On the convective nature of the instability of a front undergoing a supercritical Turing bifurcation

Fronts are traveling waves in spatially extended systems that connect two different spatially homogeneous rest states. If the rest state behind the front undergoes a supercritical Turing instability, then the front will also destabilize. On the linear level, however, the front will be only convectively unstable since perturbations will be pushed away from the front as it propagates. In other words, perturbations may grow but they can do so only behind the front. It is of interest to show that this behavior carries over to the full nonlinear system. It has been successfully done in a case study by Ghazaryan and Sandstede [A. Ghazaryan, B. Sandstede, Nonlinear convective instability of Turing-unstable fronts near onset: a case study, SIAM J. Appl. Dyn. Syst. 6 (2007) 319–347]. In the present paper, analogous results are obtained for the same system as in Ghazaryan and Sandstede (2007), but for a different parameter regime.