CRM-McGill Applied Math Seminar
Tuesday February 26th, 3:35pm
McGill, Burnside Hall Room 1205


Bistable Waves in Discrete Inhomogeneous Media

A crude model for electrical conduction in the nervous system is the 
spatially discrete Nagumo equation. Employing a piecewise linear 
approximation of the nonlinearity, one can derive exact solutions of this 
system such that a portion of the medium for conduction is deteriorated, 
characteristic of diseases that affect the nervous system.  Using Jacobi 
operator theory and transform techniques, wave-like solutions are 
constructed for problems with essentially arbitrary inhomogeneous discrete 
diffusion, and these solutions directly correspond to monotone traveling 
wave solutions in the case of homogeneous diffusion. A thorough study of 
the steady state solutions provides necessary and sufficient conditions 
for traveling waves to fail to propagate due to inhomogeneities in the 
medium.  Solutions with nonzero wave speed are also derived, and we learn 
how the wave speed and wave form are affected by the inhomogeneities.  
The case of one defect in the medium is considered as an example, and due 
to the complexity of the solutions, a numerical analysis becomes an 
essential part of this study.