The stability of certain localized structures in the perturbed Allen-Cahn and Gierer-Meinhardt systems

In this thesis, we study the stability of certain localized structures in two families of reaction-diffusion equations. Specifically we study the stability of asymmetric spike solutions to the Gierer-Meinhardt system and of curved front solutions to a perturbed Allen-Cahn model.;We study the spectra of asymmetric multispike solutions near the point at which they bifurcate off a symmetric multispike branch, (1) for the case in which the reaction-time constant tau is zero, and (2) for the case in which tau > 0. For the tau ≥ 0 case, we derive an expression for the small eigenvalues. We confirm that all such asymmetric spike solutions are unstable in a neighbourhood of the bifurcation point and we derive an explicit expression for the leading order terms of the critical eigenvalues. For the tau ≥ 0 case we study the impact on the value of tau at which Hopf bifurcation occurs, as asymmetric spike solutions bifurcate off symmetric spike solutions. We find that tau can either increase or decrease, depending upon values the of the Gierer-Meinhardt parameters.;Using singular perturbation techniques, we characterize the stability of curved front solutions to a perturbed Allen-Cahn model in terms of a geometric eigenvalue problem, and give a simple geometric interpretation of the results.