Abstract
In this paper, the accurate closed-form solutions of the Manakov-system are extracted via the extended -expansion method, the -expansion method and generalized Kudryashov method. The solutions are obtained in the form of trigonometric, hyperbolic and rational functions. The dynamical behaviour of obtained solutions is discussed by plotting 3D graphs, 2D contour graphs and density graphs. The symbolic softwares such as Maple and Mathematica are used to plot the graphs of retrieved solutions. The modulation instability (MI) of Manakov-system is also discussed in this paper.
1. Introduction
The Lorentz force law and a set of coupled partial differential equations known as Maxwell's equations form the basis of classical electromagnetism, classical optics and electric circuits. The coupled nonlinear Schrödinger equations are obtained by converting Maxwell's equations into cylindrical coordinates and accounting for an optical fibre's boundary conditions. When the inverse scattering transform is applied to the resultant equations, the Manakov-system is obtained.
Manakov-system [Citation1,Citation2] has been studied many researchers in recent years and novel results have been achieved. It can be written as (1) (1) (2) (2) where and are complex valued functions and 's and 's are constants.
The Manakov-system describes the wave propagation in two mode optical fibres and photo refractive materials [Citation3]. Multicolour solitons are vector solitons that maintain their form throughout propagation and have two or more polarization components coupled to each other. Only coupled nonlinear Schrödinger equations can explain the behaviour of both vector and multicolour solitons. Manakov initially presented the vector soliton in 1974, when he deduced the one-soliton solution and performed an asymptotic analysis for the two-soliton solution using the inverse scattering transform (IST) approach. The auxiliary equation method is used to obtain the V-shaped, periodic singular, periodic bright, etc., soliton solutions of Manakov-system [Citation2]. The physical interpretation of Manakov-system is confirmed in 1996 in the experimental observation of Manakov-solitons [Citation4]. The Manakov-system is a special case of coupled nonlinear Schrödinger equation [Citation5–7].
Many physical phenomena and dynamical processes are modelled by nonlinear evolution equations. The graphical demonstration and stability of NLEEs urged many researchers to find exact solutions using different exact methods. The multiple lump and rogue waves solutions of some nonlinear coupled equations are obtained by different exact methods. For example, the solution of nematic liquid crystals model is obtained by Kerr law [Citation8], the Hirota bilinear approach is used to find the solution of time-fractional resonant nonlinear Schrödinger equation [Citation9], the solution of Zabolotskaya–Khokholov model is obtained by new modified extended direct algebraic [Citation10], the solution of -dimensional generalized Kortewegde–VriesZakharovKuznetsov equation is obtained by new modified extended direct algebraic method [Citation11], to obtain some chirped periodic waves with self-steeping phenomena we investigate the resonant nonlinear Schrödinger equation [Citation12], the soliton solution of nonlinear telegraph equation is obtained by Hirota bilinear method [Citation13], the solution of chlorite iodide malonic acid chemical equation is obtained by residual power series method [Citation14], the chirped periodic wave type solutions of cubic-quintic nonlinear Schrödinger equation is obtained by Jacobi elliptic function [Citation15], the solution of Ablowitz–Kaup–Newell–Segur water waves equation is obtained by painlevé analysis [Citation16], the solution of nonlinear Biswas–Milovic coupled system is obtained in [Citation17] and the solution of time-fractional ion sound and Langmuir waves system is obtained in [Citation18].
The graphical demonstration and stability of Manakov-system urged many researchers to find its exact solutions using different exact methods. The exact solutions of Manakov-system are extracted using bilinear method [Citation19], Darboux transformation method [Citation20,Citation21], extended simplest equation method [Citation22], Hirota method [Citation23,Citation24], modified physics-informed neural network method [Citation25], Hirota's bilinearization method [Citation26], finite difference and finite element method [Citation27], complex Toda chain model [Citation28,Citation29], inverse scattering transform [Citation30], deep learning approach [Citation31], Hamiltonian boundary value method [Citation32], dressing technique [Citation33], Riemann–Hilbert method [Citation34], double-hump soliton solutions with the help of Hirota method [Citation35], trial equation method [Citation1], perturbed complex Toda chain [Citation36], integrable decomposition of Manakov-system [Citation37], polarization modulation instability in a Manakov fibre system [Citation38], extended auxiliary equation method [Citation39], generalization of Manakov equations [Citation40].
The Manakov-system is investigated in the article by employing three most favourite, reliable and authentic integrating strategies. The three integrating techniques are the -expansion method, -expansion method and the generalized Kudryashov method. In order to utilize the proposed techniques, the nonlinear PDEs are first converted into nonlinear ODEs by applying travelling wave transformation. The proposed techniques efficiently extract trigonometric function solutions, hyperbolic function solutions and rational solutions. Trigonometric function solutions can be characterized as periodic solutions. Hyperbolic function solutions can be characterized as dark soliton, kink soliton, bright soliton, singular soliton and complexitons.
Integration schemes applied in this article are most efficient and reliable, but the three have their own limitations. In fact, it is important to state here that every analytical method for nonlinear evolution equations has to satisfy certain constraint conditions. The existence and validity of the obtained solutions depend on these constraint conditions. For example, the classic inverse scattering transform cannot be extended to the cases of power law linearity, dual-power law or even log-law nonlinearity. The method of semi-inverse variational principle is restricted to the retrieval of bright soliton solutions only. Otherwise, for dark or singular solitons, the corresponding stationary integral would be rendered divergent and that prevents universal applicability.
The proposed methods have been successfully utilized in various studies to extract the soliton solutions of some important nonlinear partial differential equations (NLPDEs) such as KDV equation [Citation41], time-fractional parabolic equation [Citation42], strain wave equation [Citation43], Gerdjikov–Ivanov equation [Citation44], Lakshmanan–Porsezian–Daniel model [Citation45,Citation46], Triki–Biswas equation [Citation47], the Zakharov Kuznetsov–Benjamin Bona Mahony equation and the ill-posed Boussinesq equation [Citation48], the nonlinear Schrödinger equation [Citation49], the -dimensional nonlinear dispersive modified Benjamin–Bona–Mahony and the seventh-order Sawada–Kotera–Ito equations [Citation50], -dimensional DLW equations and Maccari's equations [Citation51], the -dimensional generalized Broer–Kaup equation [Citation52], the Korteweg–de Vries equation [Citation53], PHI-four equation and the Fisher equation [Citation54], Westervelt equation [Citation55] and -dimensional Biswas–Milovic equation [Citation56].
This paper has seven sections. Section 2 presents demarcation of Manakov-system. Suggested are elaborated in Section 3. Suggested methods are applied to extract the solutions of Manakov-system in Section 4. The modulation instability of Manakov-system is explained in Section 5. The work is summarized in Section 6.
2. Demarcation of Manakov-system
Considering the following travelling wave transformations: (3) (3) where (4) (4) and (5) (5) where and for i = 1, 2 are the amplitude and phase component, respectively. Substituting Equation (Equation3(3) (3) ) into Equations (Equation1(1) (1) ) and (Equation2(2) (2) ), following real and imaginary parts are achieved as
For imaginary part (6) (6) Real part takes the following form: (7) (7) By applying principle of balance, then (8) (8) Equation (Equation7(7) (7) ) is transformed as (9) (9)
3. Description of methods
In order to utilize the proposed techniques, the nonlinear PDEs are first converted into nonlinear ODEs by applying travelling wave transformation.
3.1. The extended -expansion method
The exact solution of transformed ODE is considered as (10) (10) where satisfies (11) (11) In Equations (Equation10(10) (10) ) and (Equation11(11) (11) ), , 's, 's are unknown constants and and are integers. The formal solutions of Equation (Equation11(11) (11) ) are given in [Citation57].
3.2. The -expansion method
According to -expansion method, solution of ODE is considered as (12) (12) where 's are constants to be determined. In Equation (Equation12(12) (12) ), satisfies the following ODE: (13) (13) where ϖ and χ are arbitrary constants. The general solutions of Equation (Equation13(13) (13) ) are given in [Citation58].
3.3. The generalized Kudryashov method
The following general solution of Kudryashov method is used to obtained the exact solution of transformed ODE (14) (14) where and are constant such that and . Consider the ODE (15) (15) where satisfied. The solution of Equation (Equation15(15) (15) ) is considered as (16) (16) where D is an integrating function.
4. Determination of travelling wave solutions
This section provides novel solutions of the proposed model via suggested integrating techniques.
4.1. The extended -expansion method
The exact solution of Equation (Equation9(9) (9) ) is investigated in Equation (Equation17(17) (17) ). We obtained N = 1 by using homogenous balance principle. The solution is considered in Equation (Equation17(17) (17) ) as follows: (17) (17) where in Equation (Equation17(17) (17) ) the unknown constants are , and . Equations (Equation17(17) (17) ) and (Equation11(11) (11) ) are substituted in Equation (Equation9(9) (9) ). The following system is obtained by collecting all the coefficients of along with i = and putting them equal to zero (18) (18) By solving these algebraic system of equations simultaneously using Mathematica, the following sets of solutions are obtained:
Family 1
According to S1 the trigonometric, hyperbolic and rational functions solution are obtained.
If (), then following periodic solutions are obtained: The graphical simulation of for choosing arbitrary values of parameters as , , , , , , , .
If (), then the following dark soliton solutions are obtained: The graphical simulation of for , , , , , , , .
If and , then where
Family 2
According to S2 the trigonometric, hyperbolic and rational functions solution are obtained.
If (), then the following periodic solutions are obtained: The graphical simulation of for , , , , , , , .
If (), then following singular soliton solutions are obtained: If and , then following rational solutions are obtained: where
Family 3
According to S3 the trigonometric, hyperbolic and rational functions solution are obtained. If(), then following periodic solutions are obtained: If (), then following dark-singular combo soliton solutions are obtained: If and , then following rational solutions are obtained: where
Family 4
According to S4 the trigonometric, hyperbolic and rational functions solution are obtained.
If(), then following periodic solutions are obtained: If (), then following dark-singular combo soliton solutions are obtained: If and , then following rational solutions are obtained: where
4.2. The -expansion method
According to -expansion method, solution of Equation (Equation9(9) (9) ) is considered as (19) (19) where the unknown constants are and . Equations (Equation19(19) (19) ) and (Equation13(13) (13) ) are substituted in Equation (Equation9(9) (9) ). The following system of algebraic equations is obtained by collecting all the coefficients of with i = −3, −2, −1, 0 and putting them equal to zero (20) (20) The following set of solution is obtained by solving these algebraic equations using mathematica:
Family 5
The following solutions of Manakov-system are obtained corresponding to S5.
If and , then The graphical simulation of for , c = 0, , , , is represented in Figure .
If and , then If and and , then The graphical simulation of for , c = 1, , , , .
If and and , then If , then where
4.3. The generalized Kudryashov method
In this section, Equation (Equation9(9) (9) ) is investigated by generalized Kudryashov method. Using homogenous balance principle we obtained N = M + 1. Choosing M = 1 we obtain N = 1. The general solution is assumed as (21) (21) where , , , and are constant. Equation (Equation21(21) (21) ) is substituted in Equation (Equation9(9) (9) ), we attain a polynomial in . By collecting all the coefficient of of same power and putting them equal to zero, the following system is obtained: (22) (22) The following set of solutions are obtained by solving above algebraic system of equations:
Family 6
The hyperbolic function solution of Manakov-system is obtained corresponding to S6. (23) (23) (24) (24)
Family 7
The hyperbolic function solution of Manakov-system is obtained corresponding to S7. (25) (25) (26) (26)
Family 8
The hyperbolic function solution of Manakov-system is obtained corresponding to S8. (27) (27) (28) (28)
5. Graphical representation
The presented graphs show the diverse wave behaviour corresponding to the Manakov-system. Figure represents the periodic solution . Periodic solutions are the solutions expressed in terms of trigonometric functions. The graphical simulations in Figure represent a dark soliton. Physically, dark soliton is a localized surface soliton that causes a temporary decrease in an associated wave amplitude. The periodic solution is represented in Figure . A bright soliton is a localized surface soliton that causes a temporary increase in an associated wave amplitude (Figure ). Thus, Figure shows a dark-bright soliton. The graphical illustrations presented in Figure exhibit a kink soliton. Figures and represent singular solitons.
6. Modulation instability
The modulation instability (MI) of Equations (Equation1(1) (1) ) and (Equation2(2) (2) ) is finalized in this section on the basis of linear stability technique. The steady-state solution of Equations (Equation1(1) (1) ) and (Equation2(2) (2) ) is given as (29) (29) (30) (30) where and represents the perturbed terms and the ratio of power incident is . Inserting Equations (Equation29(29) (29) ) and (Equation30(30) (30) ) to the Manakov-system and linearizing the result, the following equation are obtained: (31) (31) (32) (32) where and are complex conjugate of and , respectively. The wave transformations are applied as (33) (33) (34) (34) By substituting Equations (Equation33(33) (33) ) and (Equation34(34) (34) ) into Equations (Equation31(31) (31) ) and (Equation32(32) (32) ), the following system of equations is obtained: (35) (35) (36) (36) (37) (37) (38) (38) The matrix is obtained by arranging the system of equations. This matrix is solved for unknowns , , , . The result of the determinant of matrix is given as follows: (39) (39) the instability occurred and grows exponentially when the wave number is imaginary. Consequently, we must have the following condition for modulation instability to exist: (40) (40) The MI gain spectrum is retrieved as (41) (41) where (42) (42) It can be noticed that the modulation instability gain spectrum is dependent on group velocity dispersion and incident power (Figures and ) The MI gain is plotted in Figures with different parameter values. The results are increasing curve by decreasing the values of p and f and vice versa. The results are decreasing curve if the value of p increases and the value of f decreases. Similarly, the results are increasing curve if the value of p decreases and the value of f increases.
7. Conclusion
The three proposed methods namely, the extended -expansion method, the -expansion method and the generalized Kudryashov method are used to extract the travelling wave solutions of Manakov-system. Symbolic software, specifically Maple and Mathematica, is used to plot the 3D graphs, density graphs and 2D contour graphs of certain derived solutions. The obtained exact solutions are in the form of dark soliton, dark-singular combo soliton, singular soliton, Kink and anti-Kink soliton, periodic solutions and rational solutions. It is important to mention here that the solutions obtained in this paper by the application of three state of the art integrating techniques are novel and reliable. The dark-singular combo soliton solutions, Kink soliton and rational solutions have been retrieved for the first time in this article as compared with [Citation1,Citation2]. Moreover, the modulation instability analysis has been carried out for the very first time in this paper as compared with [Citation1,Citation2]. On the basis of linear stability analysis, the MI analysis of Manakov-system is investigated. To identify the region for the occurrence of instability, a dispersion relation is derived. The dispersion relation reflects stability of steady-state nature. The unstable solution arises when the wave number is complex, because the perturbation grows exponentially. Moreover, the stable solution occurs when the wave number is real. The MI gain is significantly affected by the incident power. Due to novelty of results, this article becomes a good contribution in the theory of soliton in future.
Disclosure statement
No potential conflict of interest was reported by the authors.
References
- Yıldırım Y. Optical soliton molecules of Manakov model by trial equation technique. Optik. 2019;185:1146–1151.
- Ozisik M, Secer A, Bayram M. On the examination of optical soliton pulses of Manakov system with auxiliary equation technique. Optik. 2022;268:169800.
- Stalin S, Ramakrishnan R, Senthilvelan M, et al. Nondegenerate solitons in Manakov system. Phys Rev Lett. 2019;122:043901.
- Frisquet B, Kibler B, Fatome J, et al. Polarization modulation instability in a Manakov fiber system. Phys Rev A. 2015;92:053854.
- Copie F, Randoux S, Suret P. The physics of the one-dimensional nonlinear Schrödinger equation in fiber optics: rogue waves, modulation instability and self-focusing phenomena. Rev Phys. 2020;5:100037.
- Abdel-Gawad HI, Biswas A, Alshomrani AS, et al. Optical solitons and stability analysis with coupled nonlinear Schrödingers equations having double external potentials. Res Phys. 2019;15:102707.
- Zhang G, Yan Z, Wen XY, et al. Interactions of localized wave structures and dynamics in the defocusing coupled nonlinear Schrödinger equations. Phys Rev E. 2017;95(4):042201.
- Ali K, Seadawy AR, Ahmed S, et al. Discussion on rational solutions for nematicons in liquid crystals with Kerr law. Chaos Solitons Fractals. 2022;160:112218.
- Rizvi STR, Seadawy AR, Ali K, et al. Multiple lump and rogue wave for time fractional resonant nonlinear Schrödinger equation under parabolic law with weak nonlocal nonlinearity. Optical Quantum Electron. 2022;54(4):1–14.
- Seadawy AR, Younis M, Baber MZ, et al. Nonlinear acoustic wave structures to the Zabolotskaya–Khokholov dynamical model. J Geom Phys. 2022;175:104474.
- Rehman HU, Seadawy AR, Younis M, et al. Weakly nonlinear electron-acoustic waves in the fluid ions propagated via a (3+1)-dimensional generalized Korteweg–de-Vries–Zakharov–Kuznetsov equation in plasma physics. Res Phys. 2022;33:105069.
- Akram U, Seadawy AR, Rizvi STR, et al. Applications of the resonanat nonlinear Schrödinger equation with self steeping phenomena for chirped periodic waves. Opt Quantum Electron. 2022;54(4):1–36.
- Rizvi STR, Ali K, Bekir A, et al. Investigation on the single and multiple dromions for nonlinear telegraph equation in electrical transmission line. Qual Theory Dyn Syst. 2022;21(1):1–14.
- Rizvi STR, Seadawy AR, Abbas SO, et al. Exact and numerical solutions to the system of the chlorite iodide malonic acid chemical reactions. Comput Appl Math. 2022;41(1):1–18.
- Seadawy AR, Rizvi STR, Mustafa B, et al. Chirped periodic waves for an cubic-quintic nonlinear Schrödinger equation with self steepening and higher order nonlinearities. Chaos Solitons Fractals. 2022;156:111804.
- Rizvi STR, Seadawy AR, Akram U, et al. Solitary wave solutions along with painleve analysis for the Ablowitz–Kaup–Newell–Segur water waves equation. Modern Phys Lett B. 2022;36(02):2150548.
- Rizvi STR, Seadawy A, Akram U. New dispersive optical soliton for an nonlinear Schrödinger equation with Kudryashov law of refractive index along with P-test. Opt Quantum Electron. 2022;54(5):1–23.
- Alruwaili AD, Seadawy AR, Rizvi STR, et al. Diverse multiple lump analytical solutions for ion sound and Langmuir waves. Mathematics. 2022;10(2):200.
- Jia HX. Solitons in PT symmetric Manakov system. Optik: Int J Light Election Opt. 2004;230:166223.
- Ye Y, Liu J, Bu L, et al. Rogue waves and modulation instability in an extended Manakov system. Nonlinear Dyn. 2020;102:1801–1812.
- Guan WY, Li BQ. Asymmetrical and self-similar structures of optical breathers for the Manakov system in photorefractive crystals and randomly birefringent fibers. Optik: Int J Light Election Opt. 2019;194:162882.
- Ahmed HM, El-Sheikh MMA, Arnous AHSEP, et al. Construction of the soliton solutions for the Manakov system by extended simplest equation method. Int J Appl Comput Math. 2021;7:239.
- Ma G, Zhou Q, Yu W, et al. Stable transmission characteristics of double-hump solitons for the coupled Manakov equations in fiber lasers. Nonlinear Dyn. 2021;106:2509–2514.
- Vijayajayanthi M, Kanna T, Lakshmanan M, et al. Explicit construction of single input–single output logic gates from three soliton solution of Manakov system. Commun Nonlinear Sci Numer Simul. 2016;36:391–401.
- Wu GZ, Fang Y, Wang YY, et al. Predicting the dynamic process and model parameters of the vector optical solitons in birefringent fibers via the modified PINN. Chaos Solitons Fractals. 2021;152:111393.
- Stalin S, Ramakrishnan R, Lakshmanan M. Nondegenerate soliton solutions in certain coupled nonlinear Schrödinger systems. Phys Lett A. 2020;384:126201.
- Ismail MS, Kanna T. Numerical study of bright-bright-dark soliton dynamics in the mixed coupled nonlinear Schrödinger system. Optik: Int J Light Election Opt. 2020;224:165633.
- Gerdjikov VS, Kostov NA, Doktorov EV, et al. Generalized perturbed complex Toda chain for Manakov system and exact solutions of Bose–Einstein mixtures. Math Comput Simul. 2009;80:112–119.
- Gerdjikov VS, Doktorov EV, Matsuka NP. N-soliton train and generalized complex Toda chain for the Manakov system. Theoret Math Phys. 2007;151(3):762–773.
- Kraus D, Biondini G, Kovačič G. The focusing Manakov system with nonzero boundary conditions. Nonlinearity. 2015;28(9):3101.
- Pu JC, Chen Y. Data-driven vector localized waves and parameters discovery for Manakov system using deep learning approach. Chaos Solitons Fractals. 2022;160:112182.
- Barletti L, Brugnano L, Tang Y, et al. Spectrally accurate space–time solution of Manakov systems. J Comput Appl Math. 2020;377:112918.
- Raskovalov AA, Gelash AA. Resonance interaction of breathers in the Manakov system. Theoret Math Phys. 2022;213(3):1669–1685.
- Wang D-S, Zhang D-J, Yang J. Integrable properties of the general coupled nonlinear Schrödinger equations. J Math Phys. 2010;51(2):023510.
- Ma G, Zhou Q, Yu W, et al. Stable transmission characteristics of double-hump solitons for the coupled Manakov equations in fiber lasers. Nonlinear Dyn. 2021;106(3):2509–2514.
- Gerdjikov VS, Todorov MD. Manakov model with gain/loss terms and N-soliton interactions: effects of periodic potentials. Appl Numer Math. 2019;141:62–80.
- Chen S, Zhou R. An integrable decomposition of the Manakov equation. Comput Appl Math. 2012;31:01–18.
- Frisquet B, Kibler B, Fatome J, et al. Polarization modulation instability in a Manakov fiber system. Phys Rev A. 2015;92(5):053854.
- Yilmaz EU, Khodad FS, Ozkan YS, et al. Manakov model of coupled NLS equation and its optical soliton solutions. J Ocean Eng Sci. 2022. In press.
- Mumtaz S, Essiambre R, Agrawal GP. Nonlinear propagation in multimode and multicore fibers: generalization of the Manakov equations. J Lightwave Technol. 2012;31(3):398–406.
- Hedli R, Kadem A. Exact traveling wave solutions to the fifth-order KdV equation using the exponential expansion method. Int J Appl Math. 2020;50:1.
- Korkmaz A, Hosseini K. Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods. Opt Quantum Electron. 2017;49:31.
- Hafez MG, Akbar MA. An exponential expansion method and its application to the strain wave equation in microstructured solids. Ain Shams Eng J. 2015;6:683–690.
- Kadkhoda N, Jafari H. Analytical solutions of the Gerdjikov–Ivanov equation by using exp(−ϕ(ξ))-expansion method. Optik: Int J Light Election Opt. 2017;139:72–76.
- Akram G, Sadaf M, Khan MA. Abundant optical solitons for Lakshmanan–Porsezian–Daniel model by the modified auxiliary equation method. Optik: Int J Light Election Opt. 2022;251:168163.
- Akram G, Sadaf M, Khan MA. Soliton solutions of Lakshmanan–Porsezian–Daniel model using modified auxiliary equation method with parabolic and anti-cubic law of nonlinearities. Optik: Int J Light Election Opt. 2022;252:168372.
- Akram G, Gillani SR. Sub pico-second soliton with Triki–Biswas equation by the extended (G′G2)-expansion method and the modified auxiliary equation method. Optik: Int J Light Election Opt. 2021;229:166227.
- Akbulut A, Kaplan M, Tascan F. The investigation of exact solutions of nonlinear partial differential equations by using exp (- ϕ (ξ)) method. Optik. 2017;132:382–387.
- Kaplan M, Akbulut A. A novel exploration for traveling wave solutions to the integrable equation of wave packet envelope. Partial Differ Equ Appl Math. 2022;5:100312.
- Akbulut A, Islam SMR, Rezazadeh H, et al. Obtaining exact solutions of nonlinear partial differential equations via two different methods. Int J Modern Phys B. 2022;36(05):2250041.
- Xu B, Zhang S. Exact solutions of nonlinear equations in mathematical physics via negative power expansion method. J Math Phys Anal Geo. 2021;17:369–387.
- Zhang S, Xu B. Painlevé test and exact solutions for (1+1)-dimensional generalized Broer–Kaup equations. Mathematics. 2022;10(3):486.
- Zhang S. Exact solutions of a KdV equation with variable coefficients via exp-function method. Nonlinear Dyn. 2008;52(1):11–17.
- Mahmud F, Samsuzzoha M, Akbar MA. The generalized Kudryashov method to obtain exact traveling wave solutions of the PHI-four equation and the fisher equation. Res Phys. 2017;7:4296–4302.
- Ghazanfar S, Ahmed N, Iqbal MS, et al. Imaging ultrasound propagation using the Westervelt equation by the generalized Kudryashov and modified Kudryashov methods. Appl Sci. 2022;12(22):11813.
- Ozisik M. Soliton solutions of nonlinear (2+1)-dimensional Biswas–Milovic equation via new approach of generalized Kudryashov scheme. NTMSCI 10 Special Iss. 2022:43(1):47–53.
- Sirisubtawee S, Koonprasert S. Exact traveling wave solutions of certain nonlinear partial differential equations using the G′G2-expansion method. Adv Math Phys. 2018;2018:7628651.
- Abdelrahman MAE, Zahran EHM, Khater MMA. The exp (-φ (ξ))-expansion method and its application for solving nonlinear evolution equations. Int J Modern Nonlinear Theory Appl. 2015;4(1):37.