Traveling Wave Solutions of the Burgers, Huxley and Nagumo Equations via the Variation of (G′/G)-Expansion Method: Analytical Fronts for Nerve Impulse Propagation in Excitable Media

Abstract

The Burgers–Huxley and Nagumo equations are canonical bistable reaction-diffusion models for the propagation of an action potential along a nerve fiber, distinguished from the classical Fisher–KPP equation by a cubic reaction term with three equilibria: a resting state, an excitation threshold, and an excited state. This paper applies the Variation of the -Expansion Method to both equations. For the Burgers–Huxley equation, under both the standard wave transformation  and a general transformationthe method returns only the two trivial constant equilibria  and  indicating that first-order balancing is insufficient to recover a genuine connecting front for this closure. For the Nagumo equation with an additional nonlinear convection term, the same method applied with the general transformation  yields twenty parameter families and thirty-eight explicit closed-form solutions. Critical examination of these solutions shows that exactly three of the tanh-type families are bounded, monotone kink solutions connecting each pair of the equilibria  of the underlying cubic nonlinearity, reproducing the classical bistable-front structure of the Nagumo equation, while the coth-type and composite families contain a non-removable singularity and are therefore mathematically valid but not physiologically realizable as impulses. These results clarify which of the many solutions produced by -type expansion methods correspond to genuine traveling nerve-impulse fronts, and illustrate the role of the convection and threshold parameters in setting the front speed and steepness.