| Abstract: |
| In this talk, I will present recent results on the FitzHugh-Nagumo (FHN) system of partial differential equations. We consider the FHN system in a more realistic geometric setting: on cylindrical surfaces of variable radii, rather than straight lines without internal geometric structure, as it has been extensively done. We show that, under some reasonable conditions, the solutions of the system are exponentially approximated by their radial averages. We also show that the radial averages are close, for very long times, to solutions of a 1-spatial-dimensional system involving the cylindrical profile: it is obtained from the standard FHN system by replacing the second-order derivative by the radial surface Laplace-Beltrami operator. This system is considerably simpler for mathematical analysis and numerical simulations. It offers an effective system for the pulse propagation. I will outline the analytical approach to obtain the above results, and, time permitting, I will discuss interesting related open problems and natural extensions of these results. The talk is based on a recent joint work with Israel Michael Sigal (University of Toronto, Canada) and Georgia Karali (NKUA, Greece). |
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