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

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.