This study derives novel exact traveling wave solutions for the nonlinear (1+1)-dimensional Chafee-Infante equation by synthesizing the generalized first integral method (GFIM) with Laurent polynomial expansions. As a fundamental reaction-diffusion model, the Chafee-Infante equation governs pattern formation in diverse systems—from biological to chemical and physical contexts—yet its strong nonlinearity poses persistent challenges to classical integration techniques such as the inverse scattering transform or Hirota's method. We transform the equation into an autonomous polynomial system and employ the division theorem to systematically identify its first integrals, thereby circumventing the need for auxiliary equations or ansatz-based heuristics. By introducing Laurent polynomial ansatzes of varying complexity—ranging from first-degree to higher-order expansions—we yield compact rational-exponential solutions that are both exact and computationally tractable. The validity of these solutions is confirmed through symbolic computation in Mathematica, while a detailed graphical analysis elucidates their behavior—from bounded, dissipative profiles to singular structures—across different parameter regimes, including the critical thresholds $ C = 0 $ and $ C = 2 $ where blow-up phenomena emerge. This work underscores the efficacy of merging Laurent series with algebraic methods, offering a powerful and generalized tool for extracting exact solutions from a broader class of intractable nonlinear partial differential equations (PDEs) arising in mathematical physics and applied mathematics.
Citation: Muhammad Noman Qureshi, Atif Hassan Soori, Muhammad Shoaib Arif, Kamaleldin Abodayeh. Exact traveling wave solutions for the Chafee-Infante equation via a generalized first integral method[J]. AIMS Mathematics, 2026, 11(3): 5532-5556. doi: 10.3934/math.2026228
This study derives novel exact traveling wave solutions for the nonlinear (1+1)-dimensional Chafee-Infante equation by synthesizing the generalized first integral method (GFIM) with Laurent polynomial expansions. As a fundamental reaction-diffusion model, the Chafee-Infante equation governs pattern formation in diverse systems—from biological to chemical and physical contexts—yet its strong nonlinearity poses persistent challenges to classical integration techniques such as the inverse scattering transform or Hirota's method. We transform the equation into an autonomous polynomial system and employ the division theorem to systematically identify its first integrals, thereby circumventing the need for auxiliary equations or ansatz-based heuristics. By introducing Laurent polynomial ansatzes of varying complexity—ranging from first-degree to higher-order expansions—we yield compact rational-exponential solutions that are both exact and computationally tractable. The validity of these solutions is confirmed through symbolic computation in Mathematica, while a detailed graphical analysis elucidates their behavior—from bounded, dissipative profiles to singular structures—across different parameter regimes, including the critical thresholds $ C = 0 $ and $ C = 2 $ where blow-up phenomena emerge. This work underscores the efficacy of merging Laurent series with algebraic methods, offering a powerful and generalized tool for extracting exact solutions from a broader class of intractable nonlinear partial differential equations (PDEs) arising in mathematical physics and applied mathematics.
| [1] |
Z. S. Feng, The first-integral method to study the Burgers–Korteweg–de Vries equation, J. Phys. A-Math. Gen., 35 (2002), 343–349. https://doi.org/10.1088/0305-4470/35/2/312 doi: 10.1088/0305-4470/35/2/312
|
| [2] | N. Taghizadeh, M. Mirzazadeh, Exact travelling wave solutions for Konopelchenko-Dubrovsky equation by the first integral method, Appl. Appl. Math., 6 (2011), 12–24. Available at: https://digitalcommons.pvamu.edu/aam/vol6/iss1/12. |
| [3] |
M. Y. Moghaddam, A. Asgari, H. Yazdani, Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by extended tanh–coth, sine–cosine and Exp-function methods, Appl. Math. Comput., 210 (2009), 422–435. https://doi.org/10.1016/j.amc.2009.01.002 doi: 10.1016/j.amc.2009.01.002
|
| [4] |
S. A. Khuri, Exact solutions for a class of nonlinear evolution equations: A unified ansätze approach, Chaos Soliton. Fract., 36 (2008), 1181–1188. https://doi.org/10.1016/j.chaos.2006.09.066 doi: 10.1016/j.chaos.2006.09.066
|
| [5] |
M. Ilie, J. Biazar, Z. Ayati, The first integral method for solving some conformable fractional differential equations, Opt. Quant. Electron., 50 (2018), 55. https://doi.org/10.1007/s11082-017-1307-x doi: 10.1007/s11082-017-1307-x
|
| [6] |
F. L. Hasan, First integral method for constructing new exact solutions of the important nonlinear evolution equations in physics, J. Phys. Conf. Ser., 1530 (2020), 012109. http://dx.doi.org/10.1088/1742-6596/1530/1/012109 doi: 10.1088/1742-6596/1530/1/012109
|
| [7] |
A. M. Wazwaz, The sine–cosine method for obtaining solutions with compact and noncompact structures, Appl. Math. Comput., 159 (2004), 559–576. https://doi.org/10.1016/j.amc.2003.08.136 doi: 10.1016/j.amc.2003.08.136
|
| [8] |
A. H. Khater, W. Malfliet, D. K. Callebaut, E. S. Kamel, The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction–diffusion equations, Chaos Soliton. Fract., 14 (2002), 513–522. https://doi.org/10.1016/S0960-0779(01)00247-8 doi: 10.1016/S0960-0779(01)00247-8
|
| [9] |
B. Erbaş, E. Yusufoğlu, Exp-function method for constructing exact solutions of Sharma–Tasso–Olver equation, Chaos Soliton. Fract., 41 (2009), 2326–2330. https://doi.org/10.1016/j.chaos.2008.09.003 doi: 10.1016/j.chaos.2008.09.003
|
| [10] |
M. Sadaf, S. Arshed, G. Akram, M. R. Ali, I. Bano, Analytical investigation and graphical simulations for the solitary wave behavior of Chaffee–Infante equation, Results Phys., 54 (2023), 107097. https://doi.org/10.1016/j.rinp.2023.107097 doi: 10.1016/j.rinp.2023.107097
|
| [11] |
S. Arshed, G. Akram, M. Sadaf, M. B. Riaz, A. Wojciechowski, Solitary wave behavior of (2+1)-dimensional Chaffee-Infante equation, PloS One, 18 (2023), e0276961. http://dx.doi.org/10.1371/journal.pone.0276961 doi: 10.1371/journal.pone.0276961
|
| [12] |
Z. Haider, K. Ahmad, Novel exact solutions for a biological population model using the power index method, Eur. J. Pure Appl. Math., 18 (2025), 5936. https://doi.org/10.29020/nybg.ejpam.v18i2.5936 doi: 10.29020/nybg.ejpam.v18i2.5936
|
| [13] |
X. Zhang, T. A. Nofal, A. Vokhmintsev, M. M. A. Khater, Exploring solitary waves and nonlinear dynamics in the fractional Chaffee–Infante equation: A study beyond conventional diffusion models, Qual. Theor. Dyn. Syst., 23 (2024), 270. https://doi.org/10.1007/s12346-024-01121-w doi: 10.1007/s12346-024-01121-w
|
| [14] |
A. A. Mamun, N. H. M. Shahen, S. N. Ananna, M. Asaduzzaman, Solitary and periodic wave solutions to the family of new 3D fractional WBBM equations in mathematical physics, Heliyon, 7 (2021), e07478. https://doi.org/10.1016/j.heliyon.2021.e07483 doi: 10.1016/j.heliyon.2021.e07483
|
| [15] |
H. M. Baskonus, M. S. Osman, H. ur Rehman, M. Ramzan, M. Tahir, S. Ashraf, On pulse propagation of soliton wave solutions related to the perturbed Chen–Lee–Liu equation in an optical fiber, Opt. Quant. Electron., 53 (2021), 556. https://doi.org/10.1007/s11082-021-03190-6 doi: 10.1007/s11082-021-03190-6
|
| [16] | R. M. Soliby, Solving transport-density equation with diffusion using the first integral method and the generalized hyperbolic functions method, Ph.D. Thesis, Universiti Tun Hussein Onn Malaysia, 2020. |
| [17] |
S. Arshed, G. Akram, M. Sadaf, M. Irfan, M. Inc, Extraction of exact soliton solutions of (2+1)-dimensional Chaffee-Infante equation using two exact integration techniques, Opt. Quant. Electron., 56 (2024), 988. https://doi.org/10.1007/s11082-024-06470-z doi: 10.1007/s11082-024-06470-z
|
| [18] |
E. M. E. Zayed, M. El-Horbaty, B. M. M. Saad, A. H. Arnous, Y. Yildirim, Novel solitary wave solutions for stochastic nonlinear reaction–diffusion equation with multiplicative noise, Nonlinear Dynam., 112 (2024), 20199–20213. https://doi.org/10.1007/s11071-024-10085-0 doi: 10.1007/s11071-024-10085-0
|
| [19] |
T. A. Abassy, M. A. El-Tawil, H. E. Zoheiry, Solving nonlinear partial differential equations using the modified variational iteration Padé technique, J. Comput. Appl. Math., 207 (2007), 73–91. https://doi.org/10.1016/j.cam.2006.07.024 doi: 10.1016/j.cam.2006.07.024
|
| [20] |
C. M. Khalique, O. D. Adeyemo, M. S. Monashane, Exact solutions, wave dynamics and conservation laws of a generalized geophysical Korteweg de Vries equation in ocean physics using Lie symmetry analysis, Adv. Math. Models Appl., 9 (2024), 147–172. https://doi.org/10.62476/amma9147 doi: 10.62476/amma9147
|
| [21] |
R. Cimpoiasu, Multiple explicit solutions of the 2D variable coefficients Chafee–Infante model via a generalized expansion method, Mod. Phys. Lett. B, 35 (2021), 2150312. https://doi.org/10.1142/S0217984921503127 doi: 10.1142/S0217984921503127
|
| [22] |
K. Ahmad, K. Bibi, M. S. Arif, K. Abodayeh, New exact solutions of Landau-Ginzburg-Higgs equation using power index method, J. Funct. Space., 2023 (2023), 4351698. https://doi.org/10.1155/2023/4351698 doi: 10.1155/2023/4351698
|
| [23] |
M. S. Arif, K. Abodayeh, Y. Nawaz, Numerical modeling of mixed convective nanofluid flow with fractal stochastic heat and mass transfer using finite differences, Front. Energy Res., 12 (2024), 1373079. https://doi.org/10.3389/fenrg.2024.1373079 doi: 10.3389/fenrg.2024.1373079
|
| [24] |
Y. Nawaz, M. S. Arif, K. Abodayeh, M. U. Ashraf, Modeling viscous dissipation in MHD boundary layer flow: A two-stage in time and compact in space finite difference approach, Numer. Heat Tr. B-Fund. 87 (2024), 1–30. https://doi.org/10.1080/10407790.2024.2338904 doi: 10.1080/10407790.2024.2338904
|
| [25] |
F. Bouchelaghem, H. Boulares, A. Ardjouni, F. Jarad, T. Abdeljawad, B. Abdalla, et al., Existence of solutions of multi-order fractional differential equations, Part. Differ. Equ. Appl. Math., 13 (2025), 101104. https://doi.org/10.1016/j.padiff.2025.101104 doi: 10.1016/j.padiff.2025.101104
|
| [26] |
M. Bilal, J. Iqbal, K. Shah, B. Abdalla, T. Abdeljawad, I. Ullah, Analytical solutions of the space–time fractional Kundu–Eckhaus equation by using modified extended direct algebraic method, Part. Differ. Equ. Appl. Math., 11 (2024), 100832. http://dx.doi.org/10.1016/j.padiff.2024.100832 doi: 10.1016/j.padiff.2024.100832
|
| [27] |
N. Manjunatha, M. G. Reddy, A. Aloqaily, S. Aljohani, A. R. Reddy, F. Ali, et al., Radiation effects on rotating system free convective nanofluid unsteady flow with heat source and magnetic field, Part. Differ. Equ. Appl. Math., 13 (2025), 101083. http://dx.doi.org/10.1016/j.padiff.2025.101083 doi: 10.1016/j.padiff.2025.101083
|
| [28] |
I. Ahmad, K. J. Ansari, H. Alrabaiah, D. Santina, N. Mlaiki, Study of 1+1 dimensional fractional order non-linear Benney equation using an analytical technique, Part. Differ. Equ. Appl. Math., 11 (2024), 100823. http://dx.doi.org/10.1016/j.padiff.2024.100823 doi: 10.1016/j.padiff.2024.100823
|
| [29] |
T. Y. Han, Y. Y. Jiang, H. G. Fan, Exploring shallow water wave phenomena: A fractional approach to the Whitham-Broer-Kaup-Boussinesq-Kupershmidt system, Ain Shams Eng. J., 16 (2025), 103700. http://dx.doi.org/10.1016/j.asej.2025.103700 doi: 10.1016/j.asej.2025.103700
|
| [30] |
R. Darazirar, R. M. Yaseen, A. A. Mohsen, A. Khan, T. Abdeljawad, Minimal wave speed and traveling wave in nonlocal dispersion SIS epidemic model with delay, Bound. Value Probl., 2025 (2025), 67. https://doi.org/10.1186/s13661-025-02055-1 doi: 10.1186/s13661-025-02055-1
|
| [31] |
I. Ullah, K. Shah, T. Abdeljawad, Study of traveling soliton and fronts phenomena in fractional Kolmogorov-Petrovskii-Piskunov equation, Phys. Scripta, 99 (2024), 055259. https://doi.org/10.1088/1402-4896/ad3c7e doi: 10.1088/1402-4896/ad3c7e
|
| [32] |
A. Mahmood, M. Abbas, G. Akram, M. Sadaf, M. B. Riaz, T. Abdeljawad, Solitary wave solution of (2+1)-dimensional Chaffee–Infante equation using the modified Khater method, Results Phys., 48 (2023), 106416. http://dx.doi.org/10.1016/j.rinp.2023.106416 doi: 10.1016/j.rinp.2023.106416
|
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Muhammad Noman Qureshi, Atif Hassan Soori, Muhammad Shoaib Arif, Kamaleldin Abodayeh. Exact traveling wave solutions for the Chafee-Infante equation via a generalized first integral method[J]. AIMS Mathematics, 2026, 11(3): 5532-5556. doi: 10.3934/math.2026228
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