Algebraically decaying pulses in a Ginzburg-Landau system with a neutrally stable mode.

Abstract. In this paper, we study the existence and stability of pulse solutions in a system with interacting instability mechanisms, which is described by a Ginzburg-Landau equation for an A-mode, coupled to a diffusion equation for a B-mode. Our main question is whether this coupling may stabilize solutions of the Ginzburg-Landau equation that are unstable when the interactions with the neutrally stable B-mode are not included in the model. The spatially homogeneous B-mode is supposed to be neutrally stable. This implies that the pulse solutions cannot decay exponentially, but must decay with an algebraic rate as x → ±∞. As a consequence, the methods that exist in the literature by which the stability of pulses in singularly perturbed reaction-diffusion systems can be studied need to be extended. This results in an 'algebraic NLEP approach', which is expected to be relevant beyond the setting of this paper. As in the case of a (weakly) stable B-mode (Doelman et al 2004 J. Nonlin. Sci. 14 237-78) we establish by the application of this approach that the B-mode indeed introduces a mechanism that may stabilize pulses that are unstable when the interactions with the B-mode are not taken into account.

Mathematics Subject Classification: 35B35, 37L15, 35B32, 35K57, 76E30