Bifurcation and meromorphic exact solutions of five nonlinear partial differential equations

Hongqiang Tu; Yongyi Gu; Hongqiang Tu; Yongyi Gu

doi:10.3934/math.2026797

AIMS Mathematics

2026, Volume 11, Issue 7: 19633-19658. doi: 10.3934/math.2026797

Previous Article Next Article

Research article

Bifurcation and meromorphic exact solutions of five nonlinear partial differential equations

Hongqiang Tu ¹,
Yongyi Gu ^{2
,
,}

1.
School of Mathematics, Guangdong University of Education, Guangzhou 510800, China
2.
School of Statistics and Data Science, Guangdong University of Finance and Economics, Guangzhou 510320, China

Received: 25 March 2026 Revised: 03 June 2026 Accepted: 16 June 2026 Published: 03 July 2026
MSC : 30D35, 34A05

Differential equations model how scientific systems change over time and space. They are essential tools across physics, engineering, biology, and economics. Searching for solutions to these equations is of great significance, and simultaneously seeking meromorphic exact solutions of multiple nonlinear partial differential equations (PDEs) would be innovative. In view of this, we transform five nonlinear PDEs into ordinary differential equations (ODEs) through appropriate transformations and summarize these ODEs into one equation, then explore the bifurcation of this equation with commonalities. Subsequently, we utilize the complex method and the exp$ (-\vartheta(z)) $-expansion approach to derive meromorphic exact solutions for five nonlinear PDEs at once, including the Tzizeica equation, the Zhiber–Shabat (ZS) equation, the modified Kortweg de Vries (mKdV) equation, the Tzitzeica–Dodd–Bullough (TDB) equation, and the modified double sine-Gordon (mDSG) equation. To better understand the dynamic structure of the results, two-dimensional, three-dimensional, and contour graphs are given to illustrate the solutions. The conceptual framework developed here could be extended to analyze additional nonlinear physical equations in scientific research.
- differential equation,
- complex method,
- exp$ (-\vartheta(z)) $-expansion method,
- bifurcation,
- exact solution,
- meromorphic function
Citation: Hongqiang Tu, Yongyi Gu. Bifurcation and meromorphic exact solutions of five nonlinear partial differential equations[J]. AIMS Mathematics, 2026, 11(7): 19633-19658. doi: 10.3934/math.2026797

Related Papers:

Abstract

Differential equations model how scientific systems change over time and space. They are essential tools across physics, engineering, biology, and economics. Searching for solutions to these equations is of great significance, and simultaneously seeking meromorphic exact solutions of multiple nonlinear partial differential equations (PDEs) would be innovative. In view of this, we transform five nonlinear PDEs into ordinary differential equations (ODEs) through appropriate transformations and summarize these ODEs into one equation, then explore the bifurcation of this equation with commonalities. Subsequently, we utilize the complex method and the exp$ (-\vartheta(z)) $-expansion approach to derive meromorphic exact solutions for five nonlinear PDEs at once, including the Tzizeica equation, the Zhiber–Shabat (ZS) equation, the modified Kortweg de Vries (mKdV) equation, the Tzitzeica–Dodd–Bullough (TDB) equation, and the modified double sine-Gordon (mDSG) equation. To better understand the dynamic structure of the results, two-dimensional, three-dimensional, and contour graphs are given to illustrate the solutions. The conceptual framework developed here could be extended to analyze additional nonlinear physical equations in scientific research.

References

[1]	J. G. Liu, Lump-type solutions and interaction solutions for the (2+1)-dimensional generalized fifth-order KdV equation, Appl. Math. Lett., 86 (2018), 36–41. https://doi.org/10.1016/j.aml.2018.06.011 doi: 10.1016/j.aml.2018.06.011
[2]	M. Li, W. K. Hu, C. F. Wu, Rational solutions of the classical Boussinesq-Burgers system, Nonlinear Dyn., 94 (2018), 1291–1302. https://doi.org/10.1007/s11071-018-4424-6 doi: 10.1007/s11071-018-4424-6
[3]	Y. Gu, L. Peng, Z. Huang, Y. Lai, Soliton, breather, lump, interaction solutions and chaotic behavior for the (2+1)-dimensional KPSKR equation, Chaos Solitons Fract., 187 (2024), 115351. https://doi.org/10.1016/j.chaos.2024.115351 doi: 10.1016/j.chaos.2024.115351
[4]	J. G. Liu, Soliton structures for the (3+1)-dimensional Painlevé integrable equation in fluid mediums, Sci. Rep., 14 (2024), 11581. https://doi.org/10.1038/s41598-024-62314-6 doi: 10.1038/s41598-024-62314-6
[5]	K. S. Nisar, I. E. Inan, H. Yepez-Martinez, M. Inc, Some new type optical and the other soliton solutions of coupled nonlinear Hirota equation, Results Phys., 35 (2022), 105388. https://doi.org/10.1016/j.rinp.2022.105388 doi: 10.1016/j.rinp.2022.105388
[6]	S. M. Alamry, M. S. Mohamed, T. Adel, K. K. Ali, Optical soliton solutions, stability, and modulation instability for fractional multi-components of Gross-Pitaevskii model, AIP Adv., 15 (2025), 115211. https://doi.org/10.1063/5.0304835 doi: 10.1063/5.0304835
[7]	K. K. Ali, H. Rezazadeh, R. A. Talarposhti, A. Bekir, New soliton solutions for resonant nonlinear Schrödingers equation having full nonlinearity, Int. J. Mod. Phys. B, 34 (2020), 2050032. https://doi.org/10.1142/S0217979220500320 doi: 10.1142/S0217979220500320
[8]	K. Hosseini, M. Mirzazadeh, S. Salahshour, D. Baleanu, A. Zafar, Specific wave structures of a fifth-order nonlinear water wave equation, J. Ocean Eng. Sci., 7 (2022), 462–466. https://doi.org/10.1016/j.joes.2021.09.019 doi: 10.1016/j.joes.2021.09.019
[9]	M. Inc, R. T. Alqahtani, M. S. Iqbal, Stability analysis and consistent solitary wave solutions for the reaction-diffusion regularized nonlinear model, Results Phys., 54 (2023), 107053. https://doi.org/10.1016/j.rinp.2023.107053 doi: 10.1016/j.rinp.2023.107053
[10]	H. M. Baskonus, T. A. Sulaiman, H. Bulut, On the novel wave behaviors to the coupled nonlinear Maccari's system with complex structure, Optik, 131 (2017), 1036–1043. https://doi.org/10.1016/j.ijleo.2016.10.135 doi: 10.1016/j.ijleo.2016.10.135
[11]	K. K. Ali, A. M. Wazwaz, M. S. Osman, Optical soliton solutions to the generalized nonautonomous nonlinear Schrödinger equations in optical fibers via the sine-gordon expansion method, Optik, 208 (2020), 164132. https://doi.org/10.1016/j.ijleo.2019.164132 doi: 10.1016/j.ijleo.2019.164132
[12]	Y. Guo, D. Li, J. Wang, The new exact solutions of the fifth-Order Sawada-Kotera equation using three wave method, Appl. Math. Lett., 94 (2019), 232–237. https://doi.org/10.1016/j.aml.2019.03.001 doi: 10.1016/j.aml.2019.03.001
[13]	K. Hosseini, A. Zabihi, F. Samadani, R. Ansari, New explicit exact solutions of the unstable nonlinear Schrödinger's equation using the $exp_{a}$ and hyperbolic function methods, Opt. Quantum Electron., 50 (2018), 82. https://doi.org/10.1007/s11082-018-1350-2 doi: 10.1007/s11082-018-1350-2
[14]	W. Ding, D. B. Qian, C. Wang, Z. G. Wang, Existence of periodic solutions of sublinear Hamiltonian systems, Acta. Math. Sin.-English Ser., 32 (2016), 621–632. https://doi.org/10.1007/s10114-016-4162-y doi: 10.1007/s10114-016-4162-y
[15]	Y. Kong, L. Xin, Q. Qiu, L. Han, Exact periodic wave solutions for the modified Zakharov equations with a quantum correction, Appl. Math. Lett., 94 (2019), 140–148. https://doi.org/10.1016/j.aml.2019.01.009 doi: 10.1016/j.aml.2019.01.009
[16]	Y. Gu, C. Jiang, Y. Lai, Analytical solutions of the fractional Hirota-Satsuma coupled KdV equation along with analysis of bifurcation, sensitivity and chaotic behaviors, Fractal Fract., 8 (2024), 585. https://doi.org/10.3390/fractalfract8100585 doi: 10.3390/fractalfract8100585
[17]	N. Aminakbari, Y. Gu, G. Dang, Study on soliton behaviors of the complex nonlinear Fokas-Lenells equation utilizing different methods, Mod. Phys. Lett. B, 38 (2024), 2450340. https://doi.org/10.1142/S0217984924503408 doi: 10.1142/S0217984924503408
[18]	Y. Gu, H. Zhu, Y. Li, J. Fan, Dynamical characteristic of wave solutions of the fractional (4+1)-dimensional Fokas equation, Mod. Phys. Lett. B, 40 (2026), 2650128. https://doi.org/10.1142/S0217984926501289 doi: 10.1142/S0217984926501289
[19]	H. Roshid, M. A. Rahman, The exp$(-\phi(\xi))$-expansion method with application in the (1+1)-dimensional classical Boussinesq equations, Results Phys., 4 (2014), 150–155. https://doi.org/10.1016/j.rinp.2014.07.006 doi: 10.1016/j.rinp.2014.07.006
[20]	N. Kadkhoda, H. Jafari, Analytical solutions of the Gerdjikov-Ivanov equation by using exp$(-\vartheta(\xi))$-expansion method, Optik, 139 (2017), 72–76. https://doi.org/10.1016/j.ijleo.2017.03.078 doi: 10.1016/j.ijleo.2017.03.078
[21]	K. Faisal, K. Maqbool, New soliton solutions of (2+1)-dimensional Bogoyavlensky-Konopelchenko equation via two integration techniques, Appl. Math. J. Chin. Univ., 40 (2025), 169–181. https://doi.org/10.1007/s11766-025-4527-z doi: 10.1007/s11766-025-4527-z
[22]	W. Yuan, Y. Shang, Y. Huang, H. Wang, The representation of meromorphic solutions to certain ordinary differential equations and its applications, Sci. Sin. Math., 43 (2013), 563-575.
[23]	W. Yuan, Y. Huang, Y. Shang, All traveling wave exact solutions of two nonlinear physical models, Appl. Math. Comput., 219 (2013), 6212–6223. https://doi.org/10.1016/j.amc.2012.12.023 doi: 10.1016/j.amc.2012.12.023
[24]	M. B. Hossen, H. O. Roshid, M. Z. Ali, Characteristics of the solitary waves and rogue waves with interaction phenomena in a (2+1)-dimensional breaking soliton equation, Phys. Lett. A, 382 (2018), 1268–1274. https://doi.org/10.1016/j.physleta.2018.03.016 doi: 10.1016/j.physleta.2018.03.016
[25]	Y. Gu, Y. Lai, Analytical investigation of the fractional Klein-Gordon equation along with analysis of bifurcation, sensitivity and chaotic behaviors, Mod. Phys. Lett. B, 39 (2025), 2550124. https://doi.org/10.1142/S0217984925501246 doi: 10.1142/S0217984925501246
[26]	Y. Gu, C. Wu, X. Yao, W. Yuan, Characterizations of all real solutions for the KdV equation and $W_R$, Appl. Math. Lett., 107 (2020), 106446. https://doi.org/10.1016/j.aml.2020.106446 doi: 10.1016/j.aml.2020.106446
[27]	J. Xu, X. Zhang, Some results of meromorphic solutions of second-order linear differential equations, J. Inequal. Appl., 2013 (2013), 304. https://doi.org/10.1186/1029-242X-2013-304 doi: 10.1186/1029-242X-2013-304
[28]	T. W. Ng, C. F. Wu, Nonlinear Loewy factorizable algebraic ODEs and Hayman's conjecture, Isr. J. Math., 229 (2019), 1–38. https://doi.org/10.1007/s11856-018-1791-0 doi: 10.1007/s11856-018-1791-0
[29]	Y. Gu, W. Yuan, N. Aminakbari, J. Lin, Meromorphic solutions of some algebraic differential equations related Painlev{é} equation IV and its applications, Math. Methods Appl. Sci., 41 (2018), 3832–3840. https://doi.org/10.1002/mma.4869 doi: 10.1002/mma.4869
[30]	K. Hosseini, Z. Ayati, R. Ansari, New exact solutions of the Tzitzeica type equations arising in nonlinear optics using a modified version of the improved tan$(-\vartheta(\xi)/2)$-expansion method, Opt. Quant. Electron., 49 (2017), 1.
[31]	A. M. Wazwaz, The variable separated ODE and the tanh methods for solving the combined and the double combined sinh-cosh-Gordon equations, Appl. Math. Comput., 177 (2006), 745–754. https://doi.org/10.1016/j.amc.2005.09.101 doi: 10.1016/j.amc.2005.09.101
[32]	A. M. Wazwaz, The tanh method for travelling wave solutions to the Zhiber-Shabat equation and other related equations, Commun. Nonlinear Sci. Numer. simul., 13 (2008), 584–592. https://doi.org/10.1016/j.cnsns.2006.06.014 doi: 10.1016/j.cnsns.2006.06.014
[33]	Sirendaoreji, S. Jiong, A direct method for solving sine-Gordon type equation, Phys. Lett. A, 298 (2002), 133–139.
[34]	E. Tala-Tebue, Z. I. Djoufack, D. C. Tsobgni-Fozap, A. Kenfack-Jiotsa, F. Kapche-Tagne, T. C. Kofané, Traveling wave solutions along microtubules and in the Zhiber-Shabat equation, Chinese J. Phys., 55 (2017), 939–946. https://doi.org/10.1016/j.cjph.2017.03.004 doi: 10.1016/j.cjph.2017.03.004
[35]	A. M. Wazwaz, A modified KdV-type equation that admits a variety of travelling wave solutions: Kinks, solitons, peakons and cuspons, Phys. Scr., 86 (2012), 045501. https://doi.org/10.1088/0031-8949/86/04/045501 doi: 10.1088/0031-8949/86/04/045501
[36]	D. Mothibi, C. Khalique, On the exact solutions of a modified Kortweg de Vries type equation and higher-order modified Boussinesq equation with damping term, Adv. Differ. Equ., 2013 (2013), 166. https://doi.org/10.1186/1687-1847-2013-166 doi: 10.1186/1687-1847-2013-166
[37]	A. M. Wazwaz, The tan h method: Solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations, Chaos Soliton. Fract., 25 (2005), 55–63. https://doi.org/10.1016/j.chaos.2004.09.122 doi: 10.1016/j.chaos.2004.09.122
[38]	A. Esen, N. M. Yagmurlu, O. Tasbozan, Double exp-function method for multisoliton solutions of the Tzitzeica-Dodd-Bullough equation, Acta Math. Appl. Sin. Engl. Ser., 32 (2016), 461–468. https://doi.org/10.1007/s10255-016-0572-y doi: 10.1007/s10255-016-0572-y
[39]	E. Fan, Y. C. Hona, Generalized tanh method extended to special types of nonlinear equations, Z. Naturforsch. A: Phys. Sci., 57 (2002), 692–700. https://doi.org/10.1515/zna-2002-0809 doi: 10.1515/zna-2002-0809
[40]	H. Jafari, R. Soltani, C. M. Khalique, D. Baleanu, Exact solutions of two nonlinear partial differential equations by using the first integral method, Bound. Value Probl., 2013 (2013), 117. https://doi.org/10.1186/1687-2770-2013-117 \newpage doi: 10.1186/1687-2770-2013-117
[41]	S. Lang, Elliptic functions, In: Graduate texts in mathematics, New York: Springer, 112 (1987), 5–21. https://doi.org/10.1007/978-1-4612-4752-4_1
[42]	A. E. Eremenko, L. Liao, T. W. Ng, Meromorphic solutions of higher order Briot-Bouquet differential equations, Math. Proc. Camb. Philos. Soc., 146 (2009), 197–206. https://doi.org/10.1017/S030500410800176X doi: 10.1017/S030500410800176X
[43]	N. A. Kudryashov, Meromorphic solutions of nonlinear ordinary differential equations, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 2778–2790. https://doi.org/10.1016/j.cnsns.2009.11.013 doi: 10.1016/j.cnsns.2009.11.013

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)