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Journal of Information and Optimization Sciences

(Since 1980)
Open Access ·Peer-reviewed·ISSN (Online): 2169-0103·ISSN (Print): 0252-2667
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The Journal of Information and Optimization Sciences (JIOS) is a world leading journal publishing high quality, rigorously peer-reviewed original research in all mathematically-oriented theoretical and applied topics in information sciences, optimization sciences and related areas since 1980. Subjects include but are not limited to: • Information Sciences • Optimization Sciences • Control Theory • Operational Research • Decision Sciences • Information Theory • Information Technology • Computer Networks and Communications • Mathematical Programming • Modelling and Simulation • Database Management • Applications to Engineering Sciences • Applications to Technology

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Journal of Information and Optimization Sciences
Open Access Research Article

Mathematical analysis of novel soliton solutions to a nonlinear wave equation

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* Corresponding author · click or hover a name for details

pp. 1–22Online FirstApril 2026DOI: 10.47974/JIOS-2037XML
Received:
01 Jan 2025
Published Online:
02 Apr 2026
Article type:
Research Article
Language:
EN
Article no.:
JIOS-2037
Pages:
1–22

Abstract

This paper seeks to develop comprehensive soliton solutions by employing the generalized modified equal width (MEW) equation as the governing framework. To achieve that, we have chosen the (δ’⁄δ)-expansion approach, which is an effective, reliable, and compatible method. The outcomes achieved in this instance are defined in terms of a mix of rational, trigonometric, and hyperbolic functions. The corresponding physical features of these resultant soliton solutions and influence of parameters are presented by carving two-dimensional and three-dimensional and contour graphs by means of MATLAB. For estimated values of these parameters, bell-shaped soliton, singular soliton, parabolic soliton, periodic soliton and solitons of other types are formulated. By examining the solutions across carefully chosen parameter regimes, the work offers a clearer perspective on the system’s complex behavior. The results introduce new soliton phenomena and uncover previously unreported aspects of the governing equation.

Keywords

Nonlinear evolution equations (NLEEs)Generalized modified equal width (MEW) equation(δ’⁄δ)-expansion approachSoliton solutions

Subject Classifications

35C0735C0835C0935B35

References

[1] M. Wang, X. Li, and J. Zhang, ‘The (G’/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics’, Phys Lett A, vol. 372, no. 4, pp. 417–423 (Jan. 2008), doi: 10.1016/j.physleta.2007.07.051.
[2] E. Yusufoglu and A. Bekir, ‘Numerical simulation of equal-width wave equation’, Computers & Mathematics with Applications, vol. 54, no. 7–8, pp. 1147–1153 (Oct. 2007), doi: 10.1016/j.camwa.2006.12.080.
[3] Md. S. Islam, K. Khan, and M. A. Akbar, ‘An analytical method for finding exact solutions of modified Korteweg–de Vries equation’, Results Phys, vol. 5, pp. 131–135 (2015), doi: 10.1016/j.rinp.2015.01.007.
[4] P. O. Mohammed, M. A. Alqudah, Y. S. Hamed, A. Kashuri, and K. M. Abualnaja, ‘Solving the Modified Regularized Long Wave Equations via Higher Degree B-Spline Algorithm’, Journal of Function Spaces, vol. 2021, pp. 1–10 (Mar. 2021), doi: 10.1155/2021/5580687.
[5] D. Lu, A. R. Seadawy, and A. Ali, ‘Dispersive traveling wave solutions of the Equal-Width and Modified Equal-Width equations via mathematical methods and its applications’, Results Phys, vol. 9, pp. 313–320 (Jun. 2018), doi: 10.1016/j.rinp.2018.02.036.
[6] B. Radha and C. Duraisamy, ‘Retraction Note to: The homogeneous balance method and its applications for finding the exact solutions for nonlinear equations’, J Ambient Intell Humaniz Comput, vol. 14, no. S1, pp. 181–181 (Apr. 2023), doi: 10.1007/s12652-022-04003-8.
[7] S. Zhang, J.-L. Tong, and W. Wang, ‘Exp-function method for a nonlinear ordinary differential equation and new exact solutions of the dispersive long wave equations’, Computers & Mathematics with Applications, vol. 58, no. 11–12, pp. 2294–2299 (Dec. 2009), doi: 10.1016/j.camwa.2009.03.020.
[8] B. Zheng, ‘Exp-Function Method for Solving Fractional Partial Differential Equations’, The Scientific World Journal, vol. 2013, pp. 1–8 (2013), doi: 10.1155/2013/465723.
[9] A. Bekir, ‘The Exp-function method for Ostrovsky equation’, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 6 (Jan. 2009), doi: 10.1515/IJNSNS.2009.10.6.735.
[10] Q. M. Al-Mdallal and M. I. Syam, ‘Sine–Cosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation’, Chaos Solitons Fractals, vol. 33, no. 5, pp. 1610–1617 (Aug. 2007), doi: 10.1016/j.chaos.2006.03.039.
[11] A.-M. Wazwaz, ‘A sine-cosine method for handlingnonlinear wave equations’, Math Comput Model, vol. 40, no. 5–6, pp. 499–508 (Sep. 2004), doi: 10.1016/j.mcm.2003.12.010.
[12] B. Radha and C. Duraisamy, ‘RETRACTED ARTICLE: The homogeneous balance method and its applications for finding the exact solutions for nonlinear equations’, J Ambient Intell Humaniz Comput, vol. 12, no. 6, pp. 6591–6597 (Jun. 2021), doi: 10.1007/s12652-020-02278-3.
[13] E. Fan and H. Zhang, ‘A note on the homogeneous balance method’, Phys Lett A, vol. 246, no. 5, pp. 403–406 (Sep. 1998), doi: 10.1016/S0375-9601(98)00547-7.
[14] M. Kaplan, A. Bekir, and A. Akbulut, ‘A generalized Kudryashov method to some nonlinear evolution equations in mathematical physics’, Nonlinear Dyn, vol. 85, no. 4, pp. 2843–2850 (Sep. 2016), doi: 10.1007/s11071-016-2867-1.
[15] F. Mahmud, M. Samsuzzoha, and M. A. Akbar, ‘The generalized Kudryashov method to obtain exact traveling wave solutions of the PHI-four equation and the Fisher equation’, Results Phys, vol. 7, pp. 4296–4302 (2017), doi: 10.1016/j.rinp.2017.10.049.
[16] P. N. Ryabov, D. I. Sinelshchikov, and M. B. Kochanov, ‘Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations’, Appl Math Comput, vol. 218, no. 7, pp. 3965–3972 (Dec. 2011), doi: 10.1016/j.amc.2011.09.027.
[17] Md. A. Islam, N. N. Rimu, and P. Dey, ‘Fractional order effects on solitary waves and chaotic regimes in the mKdV Burgers equation’, Sci Rep, vol. 15, no. 1, p. 41425 (Nov. 2025), doi: 10.1038/s41598-025-25340-6.
[18] Md. A. Islam, N. N. Rimu, S. Sarker, P. Dey, and M. A. Akbar, ‘Dynamics of soliton solutions, bifurcation, chaotic behavior, stability, and sensitivity analysis of the time-fractional Gardner equation’, AIP Adv, vol. 15, no. 9 (Sep. 2025), doi: 10.1063/5.0285186.
[19] A.-A. Hyder and A. H. Soliman, ‘An extended Kudryashov technique for solving stochastic nonlinear models with generalized conformable derivatives’, Commun Nonlinear Sci Numer Simul, vol. 97, p. 105730 (Jun. 2021), doi: 10.1016/j.cnsns.2021.105730.
[20] Z. Jia-Min, ‘Hyperbolic function method for solving nonlinear differential-different equations’, Chinese Physics, vol. 14, no. 7, pp. 1290–1295 (Jul. 2005), doi: 10.1088/1009-1963/14/7/004.
[21] Y. Shang, ‘The extended hyperbolic function method and exact solutions of the long–short wave resonance equations’, Chaos Solitons Fractals, vol. 36, no. 3, pp. 762–771 (May 2008), doi: 10.1016/j.chaos.2006.07.007.
[22] W. Malfliet, ‘The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations’, J Comput Appl Math, vol. 164–165, pp. 529–541 (Mar. 2004), doi: 10.1016/S0377-0427(03)00645-9.
[23] M. A. Abdou, ‘The extended tanh method and its applications for solving nonlinear physical models’, Appl Math Comput, vol. 190, no. 1, pp. 988–996 (Jul. 2007), doi: 10.1016/j.amc.2007.01.070.
[24] E. Fan, ‘Extended tanh-function method and its applications to nonlinear equations’, Phys Lett A, vol. 277, no. 4–5, pp. 212–218 (Dec. 2000), doi: 10.1016/S0375-9601(00)00725-8.
[25] N. Taghizadeh, M. Mirzazadeh, and F. Tascan, ‘The first-integral method applied to the Eckhaus equation’, Appl Math Lett, vol. 25, no. 5, pp. 798–802 (May 2012), doi: 10.1016/j.aml.2011.10.021.
[26] B. Zheng and Q. Feng, ‘The Jacobi Elliptic Equation Method for Solving Fractional Partial Differential Equations’, Abstract and Applied Analysis, vol. 2014, pp. 1–9 (2014), doi: 10.1155/2014/249071.
[27] S. Liu, Z. Fu, S. Liu, and Q. Zhao, ‘Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations’, Phys Lett A, vol. 289, no. 1–2, pp. 69–74 (Oct. 2001), doi: 10.1016/S0375-9601(01)00580-1.
[28] Y. Chen and Q. Wang, ‘Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1 + 1)-dimensional dispersive long wave equation’, Chaos Solitons Fractals, vol. 24, no. 3, pp. 745–757 (May 2005), doi: 10.1016/j.chaos.2004.09.014.
[29] F. Dusunceli, E. Celik, M. Askin, and H. Bulut, ‘New exact solutions for the doubly dispersive equation using the improved Bernoulli sub-equation function method’, Indian Journal of Physics, vol. 95, no. 2, pp. 309–314 (Feb. 2021), doi: 10.1007/s12648-020-01707-5.
[30] W.-X. Ma, ‘Soliton solutions by means of Hirota bilinear forms’, Partial Differential Equations in Applied Mathematics, vol. 5, p. 100220 (Jun. 2022), doi: 10.1016/j.padiff.2021.100220.
[31] K. Konno and M. Wadati, ‘Simple Derivation of Backlund Transformation from Riccati Form of Inverse Method’, Progress of Theoretical Physics, vol. 53, no. 6, pp. 1652–1656 (Jun. 1975), doi: 10.1143/PTP.53.1652.
[32] H. Naher, F. A. Abdullah, and M. A. Akbar, ‘The (G’/G) -Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation’, Math Probl Eng, vol. 2011, pp. 1–11 (2011), doi: 10.1155/2011/218216.
[33] R. Abazari and R. Abazari, ‘Hyperbolic, Trigonometric, and Rational Function Solutions of Hirota-Ramani Equation via (G’/G)-Expansion Method’, Math Probl Eng, vol. 2011, pp. 1–11 (2011), doi: 10.1155/2011/424801.
[34] T. Öziş and İ. Aslan, ‘Application of the  G’/G-expansion method to Kawahara type equations using symbolic computation’, Appl Math Comput, vol. 216, no. 8, pp. 2360–2365 (Jun. 2010), doi: 10.1016/j.amc.2010.03.081.
[35] M. A. Akbar, N. Hj. Mohd. Ali, and E. M. E. Zayed, ‘A Generalized and Improved (G’/G)-Expansion Method for Nonlinear Evolution Equations’, Math Probl Eng, vol. 2012, pp. 1–22 (2012), doi: 10.1155/2012/459879.
[36] A. Jabbari, H. Kheiri, and A. Bekir, ‘Exact solutions of the coupled Higgs equation and the Maccari system using He’s semi-inverse method and (G’/G)-expansion method’, Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2177–2186 (Sep. 2011), doi: 10.1016/j.camwa.2011.07.003.
[37] A. Borhanifar and A. Z. Moghanlu, ‘Application of the (G’/G)-expansion method for the Zhiber–Shabat equation and other related equations’, Math Comput Model, vol. 54, no. 9–10, pp. 2109–2116 (Nov. 2011), doi: 10.1016/j.mcm.2011.05.020.
[38] J. Feng, W. Li, and Q. Wan, ‘Using (G’/G)-expansion method to seek the traveling wave solution of Kolmogorov–Petrovskii–Piskunov equation’, Appl Math Comput, vol. 217, no. 12, pp. 5860–5865 (Feb. 2011), doi: 10.1016/j.amc.2010.12.071.
[39] S. Sarker, G. S. Said, M. M. Tharwat, R. Karim, M. A. Akbar, N. S. Elazab, M. S. Osman, and P. Dey, ‘Soliton solutions to a wave equation using the (ϕ’/ϕ)– expansion method’, Partial Differential Equations in Applied Mathematics, vol. 8, p. 100587 (Dec. 2023), doi: 10.1016/j.padiff.2023.100587.
[40] L. Li, E. Li, and M. Wang, ‘The (G′/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations’, Applied Mathematics-A Journal of Chinese Universities, vol. 25, no. 4, pp. 454–462 (Dec. 2010), doi: 10.1007/s11766-010-2128-x.
[41] M. Mamun Miah, H. M. Shahadat Ali, M. Ali Akbar, and A. Majid Wazwaz, ‘Some applications of the (G′/G, 1/G)-expansion method to find new exact solutions of NLEEs’, The European Physical Journal Plus, vol. 132, no. 6, p. 252 (Jun. 2017), doi: 10.1140/epjp/i2017-11571-0.
[42] P. Dey, L. H. Sadek, M. M. Tharwat, S. Sarker, R. Karim, M. A. Akbar, N. S. Elazab, and M. S. Osman, ‘Soliton solutions to generalized (3 + 1)-dimensional shallow water-like equation using the (ϕ’/ϕ,1/ϕ)-expansion method’, Arab J Basic Appl Sci, vol. 31, no. 1, pp. 121–131 (Dec. 2024), doi: 10.1080/25765299.2024.2313245.
[43] S. Sarker, R. Karim, M. A. Akbar, M. S. Osman, and P. Dey, ‘Soliton solutions to a nonlinear wave equation via modern methods’, Journal of Umm Al-Qura University for Applied Sciences (Mar. 2024), doi: 10.1007/s43994-024-00137-x.
[44] Md. A. Islam, N. N. Rimu, M. A. Akbar, S. Sarker, and P. Dey, ‘Stability analysis, bifurcation and chaotic dynamics, and analytical soliton solutions of the modified RLW-Burgers equation using an expansion scheme’, AIP Adv, vol. 15, no. 12 (Dec. 2025), doi: 10.1063/5.0306723.
[45] W. M. Taha and M. S. M. Noorani, ‘Application of the (G’/G)-expansion method for the generalized Fisher’s equation and modified equal width equation’, Journal of the Association of Arab Universities for Basic and Applied Sciences, vol. 15, no. 1, pp. 82–89 (2014), doi: 10.1016/j.jaubas.2013.05.006.

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