| The difficulty in the problem of determining stability of periodic travelling waves under the most general class of bounded continuous perturbations resides in the fact that the spectrum associated with the linearized operator is composed of bands, or more generally in the non-adjoint case, of "loops", rather than discrete spectrum as is the case when considering the more restricted case of periodic perturbations. In particular, we are interested in systems exhibiting a singular perturbation structure. We carry out this program for the celebrated Fitz-Hugh Nagumo system, ut=uxx+fu -w wt=eu-gw . where epsilon is a small parameter and f( u) = u(1 - u)(u - a). This equation is a simplified version of the Hodgkin-Huxley equations that model nerve pulse transmission through neurons. It is well known that there is a one-parameter family of periodic solutions for all 0 < epsilon < epsilon0, epsilon0 small enough. In this work, the precise structure of the continuous spectrum of the linearized operator about these nonlinear waves is obtained (as computed with respect to the space Linfinity. ) It will be established that this spectral set lies on the (open) left-half plane with the only exception of the eigenvalue at the origin, which is present due to the translation invariance of the waves. From this precise structure of the spectral set, nonlinear stability with respect long-wavelength periodic perturbations will follow.; The technical contributions of this work lie on three different fronts: (1) Matched asymptotic analysis that allow for the computation of the spectrum of the linearized operator about the nonlinear waves near the origin. (2) An extension of the Elephant-Trunk Lemma as studied in connection to the stability analysis of the pulse solutions associated with the Gray-Scott model. (3) The discovery of pole/zero cancellations in the factorization of the Evans Function. |
