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Volume 18
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Article

Analysis of Reaction-Diffusion Systems: Painlevé Properties and Conservation Laws of the Convective Fisher–Kolmogorov Equations

by
Ali Raza
1,2,*,
Alhussein Mohamed Alhussein Ahmed
3 and
Abdul Hamid Kara
4
1
Department of Mathematical Sciences, Stellenbosch University, Stellenbosch 7602, South Africa
2
Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan
3
School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg 2050, South Africa
4
School of Mathematics, University of the Witwatersrand, Johannesburg 2050, South Africa
*
Author to whom correspondence should be addressed.
Symmetry 2026, 18(2), 377; https://doi.org/10.3390/sym18020377
Submission received: 10 January 2026 / Revised: 12 February 2026 / Accepted: 16 February 2026 / Published: 18 February 2026
(This article belongs to the Special Issue Symmetry in Lie Groups and Lie Algebras)
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Abstract

This article investigates the widespread occurrence of wave-like behavior in biological systems and reflects on the importance of reaction–diffusion frameworks to explain fast information transfer. Focusing on the classical Fisher–Kolmogorov equation, it demonstrates how the combination of nonlinear reaction dynamics and diffusion gives rise to stable traveling wavefronts that propagate at a constant speed while preserving their shape. We adopt a number of approaches (some novel) in dealing with the analyses, namely the symmetry method, soliton approach (algebraic) and Painlevé analysis, among others.
Keywords: biological waves; eaction–diffusion systems; Fisher–Kolmogorov equation; conservation laws; Painlevé analysis biological waves; eaction–diffusion systems; Fisher–Kolmogorov equation; conservation laws; Painlevé analysis

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MDPI and ACS Style

Raza, A.; Ahmed, A.M.A.; Kara, A.H. Analysis of Reaction-Diffusion Systems: Painlevé Properties and Conservation Laws of the Convective Fisher–Kolmogorov Equations. Symmetry 2026, 18, 377. https://doi.org/10.3390/sym18020377

AMA Style

Raza A, Ahmed AMA, Kara AH. Analysis of Reaction-Diffusion Systems: Painlevé Properties and Conservation Laws of the Convective Fisher–Kolmogorov Equations. Symmetry. 2026; 18(2):377. https://doi.org/10.3390/sym18020377

Chicago/Turabian Style

Raza, Ali, Alhussein Mohamed Alhussein Ahmed, and Abdul Hamid Kara. 2026. "Analysis of Reaction-Diffusion Systems: Painlevé Properties and Conservation Laws of the Convective Fisher–Kolmogorov Equations" Symmetry 18, no. 2: 377. https://doi.org/10.3390/sym18020377

APA Style

Raza, A., Ahmed, A. M. A., & Kara, A. H. (2026). Analysis of Reaction-Diffusion Systems: Painlevé Properties and Conservation Laws of the Convective Fisher–Kolmogorov Equations. Symmetry, 18(2), 377. https://doi.org/10.3390/sym18020377

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