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July  2009, 23(3): 617-638. doi: 10.3934/dcds.2009.23.617

Front propagation in a noisy, nonsmooth, excitable medium

Mohar Guha 1, and  Keith Promislow 2,

1. 

Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, United States
2. 

Department of Mathematics, Michigan State University, East Lansing, MI, 48824, United States

Received  October 2007 Revised  August 2008 Published  November 2008

We consider the impact of noise on the stability and propagation of fronts in an excitable media with a piece-wise smooth, discontinuous ignition process. In a neighborhood of the ignition threshold the system interacts strongly with noise, the front can loose monotonicity, resulting in multiple crossings of the ignition threshold. We adapt the renormalization group methods developed for coherent structure interaction, a key step being to determine pairs of function spaces for which the the ignition function is Frechet differentiable, but for which the associated semi-group, $S(t)$, is integrable at $t=0$. We parameterize a neighborhood of the front solution through a dynamic front position and a co-dimension one remainder. The front evolution and the asymptotic decay of the remainder are on the same time scale, the RG approach shows that the remainder becomes asymptotically small, in terms of the noise strength and regularity, and the front propagation is driven by a competition between the ignition process and the noise.
Keywords: Renormalization Group, Front Dynamics, Excitable Media, Noise, Nonsmooth Ignition Systems.
Mathematics Subject Classification: Primary: 35K55, 35B40; Secondary: 74J30.
Citation: Mohar Guha, Keith Promislow. Front propagation in a noisy, nonsmooth, excitable medium. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 617-638. doi: 10.3934/dcds.2009.23.617
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