The Burgers-Fisher Equation: Convection, Diffusion, and Reaction in a Single Model -- A Comprehensive Review

Abstract

The Burgers-Fisher equation unites the two most influential scalar models of nonlinear science: the Burgers equation, which captures the competition between convection and diffusion, and the Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP) equation, which describes the propagation of reaction fronts. The resulting convection-reaction-diffusion equation is a prototypical model for travelling fronts in population genetics, combustion, plasma physics, and biology, and it is one of the few nonlinear partial differential equations for which exact travelling-wave solutions are known in closed form. This review provides a comprehensive and critical assessment of the literature on the Burgers-Fisher equation and its generalizations. We survey the analytical theory of travelling waves, including the exact solutions that make the equation a benchmark, and the rigorous results on existence, uniqueness, and stability of these waves. We examine the broad family of numerical methods that have been devised to approximate its solutions, from finite difference and finite element schemes to B-spline collocation, spectral, meshless, lattice Boltzmann, and operator-splitting methods. We discuss the fractional-order extensions of the equation, its relation to the Burgers-Huxley and FitzHugh-Nagumo equations, and the wide range of applications in biology, chemistry, physics, and engineering. Particular attention is given to recent developments, including machine-learning approaches such as physics-informed neural networks and deep learning algorithms, and to the newest results published in 2025 and 2026. We close by identifying open problems and future directions.