The effect of perturbed advection on a class of solutions of a non-linear reaction-diffusion equation

In this work, the traveling wave solutions of a one-dimensional reaction-diffusion equation with advection are studied. The traveling wave solutions are obtained using the G′/G-expansion method. The shock thickness and spectral stability have been discussed for the obtained solution in the parameter interval. The essential spectra of the perturbed and linearized differential operator about the traveling antikink and kink solutions at the equilibrium states are obtained. The point spectrum is calculated using Evans function with Lie midpoint method and Magnus method. It is shown that, for a symmetric potential well, the traveling kink and antikink solutions which connect the stable equilibrium states of the system are stable. It is observed that the perturbation on the advection exhibits contrasting effect on the solution properties (shock thickness and the eigenvalue) of kink and antikink solutions. Variation of the reaction coefficient leads to instability of the solutions, unlike the diffusion coefficient which enhances the stability. On the other hand, the variation of reaction and diffusion coefficients show the monotonic effect on the shock thickness of the traveling kink and antikink solutions. This study is expected to be useful in analyzing the slow or fast invasion and stability of the population movement in different steady states.