Abundance of Exact Solutions of a Nonlinear Forced (2 + 1)-Dimensional Zakharov–Kuznetsov Equation for Rossby Waves

renmandula, Na; Yin, Xiaojun

doi:https://doi.org/10.1155/2023/6983877

Journal of Mathematics

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Abstract Introduction Analysis and Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 6983877 | https://doi.org/10.1155/2023/6983877

Abundance of Exact Solutions of a Nonlinear Forced (2 + 1)-Dimensional Zakharov–Kuznetsov Equation for Rossby Waves

Na renmandula¹and Xiaojun Yin¹

Academic Editor: Firdous A. Shah

Received27 Oct 2022

Revised23 Jan 2023

Accepted03 Feb 2023

Published28 Mar 2023

Abstract

In this paper, an improved tan (φ/2) expansion method is used to solve the exact solution of the nonlinear forced (2 + 1)-dimensional Zakharov–Kuznetsov equation. Firstly, we analyse the research status of the improved tan (φ/2) expansion method. Then, exact solutions of the nonlinear forced (2 + 1)-dimensional Zakharov–Kuznetsov equation are obtained by the perturbation expansion method and the multi-spatiotemporal scale method. It is shown that the improved tan (φ/2) expansion method can obtain more exact solutions, including exact periodic travelling wave solutions, exact solitary wave solutions, and singular kink travelling wave solutions. Finally, the three-dimensional figure and the corresponding plane figure of the corresponding solution are given by using MATLAB to illustrate the influence of external source, dimension variable y, and dispersion coefficient on the propagation of the Rossby wave.

1. Introduction

In ocean and atmospheric motion, researchers have found that Rossby wave propagation is affected by nonlinear action, and sometimes linear theory cannot explain well the role of Rossby waves at energy conversion and climate change. They constructed mathematical models to reveal the dynamic characteristics of the Rossby wave [1–7]. Also, the exact solutions of these equations are crucial to studying the propagation of Rossby waves [8–11]. So, many methods have been proposed on how to solve the exact solutions to nonlinear equations, for instance, the Hirota method [12–14], the Jacobi elliptic function expansion method [15–18], the G′/G-expansion method [19–23], the Exp (−Φ(ξ))-expansion method [24, 25], the generalised exponential rational function method [26–28], the negative power expansion method [29], the hyperbolic function expansion method [30–33], the extended sub-equation method [34], -expansion method [35], the improved sub-ODE method [36], the Riccati–Bernoulli sub-ODE method [37–40], the Lie symmetry technique [41–47], the fractional sub-equation [48] etc. These are valid methods and tools for computing nonlinear equations.

Manafian et al. proposed a new method to solve nonlinear partial differential equations, namely, the improved tan (φ/2) expansion method [49]. With the help of this method, many classical nonlinear partial differential equations have been investigated and abundant exact solutions have been obtained [50–69]. Mohyud-Din and Irshad used this method to construct an exact solution for the generalised KP equation and explained that it can provide better help for the study of generalised KP equations [60]. Foroutan et al. utilised the improved tan (φ/2) expansion method to study the soliton perturbation in inverse-cubic nonlinear optical metamaterials and confirmed that the soliton is in a superconducting presence in the material by obtaining bright soliton, dark soliton, and strange soliton solutions [64]. Sendi et al. solved nonlinear partial differential equations with the help of an improved tan (φ/2) expansion method. Exact periodic travelling wave solution, exact singular kink travelling wave solution, soliton and so on are obtained, and three-dimensional graphs corresponding to some exact solutions are depicted [66]. Zkan et al. used the improved tan (φ/2) expansion method to obtain the exact solution to the (2 + 1)-dimensional KdV equation and explained the influence of different parameters on wave propagation through three-dimensional graphs and tables [69]. Compared to the negative power expansion method [29], the extended subequation method [34], and the improved sub-ODE method [36], we can obtain more formal solutions using the improved tan (φ/2) expansion method, which is one of the efficient mathematical methods and tools that are widely used and easy to implement.

In reference [11], Yin et al. deduced that the amplitude of the large-amplitude Rossby long waves satisfies the forced Zakharov–Kuznetsov (ZK) equation based on the potential vorticity equation by utilising the perturbation expansion method and the coordinate change method. In this study, we count the case that the external source of ZK equation is constant as follows:

Here, the third derivative term represents the dispersion effect, denotes the longitude variable, and represents the influence of the external source. The research shows that these three factors have a certain weight on wave propagation. Equation (1) reflects the characteristics of the large-amplitude Rossby long waves and describes two-dimensional Rossby waves, which can show the propagation of Rossby waves more comprehensively. At , equation (1) can be simplified to the forced KdV equation, described by one-dimensional Rossby solitary waves.

The 1-soliton and 2-soliton solution of the equation is derived using the coupled Burgers equation without considering the solution of external source, and the solitary waves are obtained to explain the influence of external source on the propagation of Rossby waves with the help of the extended Jacobi expansion method of elliptic functions in [11]. Other solutions of equation (1) have not been reported.

To better study the characteristics of the Rossby long wave reflected by equation (1), we use the improved tan (φ/2) expansion method with the help of mathematics to obtain exact solutions. More forms of exact solutions, including exact periodic solutions, exact solitary wave solutions, and singular kink travelling wave solutions, are derived. Then, with the help of MATLAB image processing software, we draw the three-dimensional figure and plane figures corresponding to the partial solution to equation (1). What is more, we analyse the effects of external sources, longitude variables, and dispersion coefficients on Rossby wave propagation in more detail with graphs, and give the corresponding conclusions.

2. Use of an Improved tan (φ/2) Expansion Method

Using the transform , we substitute into equation (1), which can be simplified to the following ordinary differential equation:

We integrate once on both sides of the equation (2) with respect to , getting

We assume that can be expanded into the form of a power series with respect to , namely,where satisfies the following ordinary differential equation:

For details of , we refer to [49].

Using the homogeneous equilibrium method for equation (3), considering and we obtain thus

For formula (4), we consider , then it can be transformed into

Substituting equations (5) and (6) into (3), and using the numerical calculation software Mathematics, we can get the following conclusions.

Case 1. Referring to the 19 kinds of results in [66], we obtain the trigonometric functional solutions, hyperbolic functional solutions, exponential functional solutions, and rational functional solutions of equation (3).

Case 2. With 19 kinds of results of reference [49], we obtain solutions of equation (3)

Case 3. From the 19 results of reference [66], we can obtain solutions of equation (3) of the formHere

3. Analysis and Discussion

In this section, we perform numerical simulations for the exact solution to the forced ZK equation. We use MATLAB drawing software to draw the three-dimensional and plane figures of the partial solutions. From the graph, the following conclusions can be drawn:

By depicting the exact periodic travelling wave solution to equation (1), we can clearly see the effect of external sources and terms with coefficients on wave propagation. When the parameter is taken as and the external source is not considered, the corresponding wave amplitude height to Figure 1 is 54. When the external source is considered, that is, and , it can be seen from the corresponding three-dimensional figure and plane figure in Figures 2 and 3, the amplitude heights are 53.9942 and 53.9663, respectively, so it can be concluded that with the increase in the external source , the wave amplitude height of the same time wave is decreasing. In Figure 4, we observe the influence of the longitude variable on the wave propagation without considering the external source. Compared to the height of the moment in Figure 1, the height changes from 54 to 52.803, which indicates that the wave amplitude increases with the increase in the longitude variable and the height being reduced, which is consistent with the propagation effect of the external source on the wave amplitude. Through Figure 5, we also observed the case of considering the change of external source and longitude variable at the same time. Compared to the height of the moment in Figure 1, the height changed from 54 to 52.7645, which indicates that when the external source and longitude variables change at the same time, the drop in the height of the wave amplitude is obvious; compared with only changing the external source, the height change is more obvious. Combining the above analysis, we can conclude that both the external source and longitude variables play a very important role in the propagation of Rossby waves. Figures 1–5 and 6 correspond to and the three-dimensional and planar graphs of the solutions, which are the exact periodic solutions to the forced ZK equation (1).

We consider an external source and variable , Figures 7 and 8, respectively, represent, when and the corresponding three-dimensional graphs and with the plane graphs of the exact solitary solution of the forced ZK equation (1). It can be seen from the graphs that with the increase in the dispersion coefficient , the wave amplitude height changed from 8.54017 to −9.2352, and the height obviously decreased, which was consistent with the previous results. Figures 7, 8, 9 and 10 show the corresponding three-dimensional graphs and plane graphs of the exact solitary-like solutions , and , respectively, for forcing the ZK equation (1).

We also use Figures 11 and 12 to plot the three-dimensional graphs and the corresponding planar graphs at different times for and of the exact singular kink-type travelling wave solutions and of the forced ZK. From Figure 11, it can be seen from the graph that with the change of time, the propagation height of the wave is changing, especially from to the propagation height of the arriving wave has dropped significantly, indicating that the propagation of the wave is significantly affected by the time variable during this period, while the propagation of the arriving wave is significantly affected by the time variable. But from to , the height drop is not very obvious, indicating that the wave propagation is not significantly affected by time variables during this period and basically tends to be stable.

4. Conclusions

In this paper, the method of the auxiliary equation is used to solve nonlinear equations, and the improved tan (φ/2) expansion method is used to obtain various exact solutions of the forced ZK equation. More new forms of exact solutions, including exact periodic solutions, exact solitary waves solutions, rational function solutions, and singular kink-type travelling wave solutions have been obtained. This shows that the improved tan (φ/2) expansion method is an effective mathematical method and tool for solving nonlinear partial differential equations. Then, with the help of MATLAB image processing software, we depict the three-dimensional graphics and plane graphics corresponding to these exact solutions, analyse the influence of external sources, longitude variables, and dispersion coefficient on Rossby wave propagation, and illustrate the influence of y and dispersion coefficient on Rossby wave amplitude. Compared to reference [11], we only use one solution method to take into account the influence of external sources on wave propagation and describe the influence of external sources on wave propagation more vividly through graphics, and we obtained many forms of solutions. These results have a certain practical significance for researchers exploring Rossby propagation of oceanography and atmospheric motions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (Grant no. 12262025 and 12102205), the Scientific Research Ability of Youth Teachers of Inner Mongolia Agricultural University, the Inner Mongolia Natural Science Foundation project (Grants no. 2022QN01003), the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (Grant no. NJYT23099, NJZY23114 and NJZY22510), the Basic Science Research Foundation of Inner Mongolia Agricultural University (Grant no. JC2018003), and the Program for improving the Scientific Research Ability of Youth Teachers of Inner Mongolia Agricultural University (Grant no. BR220126).

References

R. Long, “Solitary waves in the westerlies,” Journal of the Atmospheric Sciences, vol. 21, no. 2, pp. 197–200, 1964.
View at: Publisher Site | Google Scholar
H. Ono, “Algebraic Rossby wave soliton,” Journal of the Physical Society of Japan, vol. 50, no. 8, pp. 2757–2761, 1981.
View at: Publisher Site | Google Scholar
H. W. Yang, D. Z. Yang, Y. Shi, S. Jin, and B. Yin, “Interaction of algebraic Rossby solitary waves with topography and atmospheric blocking,” Dynamics of Atmospheres and Oceans, vol. 71, pp. 21–34, 2015.
View at: Publisher Site | Google Scholar
W. Y. Hong, G. Min, and H. He, “Conservation laws of space-time fractional mZK equation for Rossby solitary waves with complete coriolis force,” International Journal of Nonlinear Sciences and Numerical Stimulation, vol. 20, pp. 1–16, 2019.
View at: Google Scholar
H. Yang, J. Sun, and C. Fu, “Time-fractional Benjamin-Ono equation for algebraic gravity solitary waves in baroclinic atmosphere and exact multi-soliton solution as well as interaction,” Communications in Nonlinear Science and Numerical Simulation, vol. 71, pp. 187–201, 2019.
View at: Publisher Site | Google Scholar
L. Chen and L. Yang, “A new three-dimensional dissipative boussinesq equation for Rossby waves and its multiple soliton solutions,” Results in Physics, vol. 26, pp. 1–6, 2021.
View at: Google Scholar
X. Q. Yang, E. Fan, and N. Zhang, “Propagation and modulational instability of Rossby waves in stratified fluids,” Chinese Physics B, vol. 13, pp. 96–108, 2022.
View at: Google Scholar
R. Zhang, L. Yang, J. Song, and H. Yang, “(2+1) dimensional Rossby waves with complete Coriolis force and its solution by homotopy perturbation method,” Computers & Mathematics with Applications, vol. 73, no. 9, pp. 1996–2003, 2017.
View at: Publisher Site | Google Scholar
B. J. Zhao, R. Y. Wang, and W. J. Sun, “Combined ZK-mZK equation for Rossby solitary waves with complete Coriolis force and its conservation laws, as well as exact solutions,” Advances in Difference Equations, vol. 2018, no. 1, pp. 1–16, 2018.
View at: Google Scholar
Z. Yu, Z. Zhang, and H. Yang, “(2+1) dimensional coupled Boussinesq equations for Rossby waves in a two-layer cylindrical fluid,” Communications in Theoretical Physics, vol. 73, no. 11, pp. 1–12, 2021.
View at: Google Scholar
X. J. Yin, Q. S. Liu, and S. T. Bai, “The multiple kink solutions and interaction mechanism with help of the coupled Burgers’ equation,” Chinese Journal of Physics, vol. 77, pp. 335–349, 2022.
View at: Publisher Site | Google Scholar
S. Kumar and B. Mohan, “A study of multi-soliton solutions, breather, lumps, and their interactions for the Kadomtsev-petviashvili equation with variable time coefficient using the hirota method,” Physica Scripta, vol. 96, no. 12, pp. 1–11, 2021.
View at: Google Scholar
P. Clarkson, “THE direct method in soliton theory (cambridge tracts in mathematics 155) by ryogo hirota (translated and edited by atsushi nagai, jon nimmo and claire gilson): 200 pp., £40.00 (US$65.00), ISBN 0-521-83660-3 (cambridge university press, 2004),” Bulletin of the London Mathematical Society, vol. 38, no. 06, pp. 1054–1056, 2006.
View at: Publisher Site | Google Scholar
S. Kumar, B. Mohan, and K. Amit, “Generalized fifth-order nonlinear evolution equation for the Sawada-Kotera, Lax, and Caudrey-Dodd-Gibbon equations in plasma physics: painlevé analysis and multi-soliton solutions,” Physica Scripta, vol. 97, no. 3, pp. 1–8, 2022.
View at: Google Scholar
Z. Fu, S. Liu, S. Liu, and Q. Zhao, “New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations,” Physics Letters A, vol. 290, no. 1-2, pp. 72–76, 2001.
View at: Publisher Site | Google Scholar
J. B. Liu, L. Yang, and K. Q. Yang, “Jacobi elliptic function solutions of some nonlinear PDEs,” Physics Letters A, vol. 325, no. 3-4, pp. 268–275, 2004.
View at: Publisher Site | Google Scholar
Q. Liu and J. M. Zhu, “Exact Jacobian elliptic function solutions and hyperbolic function solutions for Sawada–Kotere equation with variable coefficient,” Physics Letters A, vol. 352, no. 3, pp. 233–238, 2006.
View at: Publisher Site | Google Scholar
H. L. Wang and C. H. Xiang, “Jacobi elliptic function solutions for the modified Korteweg–de Vries equation,” Journal of King Saud University Science, vol. 25, no. 3, pp. 271–274, 2013.
View at: Publisher Site | Google Scholar
M. Wang, X. Li, and J. Zhang, “The -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008.
View at: Publisher Site | Google Scholar
E. M. E. Zayed and K. A. Gepreel, “The G´/G-expansion method for finding the traveling wave solutions of nonlinear partial differential equations in mathematical physics,” Journal of Mathematical Physics, vol. 50, no. 1, pp. 1–14, 2009.
View at: Google Scholar
L. Chun-Ping, “(G′/G)-Expansion method equivalent to extended tanh function method,” Communications in Theoretical Physics, vol. 51, no. 6, pp. 985–988, 2009.
View at: Publisher Site | Google Scholar
M. N. Alam, M. A. Akbar, and S. T. Mohyud-Din, “A novel (G´/G)-expansion method and its application to the Boussinesq equation,” Chinese Physics B, vol. 2, no. 23, pp. 1–10, 2014.
View at: Google Scholar
J. Manafian and M. Lakestani, “Solitary wave and periodic wave solutions for Burgers, Fisher, Huxley and combined forms of these equations by the (G′/G)-expansion method,” Pramana, vol. 85, no. 1, pp. 31–52, 2015.
View at: Publisher Site | Google Scholar
K. Khan and M. A. Akbar, “Application of exp(−Φ(η))-expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation,” World Applied Sciences Journal, vol. 24, pp. 1373–1377, 2013.
View at: Google Scholar
H. O. Roshid, M. R. Kabir, R. C. Bhowmik, and B. K. Datta, “Investigation of Solitary wave solutions for Vakhnenko-Parkes equation via exp-function and Exp(−ϕ(ξ))-expansion method,” SpringerPlus, vol. 3, no. 1, p. 692, 2014.
View at: Publisher Site | Google Scholar
K. Nonlaopon, N. Mann, S. Kumar, S. Rezaei, and M. A. Abdou, “A variety of closed-form solutions, Painlevé analysis and solitary wave profiles for the modified KdV–Zakharov–Kuznetsov equation in (3 + 1)-dimensions,” Results in Physics, vol. 36, pp. 1–12, 2022.
View at: Google Scholar
S. Kumar and D. Kumar, “Analytical soliton solutions to the generalised (3 + 1) - dimensional shallow water wave equation,” Modern Physics Letters B. Condensed Matter Physics, Statistical Physics, Applied Physics, vol. 36, no. 2, pp. 1–16, 2022.
View at: Google Scholar
S. Kumar, I. Hamid, and M. A. Abdou, “Specific wave profiles and closed-form soliton solutions for generalized nonlinear wave equation in (3+1)-dimensions with gas bubbles in hydrodynamics and fluids,” Journal of Ocean Engineering and Science, vol. 8, no. 1, pp. 91–102, 2023.
View at: Publisher Site | Google Scholar
B. Xu and S. Zhang, “Exact solutions of nonlinear equations in mathematical physics using the negative power expansion method,” Journal of Mathematical Physics, Analysis, Geometry, vol. 3, no. 17, pp. 369–387, 2021.
View at: Google Scholar
D. Siren, “Constructing an infinite number of exact travelling wave solutions of nonlinear evolution equations via an extended tanh function method,” International Journal of Nonlinear Sciences and Numerical Stimulation, vol. 24, no. 3, pp. 161–168, 2017.
View at: Google Scholar
Η. A. Abdusalam, “On an improved complex tanh-function method,” International Journal of Nonlinear Sciences and Numerical Stimulation, vol. 6, no. 2, pp. 99–106, 2005.
View at: Google Scholar
A. M. Wazwaz, “The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1196–1210, 2005.
View at: Publisher Site | Google Scholar
E. Parkes and B. Duffy, “An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations,” Computer Physics Communications, vol. 98, no. 3, pp. 288–300, 1996.
View at: Publisher Site | Google Scholar
L. Ouahid, M. A. Abdou, and S. Kumar, “Analytical soliton solutions for cold bosonic atoms (CBA) in a zigzag optical lattice model employing efficient methods,” Modern Physics Letters B. Condensed Matter Physics, Statistical Physics, Applied Physics, vol. 36, no. 7, pp. 1–16, 2022.
View at: Google Scholar
E. M. E. Zayed, “The modified (w/g) expansion method and its applications for solving the modified generalised Vakhenko equation,” Italian Journal of Pure and Applied Mathematics, vol. 32, pp. 477–492, 2014.
View at: Google Scholar
S. Zhang, W. Wang, and J. L. Tong, “The improved sub-ODE method for a generalized KdV–mKdV equation with nonlinear terms of any order,” Physics Letters A, vol. 372, no. 21, pp. 3808–3813, 2008.
View at: Publisher Site | Google Scholar
X.-F. Yang, Zi-C. Deng, and W. Yi, “A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application,” Advances in Difference Equations, vol. 2015, no. 1, pp. 1–17, 2015.
View at: Google Scholar
M. Mirzazadeh, R. T. Alqahtani, and A. Biswas, “Optical soliton perturbation with quadratic-cubic nonlinearity by Riccati-Bernoulli sub-ODE method and Kudryashov’s scheme,” Optik, vol. 145, pp. 74–78, 2017.
View at: Publisher Site | Google Scholar
S. M. S. Maha, “A new solitary wave solution of the perturbed nonlinear Schrodinger equation using a Riccati-Bernoulli Sub-ODE method,” International Journal of the Physical Sciences, vol. 11, no. 6, pp. 80–84, 2016.
View at: Publisher Site | Google Scholar
M. A. E. Abdelrahman, “A note on Riccati-Bernoulli Sub-ODE method combined with complex transform method applied to fractional differential equations,” Nonlinear Engineering, vol. 7, no. 4, pp. 279–285, 2018.
View at: Publisher Site | Google Scholar
S. Kumar Dhiman and S. Kumar, “Different dynamics of invariant solutions to a generalized (3+1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili equation arising in shallow water-waves,” Journal of Ocean Engineering and Science, pp. 1–17, 2022.
View at: Google Scholar
S. Kumar and I.nullah Hamid, “Dynamics of closed-form invariant solutions and diversity of wave profiles of (2+1)-dimensional Ito integro-differential equation via Lie symmetry analysis,” Journal of Ocean Engineering and Science, pp. 1–12, 2022.
View at: Google Scholar
S. Kumar and S. Rani, “Invariance analysis, optimal system, closed-form solutions, and dynamical wave structures of a (2 + 1) dimensional dissipative long-wave system,” Physica Scripta, vol. 96, no. 12, pp. 1–19, 2021.
View at: Google Scholar
S. Kumar and S. Rani, “Lie symmetry reductions and dynamics of soliton solutions of (2 + 1)-dimensional Pavlov equation,” Pramana, vol. 94, no. 1, p. 116, 2020.
View at: Publisher Site | Google Scholar
M. R. Ali, W. X. Ma, and R. Sadat, “Lie symmetry analysis and wave propagation in variable-coefficient nonlinear physical phenomena,” East Asian Journal on Applied Mathematics, vol. 12, no. 1, pp. 201–212, 2022.
View at: Publisher Site | Google Scholar
R. Sadat, P. Agarwal, R. Saleh, and R. A. Mohamed, “Lie symmetry analysis and invariant solutions of 3D Euler equations for axisymmetric, incompressible, and inviscid flow in the cylindrical coordinates,” Advances in Difference Equations, vol. 2021, no. 1, pp. 1–16, 2021.
View at: Google Scholar
W. X. Ma, Y. Enas, E. S. Abu, M. R. Ali, and R. Sadat, “Dynamical behaviour and Wave Speed Perturbations in the (2 + 1) pKP equation qualitative,” Theory of Dynamical Systems, vol. 22, no. 1, pp. 1–16, 2022.
View at: Google Scholar
S. Zhang and H. Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, vol. 375, no. 7, pp. 1069–1073, 2011.
View at: Publisher Site | Google Scholar
J. Manafian, M. Lakestani, and A. Bekir, “Study of the analytical treatment of the (2+1)-dimensional zoomeron, the duffing and the SRLW equations via a new analytical approach,” International Journal of Algorithms, Computing and Mathematics, vol. 2, pp. 243–268, 2016.
View at: Publisher Site | Google Scholar
J. Manafian and M. Lakestani, “Optical soliton solutions for the Gerdjikov–Ivanov model via tan(ϕ/2)-expansion method,” Optik, vol. 127, no. 20, pp. 9603–9620, 2016.
View at: Publisher Site | Google Scholar
S. T. Mohyud-Din and A. Irshad, “On exact solutions of modified KdV-ZK equation,” Alexandria Engineering Journal, vol. 55, no. 4, pp. 3253–3265, 2016.
View at: Publisher Site | Google Scholar
J. Manafian and M. Lakestani, “Application of tan(ϕ/2)-expansion method for solving the Biswas–Milovic equation for Kerr law nonlinearity,” Optik, vol. 127, no. 4, pp. 2040–2054, 2016.
View at: Publisher Site | Google Scholar
M. F. Aghdaei and J. Manafian, “Optical soliton wave solutions to the resonant Davey–Stewartson system,” Optical and Quantum Electronics, vol. 48, no. 8, p. 413, 2016.
View at: Publisher Site | Google Scholar
J. Manafian and M. Lakestani, “New improvement of the expansion methods for solving the generalized fitzhugh-nagumo equation with time-dependent coefficients,” International Journal of Engineering Mathematics, vol. 2015, pp. 1–35, 2015.
View at: Publisher Site | Google Scholar
J. Manafian and M. Lakestani, “Abundant soliton solutions for the Kundu–Eckhaus equation via tan(ϕ(ξ))-expansion method,” Optik, vol. 127, no. 14, pp. 5543–5551, 2016.
View at: Publisher Site | Google Scholar
H. Z. Liu and T. Zhang, “A note on the improved tan(ϕ(ξ)/2)-expansion method,” Optik, vol. 131, pp. 273–278, 2017.
View at: Publisher Site | Google Scholar
J . Manafian, “Optical soliton solutions for Schrdinger type nonlinear evolution equations by the tan Φ(ξ)/2)expansion method - ScienceDirect,” Optik, vol. 127, no. 10, pp. 4222–4245, 2016.
View at: Google Scholar
J. Manafian, M. F. Aghdaei, and M. Zadahmad, “Analytic study of the sixth-order thin-film equation by the tan (Φ/2) expansion method,” Optical and Quantum Electronics, vol. 48, no. 8, pp. 410.1–410.14, 2016.
View at: Google Scholar
J. Manafian and M. Lakestani, “Dispersive dark optical soliton with Tzitzéica-type nonlinear evolution equations arising in nonlinear optics,” Optical and Quantum Electronics, vol. 48, no. 2, pp. 1–32, 2016.
View at: Google Scholar
S. T. Mohyud-Din, A. Irshad, N. Ahmed, and U. Khan, “Exact solutions of (3 + 1)-dimensional generalized KP equation arising in physics,” Results in Physics, vol. 7, pp. 3901–3909, 2017.
View at: Publisher Site | Google Scholar
M. Lakestani and J. Manafian, “Application of ITEM for the modified dispersive water-wave system,” Optical and Quantum Electronics, vol. 49, no. 4, pp. 1–17, 2017.
View at: Google Scholar
B. Karaagac, N. M. Yagmurlu, and A. Esen, “A study on the improved tan(ϕ (ξ)/2) −expansion method,” ITM Web of Conferences, vol. 13, Article ID 01031, 2017.
View at: Publisher Site | Google Scholar
F. A. Mehdi, “Kerr-law nonlinearity of the resonant nonlinear Schrodinger’s equation with time-dependent coefficients,” Optical and Quantum Electronics, vol. 49, no. 7, p. 245, 2017.
View at: Google Scholar
M. Foroutan, J. Manafian, and A. Ranjbaran, “Solitons in optical metamaterials with anticubic law of nonlinearity by ETEM and IGEM,” Journal of the European Optical Society Rapid Publications, vol. 14, no. 1, pp. 1–12, 2018.
View at: Google Scholar
K. Hosseini, E. Y. Bejarbaneh, P. Mayeli, and Q. Zhou, “Travelling wave solutions of the Korteweg-de Vries equation with dual-power law nonlinearity using the improved tan(ϕ(ξ)/2)-expansion method,” Optik, vol. 156, pp. 498–504, 2018.
View at: Publisher Site | Google Scholar
C. T. Sendi, J. Manafian, H. Mobasseri, M. Mirzazadeh, Q. Zhou, and A. Bekir, “Application of the ITEM for solving three nonlinear evolution equations arising in fluid mechanics,” Nonlinear Dynamics, vol. 95, no. 1, pp. 669–684, 2019.
View at: Publisher Site | Google Scholar
Y. S. Zkan and E. Yaar, “On the exact solutions of nonlinear evolution equations by the improved tan(φ/2)-expansion method,” Pramana, vol. 94, no. 1, pp. 366–433, 2020.
View at: Google Scholar
H. Günerhan, “Exact traveling wave solutions of the gardner equation by the improved tan Θ -expansion method and the wave ansatz method,” Mathematical Problems in Engineering, vol. 2020, no. 13, Article ID 5926836, 9 pages, 2020.
View at: Google Scholar
Y. S. Özkan, E. Yasar, and N. Celik, “On the exact and numerical solutions to a new (2 + 1)-dimensional Korteweg-de Vries equation with conformable derivative,” Nonlinear Engineering, vol. 10, no. 1, pp. 46–65, 2021.
View at: Publisher Site | Google Scholar

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Copyright © 2023 Na renmandula and Xiaojun Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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