https://orcid.org/0000-0002-1143-2699
Figures
Abstract
The (3+1)-dimensional mKdV−ZK model is an important framework for studying the dynamic behavior of waves in mathematical physics. The goal of this study is to look into more generic travelling wave solutions (TWSs) for the generalized ion-acoustic scenario in three dimensions. These solutions exhibit a combination of rational, trigonometric, hyperbolic, and exponential solutions that are concurrently generated by the new auxiliary equation and the unified techniques. We created numerous soliton solutions, including kink-shaped soliton solutions, anti-kink-shaped solutions, bell-shaped soliton solutions, periodic solutions, and solitary soliton solutions, for various values of the free parameters in the produced solutions. The attained solutions are displayed geometrically in the surface plot (3-D), contour, and combined two-dimensional (2-D) figures. The combined 2-D figure would make it easier to understand the impact of the speed of the wave. Based on time, the influence of the nonlinear parameter β on wave type is comprehensively investigated using various figures, demonstrating the significant impact of nonlinearity. These graphical representations are based on specific parameter settings, which help to grasp the model’s intricate general behavior. However, the results of this research are compared with the outcomes obtained in published literature executed by other scholars. The results indicate the approach’s effectiveness and reliability, making it suitable for widespread use in a range of sophisticated nonlinear models. These techniques successfully generate inventive soliton solutions for various nonlinear models, which are crucial in mathematical physics.
Citation: Arafat SMY, Asif M, Rahman MM (2025) Nonlinear dynamic wave properties of travelling wave solutions in in (3+1)-dimensional mKdV−ZK model. PLoS ONE 20(1): e0306734. https://doi.org/10.1371/journal.pone.0306734
Editor: Ghulam Rasool, Beijing University of Technology, CHINA
Received: June 23, 2024; Accepted: September 3, 2024; Published: January 8, 2025
Copyright: © 2025 Arafat et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The data are all contained within the manuscript.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
In a wide range of scientific, technical, and technological domains, theoretical outcomes are superior to experimental results in comprehending physical processes. Real characteristic phenomena of our world are described with the help of nonlinear evolution models (NLEMs). There are a lot of applications of NLEMs in several domains of physics, mathematics, and engineering. For instance, plasma physics [1,2], fluid dynamics [3,4], biomathematics [5], nonlinear optics [6,7], shallow water waves [8,9], and many others [10–15]. Many researchers also discussed phase portraits, maximal Lyapunov exponents, Neimark–Sacker bifurcation, period-doubling bifurcation diagrams, stability analysis, exploring fractional order and many other physical phenomena [16–27]. It is a very challenging task to find the exact solution for NLEMs, but researchers are very interested in discovering potent and effective techniques. Because of the widespread applications and importance of nonlinear models, many scholars have developed a variety of useful tools and approaches to analysis partial differential equations (PDEs) such as, -expansion approach [28,29], F-expansion method [30], tanh-coth method [31], first integral technique [32], direct algebraic method [33], inverse scattering method [34], sin-Gordon expansion [35], new auxiliary equation approach [36], modified simple equation method [37], Hirota bilinear method [38,39], unified method [40], modified modified exp-function method [41–43] and so on.
The (3+1)-dimensional mKdV−ZK model describes the weakly nonlinear ion-acoustic waves in a magnetized electron-positron plasma with equally hot and cool components [44]. In addition, it describes the evolution of ion-acoustic disturbances in a magnetized plasma with two negative ion components of different temperatures. Therefore, the (3+1)-dimensional mKdV−ZK model provides a base for theoretical research and application across a wide range of scientific areas. The (3+1)-dimensional mKdV−ZK [45] model is given in the following form:
(1.1)
where β is an arbitrary constant that measures the nonlinearity’s strength, u is a wave function with x,y,z, and t are the independent variables, the subscripts stipulate the partial derivatives. Also, u2ux is a nonlinear term known as cubic nonlinear that indicates the wave profile effect on linear wave propagation and represents cubic relationships.
Throughout history, numerous efficient and compatible methodologies have been used to analyze the mKdV−ZK model such as, the improved fractional sub-equation method [46], the modified Riemann–Liouville derivative, the exp-function, the -expansion, and the generalized Kudryashov method [47], the improved generalized Riccati equation mapping method [48], the improved Bernoulli sub-equation method [49], the Sardar-sub equation method [50]. the variable separated ODE method [51] and more.
Several researchers have investigated the new auxiliary equation and unified methodologies to examine the exact TWSs of a large number of NLEMs. To the best of our knowledge, researcher has not been found TWSs to the (3+1)-dimensional mKdV−ZK model through new auxiliary equation and unified methods.
Therefore, in the above-mentioned academic study, the core determination of our article is to provide standard, compatible, and stable TWSs to the specified model using these methods. Moreover, we represent the physical illustration as well as the 3-dimensional, contour and combined 2- dimensional graphical depiction are displayed to the investigated solutions. Understanding the (3+1)-dimensional mKdV-ZK model is theoretical knowledge; its understanding of wave behavior also enables scientists and engineers to create more efficient communication networks and better understand wave interactions in natural environments. The outcomes of this study will be significant in clarifying the meaning of the difficult physical phenomena in fluid mechanics, high-frequency plasma physics, marine engineering, ocean physics, and solitary wave theory.
The rest of the manuscript is organized as follows: In part 2, we discuss the methodology including new auxiliary equation unified methods as well as discuss the application of the stated model via abovementioned methods in part 3. The outcomes of these model are graphically present and give physical explanation with real life significant in part 4. Finally, the paper’s conclusion is reached.
2.Overview the schemes
We will briefly describe two analytical techniques for exploring some soliton solutions of the PDEs in this section. We suppose the PDEs with variables z,y,z, and t of a function F in the desired construction:
(2.1)
in this case, F is a nonlinear polynomial function covering wave function u(x,y,z,t), including its various partial derivatives respect to x,y,z, and t. By using a proper wave transform
(2.2)
Eq (2.1) becomes
(2.3)
The symbol (′) indicates the derivative with respect to ξ.
2.1 The new auxiliary equation scheme
In this sub-section, we describe the total procedure to the new auxiliary equation method [36]. We assume that the TWSs to the Eq (2.3) are the following form:
(2.4)
where kj(j = 0,1,2,…,S) are constants to be compute, such that kj≠0 and g(ξ) satisfies the following equation
(2.5)
we determine the positive integer S applying the balancing between the highest order derivatives and highest order nonlinear terms in Eq (2.3). Substituting Eqs (2.4) and (2.5) into Eq (2.3), producing an algebraic equation where the left and right sides are determined by the powers of
. After resolving these equations, we get a system of algebraic equation and calculate the values of
and other variables. Finally, the real constants
and g(ξ) putting into Eq (2.4), yield many TWSs from the Eq (2.1).
Case-1: when n2−4ml<0 and l≠0,
or
Case-2: when n2−4ml>0 and
or
Case-3: when and l = −m,
or
Case-4: when and l = −m,
or
Case-5: when n2−4m2<0 and
or
Case-6: when n2+4m2>0 and
or
Case-7: when
Case-8: when ml<0, n = 0 and
or
Case-9: when n = 0 and m = −l,
Case-10: when m = l = 0,
Case-11: when m = n = C and
Case-12: when n = l = ϕ and
Case-13: when .
Cas-14: when
Case-15: when
Case-16: when
Case-17: when l = n = 0,
Case-18: when
Case-19: when l = m and
Case-20: when
2.2 The unified scheme
In this sub-section, we describe the total procedure to the unified method [52]. Assume that the TWSs to the Eq (2.3) is represented in the following form:
(2.6)
where kj(j = 1,2,…,S) and lj(j = 1,2,…,S), ks and ls cannot both be zero simultaneously due to constants that will be examined further.
satisfy the Riccati differential equation.
(2.7)
we determine the positive integer S applying the balancing between the highest order derivatives and highest order nonlinear terms in Eq (2.3). Substituting Eqs (2.6) and (2.7) into Eq (2.3), producing the same powers of aj, (j = 0,1,2,…), then setting each coefficient of aj be zero yield a set of algebraic equation in terms of lj, cj and λ. Substituting lj, cj and λ into (2.6) with the help of (2.7) we obtained the TWSs of Eq (2.1) for various condition of λ.
Case-03: when
where ℬ and ℛ are two real type arbitrary parameters, and h is also arbitrary constant.
3. Mathematical formulation of the model
In this section, we will apply the new auxiliary equation method and the unified method to mKdV−ZK model to explore TWSs and mathematical analysis. Let us consider the travelling wave transformation
(3.1)
Utilizing Eq (1.1) with the help of Eq (3.1), then we get the form:
(3.2)
Integrating and simplifying of Eq (3.2), the required form:
(3.3)
By using the balancing procedure in (3.3), we find S = 1.
3.1 Solution analysis through the new auxiliary equation method
We attain the balance value of S from (3.3) the general solution of (2.4) takes the following form:
(3.4)
where k0, and k1 are constants and to be evaluated latter. By inserting Eq (3.4) and Eq (2.5) into Eq (3.3), and then setting the coefficients of ag(ξ) to zero, we may build a set of algebraic equations that Maple can solve to reach the following solution sets:
Set-1:
Set-2:
Inserting the Set-1 values in Eq (3.4) along with Eq (3.3), we can attain the following solutions as the mKdV−ZK model.
When n2−4ml<0 and l≠0,
or
When n2−4ml>0 and l≠0,
or
When n2+4m2<0, l≠0 and l = −m,
or
When n2+4m2>0, l≠0 and l = −m,
or
When n2−4m2<0 and l = m,
or .
When n2+4m2>0 and l = m,
or
When n2 = 4ml,
When ml<0, n = 0 and l≠0,
or
When n = 0 and m = −l,
When n = l = ϕ and m = 0,
When n = (m+l),
When n = −(m+l),
When m = 0,
When l = n = m≠0,
When m = n = 0,
When l = m and n = 0,
Inserting the Set-2 values in Eq (3.4) along with Eq (3.3), we can attain the following solutions as the mKdV−ZK model.
When n2−4ml<0 and l≠0,
or
When n2−4ml>0 and l≠0,
or
When n2+4ml2<0, l≠0 and l = −m,
or
When n2+4m2>0, l≠0 and l = −m,
or
When n2−4m2<0 and l = m,
or
When n2+4m2>0 and l = m,
or
When n2 = 4ml, .
When ml<0, n = 0 and l≠0,
or
When n = 0 and m = −l,
When n = l = ϕ and m = 0,
When n = (m+1),
When n = −(m+1),
When m = 0,
When l = n = m≠0,
When m = n = 0,
When l = m and n = 0,
3.2 Solution analysis through the unified method
Based upon the number of balance principle S the trial solution of Eq (3.3) become of the following form:
(3.5)
Where k0, k1 and l1 are arbitrary constant and k1, l1 cannot both be zero simultaneously. Inserting Eq (3.5) and Eq (2.7) into Eq (3.3), and after that, adjusting the φ(ξ) factors to zero, we are able to develop following set of collection:
Set-1:
Set-2:
Set-3:
Set-4:
Set-5:
Set-6:
Where
Inserting the Set-1 values in Eq (3.5) along with Eq (3.3), we can attain the following solutions as the mKdV−ZK model.
Inserting the Set-2 values in Eq (3.5) along with Eq (3.3), we can attain the following solutions as the mKdV−ZK model.
Inserting the Set-3 values in Eq (3.5) along with Eq (3.3), we can attain the following solutions as the mKdV−ZK model.
Inserting the Set-4 values in Eq (3.5) along with Eq (3.3), we can attain the following solutions as the mKdV−ZK model.
Inserting the Set-5 values in Eq (3.5) along with Eq (3.3), we can attain the following solutions as the mKdV−ZK model.
Inserting the Set-6 values in Eq (3.5) along with Eq (3.3), we can attain the following solutions as the mKdV−ZK model.
4. Result and discussion
This part is divided into two subparts. Subpart 4.1 shows a graphical illustration of the obtained solutions, and Subpart 4.2 provides a physical description. Using MATLAB software, 3D, Contour, and combined 2D wave profiles are made by choosing an appropriate undefined parameter in the (3+1)-dimensional mKdV-ZK model. Functions with two-dimensional input and one-dimensional output can be displayed using three-dimensional graphs. In data analysis, these graphs are frequently used to identify the highest and lowest levels in a multidimensional data collection. This part aims to illustrate the solutions discovered during this research. We study the physical use of the wave.
4.1 Graphically illustration
4.2. Physical description
Our main focus was on the wave function u(x,y,z,t) solutions to the (3+1)-dimensional mKdV−ZK models and how the wave profiles are influenced by the parameters. The steady propagation of each wave is depicted in 3D, contour, and combination 2D images. The 3D representation of the solution u3(x,y,z,t) for the choosing values of parameters of p = 0.3, q = 1, r = 1, m = 0.3, n = 2, l = 0.7, β = −0.1, y = z = 0. The 3D structure presents the flat kink shaped soliton of this solution, which is portrayed in Fig 1(A) and involved contour in Fig 1(B) is plotted. Fig 1(C) demonstrates the progression of the waves for different values of t = 0.1, 0.1, 1. The desired solution u5(x,y,z,t) highlights flat anti-kink shaped soliton for taking free parameter of p = −1, q = 0.3, r = −0.1, m = 0.9, n = 1, l = −0.9, β = −1, y = z = 1 as seen in Fig 2(A) and equivalent contour in Fig 2(B) are plotted respectively. Fig 2(C) indicates the progression of the waves for various values of t = 0.3,0.9,1.5. The 3D representation of the modulus solution u11(x,y,z,t) for the suitable values of parameters of p = 0.3, q = 0.2, r = 0.5, m = 0.25, n = 1.2, l = 0.25, β = −10, y = z = 1. The 3D structure presents the dark soliton or V-shaped soliton of this solution, which is displayed in Fig 3(A) and associated contour in Fig 3(B) is plotted. Fig 3(C) shows the progression of the waves for different values of t = 1, 5, 9. The 2D plot provides a better understanding the effect of free parameters for different values of t and shows various positions of these values. We depict the 3D wave structure of the solution u20(x,y,z,t) for the parameters of The 3D structure represent the smooth anti-kink shaped soliton of this solution, which is portrayed in Fig 4(A) and associated contour in Fig 4(B) is plotted. Fig 4(C) demonstrates the progression of the waves for several values of t = 1, 5, 9. The 3D surface of the solution u55(x,y,z,t) conveys the smooth anti-kink soliton for the selecting parametric values of
The 3D structure is displayed in Fig 5(A) and its corresponding contour in Fig 5(B) is plotted. Also, Fig 5(C) shows the progression of the waves for distinguished values of λ = −2,−4,−6. The desired solution u57(x,y,z,t) shows smooth kink shaped soliton for picking unrestricted parameter of
, is depicted in Fig 6(A). and related contour in Fig 6(B) are plotted respectively. Fig 6(C) demonstrates the progression of the waves for distinct values of λ = −1,−2,−3. The 3D wave structure of the modulus solution u59(x,y,z,t) signifies the bell shaped soliton (which another name bright soliton studying in optical fiber) for the free parameter of
The 3D structure is showed in Fig 7(A) and associated contour in Fig 7(B) is plotted. Fig 7(C) shows the progression of the waves for different values of t = 1.7, 1.8, 1.9. The absolute solution u62(x,y,z,t) displays M-shaped soliton for the free parameter of
is displayed in Fig 8(A). and associated contour in Fig 8(B) are plotted respectively. Fig 8(C) shows the progression of the waves for different values of λ = 0.12, 0.16, 0.2. A stable and durable wave solution is the M-shaped soliton. It can be provide far-reaching data transmission. The impact of nonlinear parameters β and λ of the obtained solution based on the time exhibits the wave amplitude of the horizontal axis.
4.3 Comparison
In this research article, we will discuss comparison between attained solutions and Al-Ghafri et. al [51] solutions. Al-Ghafri et. al [51] studied of the (3+1)-dimensional space–time fractional mKdV−ZK equation by the variable separated ODE method. If we consider α = 1, the (3+1)-dimensional space-time fractional mKdV−ZK model will be converted (3+1)-dimensional mKdV−ZK model. Using the variable separated ODE method, Al-Ghafri et. al [51] have explored Two case included twelve subcase that contains fifty-four solitons’ solutions. On the other hand, the new auxiliary equation method used to generate many wave solutions for the (3+1)-dimensional mKdV−ZK Model. Both methods have some common solutions shown in Table 1.
In addition to these solutions, we gain forty-three new TWSs u1(ξ), u2(ξ), u5(ξ)−u7(ξ) and u9(ξ)−u46(ξ) in this article that are not mentioned in Al-Ghafri et. al [51].
In addition, Zafar et. al [53] have also explored twenty solitary wave solutions from the conformable time-fractional (3+1)-dimensional mKdV−ZK equation through three integration schemes included as exp(ξ) function scheme, hyperbolic function scheme, and modified kudryashov scheme. As opposed to generate many wave solutions from the stated equation by the mentioned method in this research. Both approaches share some potential solutions, which are compared and contrasted in Table 2.
Apart from these solutions, further new forty-four new exact TWSs u1(ξ)−u8(ξ), u10(ξ) and u12(ξ)−u46(ξ) are established in this article that are not mentioned in Zafar et. al [53].
5.Conclusion
Through the use of the new auxiliary equation approach and the unified technique, we have achieved the exact and precise travelling wave soliton solutions for the (3+1)-dimensional mKdV−Zk model. Under certain scenarios, the travelling wave solutions can be expressed as rational, hyperbolic, and trigonometric functions. However, these techniques produce abundantly distinct free parametric values that depict geometrically kink-shaped soliton solutions, anti-kink-shaped solutions, bell-shaped soliton solutions, and periodic solutions. In addition, the reactions of various nonlinearities strengths, wave velocity, and other model factors are investigated. Several of the solutions found are completely fresh and have not been discussed in any of the previous research. To elucidate the attained outcomes, we have looked at surface, contour, and combined 2-D diagrams. For distinct numerical values of parameters, we demonstrated several 2-D graphs, and watching the graph makes it easy to observe the wave velocity. These extensive results can be helpful resources for investigators as they look at the geometrical structure and understand the system’s physical interpretation. The outcomes of other research studies that are currently available are compared with the findings of this work. The calculation is straightforward, yields more unique results than other techniques currently used, and has a wider range of applications due to decreased consistency and computational tasks. However, these methods fail to produce exact solutions for some fractional balance number models. To summarize, both the unified scheme and the NAE scheme are effective, compatible, and simple approaches for obtaining full wave solutions with a variety of free parameters, providing significant insights into wave profiles across varied contexts. These techniques yielded precise travelling wave soliton solutions, and they are strongly recommended for future research on nonlinear models, which hold significance in mathematical physics.
Acknowledgments
The authors would like to thank the editor of the journal and anonymous reviewers for their generous time in providing detailed comments and suggestions that helped us to improve the paper.
References
- 1. Osman MS., and Ghanbari B. New optical solitary wave solutions of Fokas-Lenells equation in presence of perturbation terms by a novel approach. Optik. 2018; 75: 328–333.
- 2. Arafat SMY., Islam SMR., Rahman MM., Saklayen MA. On nonlinear optical solitons of fractional Biswas-Arshed Model with beta derivative. Results Phys. 2023; 48: 106426.
- 3. Iqbal MA., Akbar MA., Islam MAThe nonlinear wave dynamics of fractional foam drainage and Boussinesq equations with Atangana’s beta derivative through analytical solutions. Results Phys. 2024; 56: 107251.
- 4. Adel W., Rezazadeh H., Inc M. On numerical solutions of telegraph, viscous, and modified Burgers equations via Bernoulli collocation method. Sci Iran. 2024; 31(1): 43–54.
- 5. Shao-Wen Y., Islam ME., Akbar MA., Inc M, Adel M., Osman MSAnalysis of parametric effects in the wave profile of the variant Boussinesq equation through two analytical approaches. Open Phys. 2022; 20(1): 778–794.
- 6. Arafat SMY., Khan K, Islam SMR., Rahman MM. Parametric effects on paraxial nonlinear Schrödinger equation in Kerr media. Chin J Phys. 2023; 83: 361–378.
- 7. Islam SMR. Bifurcation analysis and exact wave solutions of the nano-ionic currents equation: Via two analytical techniques. Results Phys. 2024; 58:107536.
- 8. Arafat SMY., Rahman MM., Karim MF., Amin MR. Wave profile analysis of the (2+ 1)-dimensional Konopelchenko–Dubrovsky model in mathematical physics. Partial Differ Equ Appl. 2023; 8: 100576.
- 9. Akbar MA., Abdullah FA., Islam MT., Al Sharif MAOsman MS. New solutions of the soliton type of shallow water waves and superconductivity models. Results Phys. 2023; 44: 106180.
- 10. Chen J., Pelinovsky DE., Upsal J. Modulational instability of periodic standing waves in the derivative NLS equation. J Nonlinear Sci. 2021; 31(3): 58.
- 11. Osman MS., Rezazadeh H., Eslami M. Traveling wave solutions for (3+ 1)-dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity. Nonlinear Eng. 2019; 8(1): 559–567.
- 12. Ray AK., Bhattacharjee JK. Standing and travelling waves in the shallow-water circular hydraulic jump. Phys Lett A. 2007; 371(3): 241–248.
- 13. Vivas CM., Arshed S., Sadaf M., Perveen Z., Akram G. Numerical simulations of the soliton dynamics for a nonlinear biological model: Modulation instability analysis. PloS One. 2023; 18(2): 0281318.
- 14. Nakkeeran K. Optical solitons in erbium doped fibers with higher order effects. Phys Lett A. 2000; 275(5–6): 415–418.
- 15. Akbar MA., Abdullah FA., Islam MT., Al Sharif MA, Osman MS. New solutions of the soliton type of shallow water waves and superconductivity models. Results Phys. 2023; 44: 106180.
- 16. Berkal M., Almatrafi MB. Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative. Fractal and Fractional. 2023; 7(5): 344.
- 17. Ahmed R., Almatrafi MB. Complex dynamics of a predator-prey system with Gompertz growth and herd behavior. Int J Anal Appl. 2023; 21:100.
- 18. Berkal M., Navarro JF., Almatrafi MB. Qualitative Behavior for a Discretized Conformable Fractional-Order Lotka-Volterra Model with Harvesting Effects. Int J Anal Appl. 2024.
- 19. Kai Y., Yin Z. On the Gaussian traveling wave solution to a special kind of Schrödinger equation with logarithmic nonlinearity. Mod Phys Lett B. 2022; 36(2): 2150543.
- 20. Kai Y., Yin Z. Linear structure and soliton molecules of Sharma-Tasso-Olver-Burgers equation. Phy Lett A. 2022; 452: 128430.
- 21. Uddin MJ., Almatrafi MB, Monira S. A study on qualitative analysis of a discrete fractional order prey-predator model with intraspecific competition. Commun Math Biol Neurosci. 2024.
- 22. Berkal M., Navarro JF., Almatrafi MB., Hamada MY. Qualitative behavior of a two-dimensional discrete-time plant-herbivore model. Commun Math Biol Neurosci. 2024.
- 23. Umer MA., Arshad M., Seadawy AR., Ahmed I., Tanveer M. Exploration conversations laws, different rational solitons and vibrant type breather wave solutions of the modify unstable nonlinear Schrödinger equation with stability and its multidisciplinary applications. Opt Quant Electron. 2024; 56(3): 420.
- 24. Alharbi AR., Almatrafi MB. Exact solitary wave and numerical solutions for geophysical KdV equation. J King Saud Univ Sci. 2022; 34(6): 102087.
- 25. Almatrafi MB. Solitary wave solutions to a fractional model using the improved modified extended tanh-function method. Fractal and Fractional. 2023; 7(3): 252.
- 26. Arshad M., Aldosary SF., Batool S., Hussain I.Hussain NExploring fractional-order new coupled Korteweg-de Vries system via improved Adomian decomposition method. Plos one. 2024; 19(5): e0303426. pmid:38805437
- 27. Zhu C., Al-Dossari M., Rezapour S., Shateyi S., Gunay B. Analytical optical solutions to the nonlinear Zakharov system via logarithmic transformation. Results Phys. 2024; 56: 107298.
- 28. Biswas A., Mirzazadeh M., Eslami M. Dispersive dark optical soliton with Schödinger-Hirota equation by (G′/G)-expansion approach in power law medium. Optik. 2014; 125(16): 4215–4218.
- 29. Shakeel M., Ul-Hassan QMAhmad Applications of the Novel (G′/G)‐Expansion Method for a Time Fractional Simplified Modified Camassa‐Holm (MCH) Equation. Abstr Appl Anal. 2014; 2014(1); 601961.
- 30. Yıldırım Y. Optical solitons with Biswas–Arshed equation by F-expansion method. Optik. 2021; 227: 165788.
- 31. Wazwaz AM. Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method. Appl Math Comput. 2007; 190(1): 633–640.
- 32. Akinyemi L., Mirzazadeh M., Amin BS., Hosseini K. Dynamical solitons for the perturbated Biswas–Milovic equation with Kudryashov’s law of refractive index using the first integral method. J Mod Opt. 2022; 69(3): 172–182.
- 33. Tian Y., Liu J. Direct algebraic method for solving fractional Fokas equation. Therm Sci. 2021; 25(3 Part B): 2235–2244.
- 34. Bao G., Li P., Lin J., Triki F. Inverse scattering problems with multi-frequencies. Inverse Probl. 2015; 31(9): 093001.
- 35. Raslan KR., Soliman AA., Ali KK, Gaber MAlmhdy SR. Numerical Solution for the Sin-Gordon Equation Using the Finite Difference Method and the Non-Stander Finite Difference Method. Appl Math. 2023; 17(2): 253–260.
- 36. Islam SMR., Arafat SMY., Wang H.Abundant closed-form wave solutions to the simplified modified Camassa-Holm equation. J Ocean Eng Sci. 2023; 8(3): 238–245.
- 37. Kaplan M., Bekir A. The modified simple equation method for solving some fractional-order nonlinear equations. Pramana. 2016; 87:1–5.
- 38. Wang S. Novel soliton solutions of CNLSEs with Hirota bilinear method. J Opt. 2023; 52(3): 1602–1607.
- 39. Ma WX. N-soliton solution and the Hirota condition of a (2+ 1)-dimensional combined equation. Math Comput Simul. 2021; 190: 270–279.
- 40. Xiao F., Li G., Jiang D., Xie Y., Yun J., Liu Y., Fang . An effective and unified method to derive the inverse kinematics formulas of general six-DOF manipulator with simple geometry. Mech Mach Theory. 2021; 159: 104265.
- 41. Attaullah Shakeel M., Ahmad B., Shah NA, Chung JDSolitons solution of Riemann wave equation via modified exp function method. Symmetry. 2022; 14(12): 2574.
- 42. Shakeel M., Shah NA., Chung JD. Application of modified exp-function method for strain wave equation for finding analytical solutions. Ain Shams Eng. 2023; 14(3): 101883.
- 43. Shakeel M., Attaullah , Shah NA., Chung JD. Modified exp-function method to find exact solutions of microtubules nonlinear dynamics models. Symmetry. 2023; 15(2): 360.
- 44. Das KP., Verheest F. Ion-acoustic solitons in magnetized multi-component plasmas including negative ions. J plasma phy. 1989; 41(1): 139–155.
- 45. Nonlaopon K., Mann N., Kumar S., Rezaei S., Abdou MA. A variety of closed-form solutions, Painlevé analysis, and solitary wave profiles for modified KdV–Zakharov–Kuznetsov equation in (3+ 1)-dimensions. Results Phys. 2022; 36: 105394.
- 46. Sahoo S., Ray SS. Improved fractional sub-equation method for (3+ 1)-dimensional generalized fractional KdV–Zakharov–Kuznetsov equations. Comput Math with Appl. 2015; 70(2): 158–166.
- 47. Guner O., Aksoy E., Bekir A., Cevikel AC. Different methods for (3+ 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation. Comput Math with Appl. 2016; 71(6): 1259–1269.
- 48. Rehman HU., Awan AU., Hassan AM., Razzaq S. Analytical soliton solutions and wave profiles of the (3+ 1)-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. Results Phys. 2023; 52: 106769.
- 49. Yel G., Sulaiman TA., Baskonus HM. On the complex solutions to the (3+ 1)-dimensional conformable fractional modified KdV–Zakharov–Kuznetsov equation. Mod Phys Lett B. 2020; 34(05): 2050069.
- 50. Yasin S., Khan A., Ahmad S., Osman MS. New exact solutions of (3+ 1)-dimensional modified KdV-Zakharov-Kuznetsov equation by Sardar-subequation method. Opt Quan Electron. 2024; 56(1): 90.
- 51. Al-Ghafri KS., Rezazadeh H. Solitons and other solutions of (3+ 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation. AMNS. 2019; 4(2): 289–304.
- 52. Akcagil S., Aydemir T., A new application of the unified method. New trend math sci. 2018; 6(1):185–189.
- 53. Zafar A., Bekir A. Exploring the conformable time-fractional (3+1)-dimensional modified Korteweg-Devries-Zakharov-Kuznetsov equation via three integration schemes. J Appl Anal Comput. 2021; 11(1): 161–175.