Bifurcation Analysis and Dynamical Investigation of Nonlinear DiffusionReaction Equations with Nonlinear Convective Flux Term: Stability and Applications

Abstract

The aim of this work is to examine the rich dynamics of quadratic and quartic nonlinear diffusion-reaction (DR) equations with a nonlinear convective flux term. These equations are crucial for simulating a variety of biological and physical processes, such as the dynamics of species populations. The main goal is to use the modified extended simple equation method (mESEM), a generalization of the standard simple equation method that hasn’t been used in this situation before, to extend the analytical treatment of such equations. We obtain a variety of new exact solutions using this method, such as breathers, kink and anti-kink waves, multi-peak solitons, bright-dark solitons, periodic waves, and waveforms represented by hyperbolic, trigonometric, and rational functions. Analyzing the stability and physical relevance of these solutions is another major goal of this work. Modulational instability analysis verifies the robustness of the obtained waveforms, while bifurcation analysis reveals qualitative changes in system behavior under parameter variations. The various wave structures and their dynamical characteristics are further highlighted with graphic illustrations. In general, the study highlights the potential of mESEM to reveal rich wave phenomena with applications spanning fluid dynamics, plasma physics, chemical reaction processes, population biology, neuroscience, and optical fiber communication, in addition to showcasing its effectiveness and versatility in solving nonlinear DR equations.OPEN ACCESS Received: 15/07/2025 Accepted: 18/09/2025 Published: 23/01/2026

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