Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals

If a traveling wave is stable or unstable depends essentially on the spectrum of a differential operator P obtained by linearization. We investigate how spectral properties of the all-line operator P are related to eigenvalues of finite-interval BVPs Pu(x) = su(x), x - less than or equal to x less than or equal to x(+), Ru = 0. Here R is a linear boundary operator; for which we will derive determinant conditions, and the x-interval is assumed to be sufficiently large. Under suitable assumptions, we show (a) resolvent estimates for large s; (b) if s is in the resolvent of the all-line operator P, then s is also in the resolvent for finite-interval BVPs; (c) eigenvalues of P lead to approximating eigenvalues on finite intervals. These results allow to study the stability question for traveling waves by investigating eigenvalues of finite-interval problems. We give applications to the FitzHugh-Nagumo system with small diffusion and to the complex Ginzburg-Landau equations.