pesch suggested two methods to address this. [11]. After optimization, the first approach for dispersion was indirect. The second approach, the direct methodology, was optimized after being reduced to a fine level of detail. The ordinary differential equations have been used to study the models of the non-linear systems. An algohttps://doi.org/10.1016/j.aej.2024.03.061 Received 27 December 2023; Received in revised form 23 February 2024; Accepted 20 March 2024 Alexandria Engineering Journal 95 (2024) 247–261 248 W.A. Faridi, M. Iqbal, M.B. Riaz et al. rithm for computing the generalized frequency response to the family of non-linear differential equations was developed by Peyton and Billings [12]. Victor’s research provided a comprehensive explanation of the modeling, existence, stability, importance, and uniqueness of ordinary differential equations [13]. In his paper, Ahmed proposed an ODE-PDE system that explains systems of both partial differential equations and ordinary differential equations [14]. The goal of the researchers and analysts is to connect the two ODE-PDE systems by introducing a midway path [15]. Due to their many applications across a wide range of contemporary research fields, partial differential equations are currently attracting a great deal of attention from researchers. The incorporation of the fractional theory of calculus a popular method for solving partial differential equations staggering road leading to the science world. The fractional partial differential equations revealed some shocking as well deep knowledge of real physical phenomenon as compared to classic order fractional derivatives. Due to curiosity of fractional theory, a lot of fractional order derivative and operators formulated. Some of them are presented here, Caputo fractional derivative [16], conformal fractional differential operator [17], 𝛽-fractional differential operator [18], truncated M-fractional derivative [19], Riemann-Liouville fractional operator [20], Caputo-Fabrizio fractional derivative [21], and the Atangana-Baleanu fractional derivative [22]. A numerous variety of applications with these fractional operators has been done in diverse fields of sciences. The analysis of local fractional Klein-Gordon equation conducted by Jagdeve and Baleanu [23]. Jagdeve provided the solution of differential difference and examined [24]. Baleanu examined a new fractional operator in the logistic equation [25]. Atangana analyzed the Cauchy problem as well revealed a fresh understanding of changing rate [26]. When studying the applications of such types of the fractional theory then obviously, a question arises why we use fractional theory, and what are the advantages of fractional theory. Analysis of fractional partial differential equations in sort of solitons is one of the remarkable studies in non-linear optics. The question is why we prefer the fractional-order derivative to the classic derivative. The logic is that the classic order derivative is the local operator that’s why it is ill-suited in some critical and sensitive situations where heavy tails occur and the influence of larger neighborhoods can’t be ignored anymore. On the contrary, the fractional-order derivative is a global differential operation that contrives to examine such types of influences. The fractional theory of differentiation is the generalization of the classic order derivative. For the study in sort of solitons, a proficient class of schemes, approaches, and method has been formulated such as extended trial equation method [28], the symmetry strategy [29], Lie and Buckland transformation approach [30], exp-function method [31], and tanh-coth trigonometric function technique [32]etc. A lot of research work has been conducted by various scientists and researchers in the discipline of fractional solitons. The emerging telecommunication system simulated by the higher-order cubic quintic non-linear fractional complex schrödinger equations has examined by the Mustafa and Attia [27]. Mostafa and Behzad discuss the Riemann wave propagating and examined the fractional Bogoyavlensky Konopelchenko equation and also the propagation of long-wave [33]. Qin and Mostafa examined the fractional Broer–Kaup system which simulates the bidirectional propagation of long-wave in shallow water [34]. The flow of shallow water discussed by Mostafa [35]. Abdel investigated the physical phenomenon of water waves propagation [36]. The fractional Klein– Gordon equations have investigated by Inc and Baleanu and the fractional Cahn–Allen equation [37]. The varicella-zoster virus is modeled and investigated by Qureshi [38]. Ya¸sar et al. [39,40]have been constructed the traveling wave solutions of Ostrovsky and perturbed nonlinear Schrödinger equation by using the analytical techniques. Ay et al. [41,42]have been discussed the multiple soliton solutions such as breather, kink-type and their interaction. Biswas et al. [43]have investigated the dynamical aspects of Lakshmanan–Porsezian–Daniel model. This field has multiple applications in diverse areas of physical sciences [44–49]. The propensity of the waves to “condensate” in k-space is what defines the dynamics of Langmuir turbulence [50]. In physics, particularly in the context of solid-state physics and quantum mechanics, “k-space” refers to the space of wave vectors, often denoted by the symbol k. kspace is a mathematical space used to describe the wave-like behavior of particles, particularly electrons, in periodic structures such as crystals. It plays a crucial role in understanding the electronic properties of materials and is essential for theoretical modeling and experimental characterization in condensed matter physics and materials science. This study is about the real phenomenon of ion sound and Langmuir waves. The non-linear complex system of the partial differential equation for ion sound wave and Langmuir waves is considered here to see that the influence of ponderomotive force can be quite important avenue one in the field of high-frequency [51]. The system presented is [52]: {𝑖𝑃𝑡+1 2𝑃𝑥𝑥 −𝑅𝑃 =0, 𝑅𝑡𝑡 −𝑅𝑥𝑥 −2(|𝑃|2)𝑥𝑥 =0,(1) where, 𝑃is presenting the normalized electric field (NEF) and R is normalized density perturbation (NDP) of the Langmuir oscillation. The primary nonlinear processes, which are essentially the fourplasmon scattering, the electron and ion-induced scattering, and the decay of Langmuir waves leading to ion sound, do not alter the quantity of Langmuir quanta; rather, they merely reduce their wave vectors. The long-wave region of the spectrum, where linear damping (collision and Landau damping) is minimal, is where the Langmuir oscillation’s energy concentrates as a result. This raises the question of what technique is used to release this energy. This research presents evidence that suggests the mechanism could be a special three-dimensional concentrating of the Langmuir waves, resulting in the creation of local singularities in their amplitudes. In the last decades, many scientists have found various types of outcomes of the system of partial differential equations for the ion sound and Langmuir waves [51]. Demiray obtained the exact solution by utilizing the extended trial function method of the system for ion sound and Langmuir waves [53]. Vidojevic examined the effects of Langmuir waves electric fields on the shape of spectral line [54]. Manafian has obtained the solitonic structures from the nonlinear Schrödinger evolution equation with the prosecution of tan(𝜙 2)- expansion method [55]. Mohyud-Din has been obtained the numerical soliton solutions of improved Boussinesq equation [56]. A lot of work has been done on it but the analysis of the framework to generate the Langmuir oscillation and ion sound in influence of fractional theory is not done yet. On this inspiration, we are investigating this complex system using fractional theory in sort of solitons. This study is systematized as, Section 2is devoted for necessary prerequisites, Section 3is presenting the fractional appearance of the model and some useful transformations, description of scheme and applications are in Section 4, Section 5is an explanation of graphics, and rest of the paper is conclusion and references. 2. Preliminaries Here, certain pertinent contributions and notations are highlighted, which are then used to condense novel and anarchistic ideas. 2.1. Conformal fractional operator Definition: Assume that a ℎ ∶ℝ+→ℝis differential function. The conformal derivative with fractional order 0 <𝜛≤1as [20]: 𝐷𝜛 𝑡(ℎ(𝑡)) = lim 𝜀→0 ℎ(𝑡+𝜀𝑡1−𝜛)−ℎ(𝑡) 𝜀,(2) in which time is positive. The properties of fractional derivative are given below: Alexandria Engineering Journal 95 (2024) 247–261 249 W.A. Faridi, M. Iqbal, M.B. Riaz et al. (𝟏)∶ 𝐷𝜛 𝑡𝑡𝜍=𝜍𝑡𝜍−𝜛, ∀𝜍∈. (𝟐)∶ 𝐷𝜛 𝑡(𝑘(ℎ(𝑡))) =𝑘𝐷𝜛 𝑡(ℎ(𝑡)). (𝟑)∶ 𝐷𝜛 𝑡(𝑚(ℎ(𝑡)) +𝑛(𝑔(𝑡))) =𝑚𝐷𝜛 𝑡(ℎ(𝑡)) +𝑛𝐷𝜛 𝑡(𝑔(𝑡)) ∀𝑚, 𝑛 ∈. (𝟒)∶ 𝐷𝜛 𝑡(𝑔(𝑡) ℎ(𝑡)) =(ℎ(𝑡)) 𝐷𝜛 𝑡(𝑔(𝑡))−(𝑔(𝑡)) 𝐷𝜛 𝑡(ℎ(𝑡)) (ℎ(𝑡))2. (𝟓)∶ 𝐷𝜛 𝑡(𝑔(𝑡) ∗ℎ(𝑡)) =(𝑔(𝑡)) 𝐷𝜛 𝑡(ℎ(𝑡)) ∗(ℎ(𝑡)) 𝐷𝜛 𝑡(𝑔(𝑡)). (𝟔)∶ 𝐷𝜛 𝑡ℎ(𝑡) =𝑡1−𝜛𝑑ℎ 𝑑𝑡 . (𝟕)∶ 𝐷𝜛 𝑡𝑥𝜍=Γ(1+𝜍) Γ(1+𝜍−𝜛)𝑥𝜍−𝜛,𝜍≥0. (𝟖)∶ 𝐷𝜛 𝑡(𝑐) =0 ∀𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠. (𝟗)∶ 𝐷𝜛 𝑡((ℎ(𝑥))◦(𝑔(𝑡))) =𝑡1−𝜛𝑔′(𝑡)ℎ′(𝑔(𝑡)). 2.2. Atangana-Baleanu fractional operator Definition: The Atangana-Baleanu in the sense of Riemann-Liouvilla fractional differential operator is defined as in [22]: 𝐴𝐵𝑅 0𝐷𝛼 𝑎+(ℎ(𝑡)) = 𝐴𝐵(𝛼) (1 − 𝛼) 𝑑 𝑑𝑡 𝑡 ∫ 𝑎 ℎ(𝜏)𝐸𝛼(−𝛼(𝑡−𝜏)𝛼 1−𝛼)𝑑𝜏, (3) where 𝐸𝛼is the Mittag-leffer function and 𝐴𝐵(𝛼)is the normalization. Thus: 𝐴𝐵𝑅 0𝐷𝛼 𝑎+(ℎ(𝑡)) = 𝐴𝐵(𝛼) (1 − 𝛼) ∞ ∑ 𝑛=0 (−𝛼 1−𝛼)𝑛𝑅𝐿𝐼𝛼𝑛 𝑎ℎ(𝑡).(4) 3. Different fractional forms of Eq. (1) The ion sound and Langmuir waves is transformed into fractional form by using two different fractional operators. (1):The conformable fractional operator is applied on (1)and the consequence is: {𝑖𝐷𝛼 𝑡𝑃+1 2𝐷2𝛼 𝑥𝑥𝑃−𝑅𝑃 =0, 𝐷2𝛼 𝑡𝑡 𝑅−𝐷2𝛼 𝑥𝑥𝑅−2𝐷2𝛼 𝑥𝑥(|𝑃|2)=0,(5) where 𝐷𝛼 𝑡and 𝐷𝛼 𝑥are conformable fractional operators with respect to 𝑡and 𝑥, respectively. (2): The Atangana-Baleanu fractional operator is applying on Eq. (1) thus, we acquire: {𝑖𝐴𝐵𝑅 0𝐷𝛼 𝑡𝑃+1 2 𝐴𝐵𝑅 0𝐷2𝛼 𝑥𝑥𝑃−𝑅𝑃 =0, 𝐴𝐵𝑅 0𝐷2𝛼 𝑡𝑡 𝑅−𝐴𝐵𝑅 0𝐷2𝛼 𝑥𝑥𝑅−2 𝐴𝐵𝑅 0𝐷2𝛼 𝑥𝑥(|𝑃|2)=0,(6) where, 𝐴𝐵𝑅 0𝐷𝛼 𝑡and 𝐴𝐵𝑅 0𝐷𝛼 𝑥are the Atangana-Baleanu in sense of Riemann-Liouvilla operators with reference to 𝑡and 𝑥respectively. 3.1. Operative transformations At this point realistic transformations are introduced enabling the conversion of the relevant PDE towards ODE for obtaining the answer to Eq. (5)and Eq. (6). Let’s begin by do certain negotiable changes with respect to these fractional operators that are in operation: {𝑃=𝑃(𝑥, 𝑡),where 𝑃(𝑥, 𝑡)=𝑉(𝜓)𝑒𝑖𝜑(𝑥,𝑡) 𝑅=𝑅(𝑥, 𝑡),where 𝑅(𝑥, 𝑡)=𝑈(𝜓).(7) We will pursue 𝜓and 𝜑as per concerning fractional derivative. (1):We delegate 𝜓and 𝜑as for the conformable fractional operator: ⎧ ⎪ ⎨ ⎪ ⎩ 𝜓=𝜔(𝑥𝛼 𝛼)+𝜆(𝑡𝛼 𝛼), 𝜑=𝑘(𝑥𝛼 𝛼)+𝜇(𝑡𝛼 𝛼)+𝛾0. (8) (2): We ordain 𝜓and 𝜑as for the Atangana-Baleanu in sense of Riemann-Liouvilla fractional operator: ⎧ ⎪ ⎨ ⎪ ⎩ 𝜓=𝜔𝑥 +𝜆(1−𝛼)𝑡−𝛼𝑛 𝐴𝐵(𝛼)Σ∞ 𝑛=0(− 𝛼 1−𝛼)Γ(1−𝛼𝑛), 𝜑=𝑘𝑥 +𝜇(1−𝛼)𝑡−𝛼𝑛)+𝛾0 𝐴𝐵(𝛼)Σ∞ 𝑛=0(− 𝛼 1−𝛼)Γ(1−𝛼𝑛).(9) 4. Construction of ion sound and Langmuir waves 4.1. Description of proposed technique We shall adhere to steps [57]. Consider a NLPDE of the following form: 𝑌(𝕌,𝕌𝑡,𝕌𝑥,𝕌𝑡𝑡,𝕌𝑥𝑥,⋯)=0.(10) Its NODE will be: ℚ(ℝ,ℝ′,ℝ′′,⋯)=0.(11) Consider: 𝕌(𝑥, 𝑡)=𝕌(𝜓),(12) here, 𝜓=𝑚𝑥 +𝑐𝑡. The prime symbols in Eq. (11) indicate the order of differentiation with respect to the different variables in the equation. Considering the following equation as the general solution of Eq. (11)by using MAE approach, 𝕌(𝜓)= 𝑁 ∑ 𝑖=0 [𝑎𝑖𝑓𝑖𝜙(𝜓)],(13) where, 𝜓=𝑘1(𝑥 +𝑦) +𝑘2𝑡. 𝑎𝑖𝑠are constants of the equation and the function 𝜙(𝜓)can be described by the auxiliary equation that follows: 𝜙′(𝜓)= 1 𝑙𝑛(𝑓)(𝛽𝑓−𝜙(𝜓)+𝛾𝑓𝜙(𝜓)+𝜒),(14) where 𝜎0, 𝜎1, 𝜎2, 𝜎3, ..., 𝜎𝑁are the coefficients to be known in such a way that 𝛼𝑁≠0. The principle of balancing suggests determining the value of N by setting the most significant nonlinear factor equal to the higher-order derivative of the given equation. There are several possible solutions to this equation, depending on the specific case: 1. If 𝛾≠0and 𝜒2−4𝛽𝛾 < 0, 𝑓𝜙(𝜓) 1=√4𝛽𝛾 −𝜒2tan(1 2𝜓√4𝛽𝛾 −𝜒2) 2𝛾−𝜒 2𝛾,(15) 𝑓𝜙(𝜓) 2=−√4𝛽𝛾 −𝜒2cot (1 2𝜓√4𝛽𝛾 −𝜒2) 2𝛾−𝜒 2𝛾.(16) 2. If 𝛾≠0and 𝜒2−4𝛽𝛾 > 0, 𝑓𝜙(𝜓) 3=−√𝜒2−4𝛽𝛾 tanh(1 2𝜓√𝜒2−4𝛽𝛾) 2𝛾−𝜒 2𝛾,(17) 𝑓𝜙(𝜓) 4=−√𝜒2−4𝛽𝛾 coth (1 2𝜓√𝜒2−4𝛽𝛾) 2𝛾−𝜒 2𝛾.(18) 3. If 𝛾≠0, 𝛾=−𝛽and 𝜒2+4𝛽2<0, 𝑓𝜙(𝜓) 5=𝜒 2𝛽−√−4𝛽2−𝜒2tan(1 2𝜓√−4𝛽2−𝜒2) 2𝛽,(19) 𝑓𝜙(𝜓) 6=√−4𝛽2−𝜒2cot (1 2𝜓√−4𝛽2−𝜒2) 2𝛽+𝜒 2𝛽.(20) 4. If 𝛾≠0, 𝛾=−𝛽and 𝜒2+4𝛽2>0, 𝑓𝜙(𝜓) 7=√4𝛽2+𝜒2tanh(1 2𝜓√4𝛽2+𝜒2) 2𝛽+𝜒 2𝛽,(21) Alexandria Engineering Journal 95 (2024) 247–261 250 W.A. Faridi, M. Iqbal, M.B. Riaz et al. 𝑓𝜙(𝜓) 8=√4𝛽2+𝜒2coth (1 2𝜓√4𝛽2+𝜒2) 2𝛽+𝜒 2𝛽.(22) 5. If 𝛾≠0, 𝛾=𝛽and 𝜒2+4𝛽2<0, 𝑓𝜙(𝜓) 9=√4𝛽2−𝜒2tan(1 2𝜓√4𝛽2−𝜒2) 2𝛽−𝜒 2𝛽,(23) 𝑓𝜙(𝜓) 10 =−√4𝛽2−𝜒2cot (1 2𝜓√4𝛽2−𝜒2) 2𝛽−𝜒 2𝛽.(24) 6. If 𝛾=𝛽and 𝜒2−4𝛽2>0, 𝑓𝜙(𝜓) 11 =−√𝜒2−4𝛽2tanh(1 2𝜓√𝜒2−4𝛽2) 2𝛽−𝜒 2𝛽,(25) 𝑓𝜙(𝜓) 12 =−√𝜒2−4𝛽2coth (1 2𝜓√𝜒2−4𝛽2) 2𝛽−𝜒 2𝛽.(26) 7. If 4𝛽𝛾 =𝜒2, 𝑓𝜙(𝜓) 13 =−𝜒𝜓 +2 2𝛾𝜓 .(27) 8. If 𝛽𝛾 < 0, 𝜒=0, 𝑓𝜙(𝜓) 14 =− √−𝛽 𝛾tanh(√−𝛽𝛾𝜓),(28) 𝑓𝜙(𝜓) 15 =− √−𝛽 𝛾coth (√−𝛽𝛾𝜓).(29) 9. If 𝛽=−𝛾, 𝑤𝑖𝑡ℎ𝜒 =0, 𝑓𝜙(𝜓) 16 =−𝑒−2𝛾𝜓 +1 1−𝑒−2𝛾𝜓 ,(30) 10. If 𝛽=𝛾=0, 𝑓𝜙(𝜓) 17 = sinh(𝜒𝜓)+cosh(𝜒𝜓).(31) 11. If 𝛽=𝜒=𝑘, 𝛾=0, 𝑓𝜙(𝜓) 18 =𝑒K𝜓−1.(32) 12. If 𝛾=𝜒=𝑘, 𝛽=0, 𝑓𝜙(𝜓) 19 =𝑒K𝜓 1−𝑒K𝜓.(33) 13. If 𝜒=𝛽+𝛾, 𝑓𝜙(𝜓) 20 =−1−𝛽e𝜓(𝛽−𝛾) 1−𝛾e𝜓(𝛽−𝛾).(34) 14. If 𝜒=−𝛽−𝛾, 𝑓𝜙(𝜓) 21 =𝑒𝜓(𝛽−𝛾)−𝛽 𝑒𝜓(𝛽−𝛾)−𝛾.(35) 15. If 𝛽=0, 𝑓𝜙(𝜓) 22 =𝜒e𝜒𝜓 −𝛽 1−𝛾e𝜒𝜓 .(36) 16. If 𝜒=𝛽, 𝑓𝜙(𝜓) 23 =1 2(√3tan(√3𝛽𝜓 2)−1 ).(37) 17. If 𝜒=𝛾=0, 𝑓𝜙(𝜓) 24 =𝛽𝜓. (38) 18. If 𝜒=𝛾=0, 𝑓𝜙(𝜓) 25 =− 1 𝛾𝜓 .(39) 19. If 𝜒=0, 𝛾=𝛽, 𝑓𝜙(𝜓) 26 =tan(𝛽𝜓).(40) 20. If 𝛾=0, 𝑓𝜙(𝜓) 27 =𝑒𝜒𝜓 −𝑛 𝑙.(41) 4.2. Application of new auxiliary equation method In order to obtain the solution of Eq. (5)and Eq. (6), the traveling wave transformation Eq. (7)is applying to the Eq. (5)and Eq. (6)along with the fractional definitions and get, 𝑖(𝜆+𝜔𝑘)𝑉′=0.(42) 𝑘𝑉 ′′ −(2𝜇+𝑘2)𝑉−2𝑉𝑈=0.(43) (𝜆2−𝜔2)𝑈′′ −2𝜔2(𝑉2)′′ =0.(44) Integrating twice the Eq. (44)with zero constant of integration: 𝑈(𝜓)= 2 𝑘2−1𝑉2(𝜓),𝜆=−𝜔𝑘. (45) Plugging Eq. (45) into Eq. (43), we will get: 𝜔2(𝑘2−1)𝑉′′ −(2𝜇2+𝑘2)(𝑘2−1)𝑉−4𝑉3=0.(46) The homogeneous balancing constant is one (𝑁=1)of Eq. (46). Thus, 𝑉(𝜓)=𝑎0+𝑎1(𝑓𝜙(𝜓)),(47) where, 𝜙′(𝜔)= 1 𝑙𝑛(𝑓)(𝛽𝑓−𝜙(𝜔)+𝛾𝑓𝜙(𝜔)+𝜒).(48) The Eq. (47)is plugged in Eq. (46)and get a system of algebraic equations by collecting the coefficients of distinct power of 𝑓𝜙(𝜓). The obtained algebraic equations system is solved by using Mathematica and get set of solutions, [𝑎0=±𝜔√𝑘2−1𝜒 2√2 ,𝑎 1=±2𝜔√𝑘2−1𝛾 2√2 ,𝜇=±√𝜔2(4𝛽𝛾 −𝜒2)−2𝑘2 2.] (49) The solution set (49)is plugging into Eq. (47): ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑃(𝑥, 𝑡)=(±𝜒Λ±2𝛾Λ(𝑓𝜙(𝜓) 𝑖))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅(𝑥, 𝑡)= 2 𝑘2−1 (±𝜒Λ±2𝛾Λ(𝑓𝜙(𝜓) 𝑖))2 . (50) Here Λ =𝜔√𝑘2−1 2√2and Π =𝜒2−4𝛾𝛽. It is important to notice that, one can get variety of distinct solutions by using 𝑓𝜙(𝜓) 𝑖from (15)-(41). (Family 1): Since 𝛾≠0and 𝜒2−4𝛽𝛾 < 0, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃1(𝑥, 𝑡)=(Λ√4𝛽𝛾 −𝜒2tan(1 2𝜓√4𝛽𝛾 −𝜒2))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅1(𝑥, 𝑡)= 2 𝑘2−1 (Λ√4𝛽𝛾 −𝜒2tan(1 2𝜓√4𝛽𝛾 −𝜒2))2 . (51) ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃2(𝑥, 𝑡)=(Λ√4𝛽𝛾 −𝜒2cot (1 2𝜓√4𝛽𝛾 −𝜒2))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅2(𝑥, 𝑡)= 2 𝑘2−1 (Λ√4𝛽𝛾 −𝜒2cot (1 2𝜓√4𝛽𝛾 −𝜒2))2 . (52) (Family 2): Since 𝛾≠0and 𝜒2−4𝛽𝛾 > 0, Alexandria Engineering Journal 95 (2024) 247–261 251 W.A. Faridi, M. Iqbal, M.B. Riaz et al. Fig. 1. 2D comparison and 3D wave propagation of 𝑃1(𝑥, 𝑡)and 𝑅1(𝑥, 𝑡)at 𝛼=0.1. ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃3(𝑥, 𝑡)=(Λ√𝜒2−4𝛽𝛾 tanh(1 2𝜓√𝜒2−4𝛽𝛾))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅3(𝑥, 𝑡)= 2 𝑘2−1 (Λ√𝜒2−4𝛽𝛾 tanh(1 2𝜓√𝜒2−4𝛽𝛾))2 . (53) ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃4(𝑥, 𝑡)=(Λ√𝜒2−4𝛽𝛾 coth (1 2𝜓√𝜒2−4𝛽𝛾))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅4(𝑥, 𝑡)= 2 𝑘2−1 (Λ√𝜒2−4𝛽𝛾 coth (1 2𝜓√𝜒2−4𝛽𝛾))2 . (54) (Family 3): Since 𝛾≠0, 𝛾=−𝛽and 𝜒2+4𝛽2<0, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃5(𝑥, 𝑡)=(Λ√−4𝛽2−𝜒2tan(1 2𝜓√−4𝛽2−𝜒2))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅5(𝑥, 𝑡)= 2 𝑘2−1 (Λ√−4𝛽2−𝜒2tan(1 2𝜓√−4𝛽2−𝜒2))2 . (55) ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃6(𝑥, 𝑡)=(Λ√−4𝛽2−𝜒2cot (1 2𝜓√−4𝛽2−𝜒2))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅6(𝑥, 𝑡)= 2 𝑘2−1 (Λ√−4𝛽2−𝜒2cot (1 2𝜓√−4𝛽2−𝜒2))2 . (56) (Family 4): Since 𝛾≠0, 𝛾=−𝛽and 𝜒2+4𝛽2>0, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃7(𝑥, 𝑡)=(Λ√4𝛽2+𝜒2tanh(1 2𝜓√4𝛽2+𝜒2))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅7(𝑥, 𝑡)= 2 𝑘2−1 (Λ√4𝛽2+𝜒2tanh(1 2𝜓√4𝛽2+𝜒2))2 . (57) ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃8(𝑥, 𝑡)=(Λ√4𝛽2+𝜒2coth (1 2𝜓√4𝛽2+𝜒2))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅8(𝑥, 𝑡)= 2 𝑘2−1 (Λ√4𝛽2+𝜒2coth (1 2𝜓√4𝛽2+𝜒2))2 . (58) (Family 5): Since 𝛾≠0, 𝛾=𝛽and 𝜒2−4𝛽2<0, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃9(𝑥, 𝑡)=(Λ√4𝛽2−𝜒2tan(1 2𝜓√4𝛽2−𝜒2))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅9(𝑥, 𝑡)= 2 𝑘2−1 (Λ√4𝛽2−𝜒2tan(1 2𝜓√4𝛽2−𝜒2))2 . (59) Alexandria Engineering Journal 95 (2024) 247–261 252 W.A. Faridi, M. Iqbal, M.B. Riaz et al. Fig. 2. 2D comparison and 3D wave propagation of 𝑃1(𝑥, 𝑡)and 𝑅1(𝑥, 𝑡)at 𝛼=0.7. ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃10(𝑥, 𝑡)=(Λ√4𝛽2−𝜒2cot (1 2𝜓√4𝛽2−𝜒2))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅10(𝑥, 𝑡)= 2 𝑘2−1 (Λ√4𝛽2−𝜒2cot (1 2𝜓√4𝛽2−𝜒2))2 . (60) (Family 6): Since 𝛾=𝛽and 𝜒2−4𝛽2>0, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃11(𝑥, 𝑡)=(Λ√𝜒2−4𝛽2tanh(1 2𝜓√𝜒2−4𝛽2))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅11(𝑥, 𝑡)= 2 𝑘2−1 (Λ√𝜒2−4𝛽2tanh(1 2𝜓√𝜒2−4𝛽2))2 . (61) ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃12(𝑥, 𝑡)=(Λ√𝜒2−4𝛽2coth (1 2𝜓√𝜒2−4𝛽2))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅12(𝑥, 𝑡)= 2 𝑘2−1 (Λ√𝜒2−4𝛽2coth (1 2𝜓√𝜒2−4𝛽2))2 . (62) (Family 7): Since 4𝛽𝛾 =𝜒2, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃13(𝑥, 𝑡)=(±𝜒Λ∓Λ𝜒𝜓+2 𝜓)×𝑒𝑖𝜑(𝑥,𝑡), 𝑅13(𝑥, 𝑡)= 2 𝑘2−1 (±𝜒Λ∓Λ𝜒𝜓+2 𝜓)2 . (63) (Family 8): Since 𝛽𝛾 < 0, 𝜒=0, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃14(𝑥, 𝑡)=(∓2Λ √−𝛾𝛽tanh(√−𝛽𝛾𝜓))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅14(𝑥, 𝑡)= 2 𝑘2−1 (2Λ√−𝛾𝛽tanh(√−𝛽𝛾𝜓))2 . (64) ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃15(𝑥, 𝑡)=(∓2Λ √−𝛾𝛽coth (√−𝛽𝛾𝜓))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅15(𝑥, 𝑡)= 2 𝑘2−1 (∓2Λ√−𝛾𝛽coth (√−𝛽𝛾𝜓))2 . (65) (Family 9): Since 𝛽=−𝛾, 𝑤𝑖𝑡ℎ𝜒 =0, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃16(𝑥, 𝑡)=(±2𝛾Λ(−𝑒−2𝛾𝜓+1 1−𝑒−2𝛾𝜓 ))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅16(𝑥, 𝑡)= 2 𝑘2−1 (±2𝛾Λ(−𝑒−2𝛾𝜓+1 1−𝑒−2𝛾𝜓 ))2 . (66) Alexandria Engineering Journal 95 (2024) 247–261 253 W.A. Faridi, M. Iqbal, M.B. Riaz et al. Fig. 3. 2D comparison and 3D wave propagation for 𝑃1(𝑥, 𝑡)and 𝑅1(𝑥, 𝑡)at 𝛼=0.9. (Family 10): Since 𝛾=𝜒=𝑘, 𝛽=0, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃17(𝑥, 𝑡)=(±𝛾Λ±2𝛾Λ(𝑒K𝜓 1−𝑒K𝜓))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅17(𝑥, 𝑡)= 2 𝑘2−1 (±𝛾Λ±2𝛾Λ(𝑒K𝜓 1−𝑒K𝜓))2 . (67) (Family 11): Since 𝜒=𝛽+𝛾, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃18(𝑥, 𝑡)=(±(𝛽+𝛾)Λ±2𝛾Λ(−1−𝛽e𝜓(𝛽−𝛾) 1−𝛾e𝜓(𝛽−𝛾)))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅18(𝑥, 𝑡)= 2 𝑘2−1 (±(𝛽+𝛾)Λ ± 2𝛾Λ(−1−𝛽e𝜓(𝛽−𝛾) 1−𝛾e𝜓(𝛽−𝛾)))2 . (68) (Family 12): Since 𝜒=−𝛽−𝛾, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃19(𝑥, 𝑡)=(∓(𝛽+𝛾)Λ±2𝛾Λ(𝑒𝜓(𝛽−𝛾)−𝛽 𝑒𝜓(𝛽−𝛾)−𝛾))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅19(𝑥, 𝑡)= 2 𝑘2−1 (∓(𝛽+𝛾)Λ ± 2𝛾Λ(𝑒𝜓(𝛽−𝛾)−𝛽 𝑒𝜓(𝛽−𝛾)−𝛾))2 . (69) (Family 13): Since 𝛽=0, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃20(𝑥, 𝑡)=(±𝜒Λ±2𝛾Λ(𝜒e𝜒𝜓−𝛽 1−𝛾e𝜒𝜓 ))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅20(𝑥, 𝑡)= 2 𝑘2−1 (±𝜒Λ±2𝛾Λ(𝜒e𝜒𝜓−𝛽 1−𝛾e𝜒𝜓 ))2 . (70) (Family 14): Since 𝜒=𝛽, ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝑃21(𝑥, 𝑡)=(±𝛽Λ±2𝛾Λ(1 2(√3tan(√3𝛽𝜓 2)−1 )))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅21(𝑥, 𝑡)= 2 𝑘2−1 (±𝛽Λ±2𝛾Λ(1 2(√3tan(√3𝛽𝜓 2)−1 )))2 . (71) (Family 15): Since 𝜒=0, 𝛾=𝛽, ⎧ ⎪ ⎨ ⎪ ⎩ 𝑃22(𝑥, 𝑡)=(±2𝛽Λ(tan(𝛽𝜓)))×𝑒𝑖𝜑(𝑥,𝑡), 𝑅22(𝑥, 𝑡)= 2 𝑘2−1 (±2𝛽Λ(tan(𝛽𝜓)))2. (72) Alexandria Engineering Journal 95 (2024) 247–261 254 W.A. Faridi, M. Iqbal, M.B. Riaz et al. Fig. 4. 2-D pictorial comparision for conformable derivative and AB operator of 𝑃1(𝑥, 𝑡)as along 𝑅1(𝑥, 𝑡)at 𝛼=1. We will adopt 𝜓as per concerning fractional model from Eq. (8) and Eq. (9) corresponding to conformable fractional derivative and AB fractional differential operator respectively. 5. Graphical analysis Fig. 1is illustrating the 2D graphical comparison and 3D wave propagation with two different fractional operators using the parametric numbers, 𝜒=2,𝛽=2.5,𝛾=1,𝜔 =0.01 and 𝑘 =0.5at fractional order 𝛼=0.1for the normalized electric field 𝑃1(𝑥, 𝑡)and normalized density perturbation 𝑅1(𝑥, 𝑡). Fig. 1a and Fig. 1c are presenting the 3D wave propagation of normalized electric field with conformable operator, and AtanganaBaleanu derivative respectively. Fig. 1e is displaying the 2D graphical comparison of fractional operators. Fig. 1b and Fig. 1d have been 3D anatomization of normalized density perturbation with conformable operator and Atangana-Baleanu derivative respectively. Fig. 1f is displaying the 2D graphical comparison of fractional operators. Fig. 2is illustrating the 2D graphical comparison and 3D wave propagation with two different fractional operators using the parametric numbers, 𝜒=2,𝛽=2.5,𝛾=1,𝜔 =0.01 and 𝑘 =0.5at fractional order 𝛼=0.7for the normalized electric field 𝑃1(𝑥, 𝑡)and normalized density perturbation 𝑅1(𝑥, 𝑡). Fig. 2a and Fig. 2c are presenting the 3D wave propagation of normalized electric field with conformable operator, and AtanganaBaleanu derivative respectively. Fig. 2e is displaying the 2D graphical comparison of fractional operators. Fig. 2b and Fig. 2dhave been 3D anatomization of normalized density perturbation with conformable operator and Atangana-Baleanu Alexandria Engineering Journal 95 (2024) 247–261 255 W.A. Faridi, M. Iqbal, M.B. Riaz et al. Fig. 5. 2D comparison of operators and influence of fractional order and compared with classic order for 𝑃1(𝑥, 𝑡)and 𝑅1(𝑥, 𝑡). derivative respectively. Fig. 2f is displaying the 2D graphical comparison of fractional operators. Fig. 3is illustrating the 2D graphical comparison and 3D wave propagation with two different fractional operators using the parametric numbers, 𝜒=2,𝛽=2.5,𝛾=1,𝜔 =0.01 and 𝑘 =0.5at fractional order 𝛼=0.9for the normalized electric field 𝑃1(𝑥, 𝑡)and normalized density perturbation 𝑅1(𝑥, 𝑡). Fig. 3a and Fig. 3c are presenting the 3D wave propagation of normalized electric field with conformable operator, and AtanganaBaleanu derivative respectively. Fig. 3e is displaying the 2D graphical comparison of fractional operators. Fig. 3b and Fig. 3d have been 3D anatomization of normalized density perturbation with conformable operator and Atangana-Baleanu derivative respectively. Fig. 3f is displaying the 2D graphical comparison of fractional operators. Fig. 4is illustrating the 2D graphical comparison and 3D wave propagation with two different fractional operators using the parametric numbers, 𝜒=2,𝛽=2.5,𝛾=1,𝜔 =0.01 and 𝑘 =0.5at fractional order 𝛼=1for the normalized electric field 𝑃1(𝑥, 𝑡)and normalized density perturbation 𝑅1(𝑥, 𝑡). Fig. 4a and Fig. 4c are presenting the 3D wave propagation of normalized electric field with conformable operator, and AtanganaBaleanu derivative respectively. Fig. 5a is displaying the 2D graphical comparison of fractional operators. Fig. 4b and Fig. 4d have been 3D anatomization of normalized density perturbation with conformable operator and Atangana-Baleanu derivative respectively. Fig. 5b is displaying the 2D graphical comparison of fractional operators. Fig. 5is illustrating the 2D graphical comparison of two different fractional operators using the parametric values 𝜒=2,𝛽=2.5,𝛾= 1,𝜔 =0.01 and 𝑘 =0.5at different fractional order for the normalized electric field 𝑃1(𝑥, 𝑡)and normalized density perturbation 𝑅1(𝑥, 𝑡). Fig. 5a and Fig. 5c are presenting the 2D comparison of normalized electric field with conformable operator, and Atangana-Baleanu derivative respectively. Fig. 5b and Fig. 5d are presenting the 2D comparison of normalized density perturbation with conformable operator and Atangana-Baleanu derivative respectively. Remark. The solution 𝑃1(𝑥, 𝑡)and 𝑅1(𝑥, 𝑡)are graphically presenting the anti-cusped with decay type soliton behavior for the normalized electric field and the normalized density perturbation respectively, with conformable operator and the Atangana-Baleanu operator. The Atangana-Baleanu operator is describing the different pattern from the conformable operator for NEF and NDP. The existence of cusps typically refers to sharp points or regions of rapid change in the normalized electric field and normalized density perturbation in Lagumir and ion sound oscillations. It is important to notice that, the variation of fractional order has significant impact the propagation of waves. Thus, the fractional order has tendency to control the oscillations of the electron density in a plasma and the motion of ions and the associated electric fields. Fig. 6is illustrating the 2D graphical comparison and 3D wave propagation with two different fractional operators using the parametric numbers, 𝜒=2,𝛽=0.5,𝛾=1,𝜔 =0.1and 𝑘 =1.1at fractional order 𝛼=0.1for the normalized electric field 𝑃3(𝑥, 𝑡)and normalized density perturbation 𝑅3(𝑥, 𝑡). Fig. 6a and Fig. 6c are presenting the 3D wave propagation of normalized electric field with conformable operator, and AtanganaBaleanu derivative respectively. Fig. 6e is displaying the 2D graphical comparison of fractional operators. Fig. 6b and Fig. 6d have been 3D anatomization of normalized density perturbation with conformable operator and Atangana-Baleanu derivative respectively. Fig. 6f is displaying the 2D graphical comparison of fractional operators. Fig. 7is illustrating the 2D graphical comparison and 3D wave propagation with two different fractional operators using the parametric numbers, 𝜒=2,𝛽=0.5,𝛾=1,𝜔 =0.1and 𝑘 =1.1at fractional order 𝛼=0.7for the normalized electric field 𝑃3(𝑥, 𝑡)and normalized density perturbation 𝑅3(𝑥, 𝑡). Fig. 7a and Fig. 7c are presenting the 3D wave propagation of normalized electric field with conformable operator, and AtanganaBaleanu derivative respectively. Fig. 7e is displaying the 2D graphical comparison of fractional operators. Fig. 7b and Fig. 7d have been 3D anatomization of normalized density perturbation with conformable operator and Atangana-Baleanu derivative respectively. Fig. 7f is displaying the 2D graphical comparison of fractional operators.