Spectral stability of one-dimensional reaction–diffusion equation with symmetric and asymmetric potential

Abstract

In this work, the spectral stability of a class of solutions of the one-dimensional nonlinear reaction–diffusion equation with spatial symmetry and asymmetry is studied. The solutions are first obtained using the \(G'/G\)-expansion method, and it is shown that this technique generates only kink and antikink type solutions. The essential spectra of the perturbed differential operator at the equilibrium states are obtained. The Evans function is used to determine the point spectrum using Lie midpoint method and Magnus method over a large range of spatial distance. It is shown that for a symmetric potential well, the stationary kink and antikink solutions connecting the stable equilibrium (vacuum) states of the system are stable. Interestingly, it is found that for a potential with broken symmetry, the stable stationary kink/antikink solutions transform into stable traveling kink/antikink solutions. This partially answers the question (Alfimov and Medvedeva in Phys Rev E 84:056606, 2011) on the qualitative features of the potential that are decisive for the existence/nonexistence of traveling kink/antikink solutions. The region of stability of the traveling kink/antikink solutions in the parameter space of the potential is identified. An interesting exchange of wave speeds between the kink and antikink solutions through the stationary solution state corresponding to the symmetric potential is brought out.