From Unbounded to Bounded Domains: the Fate of Point and Essential Spectrum

The existence and stability of travelling waves are typically investigated on unbounded domains even though the domains in experiments or numerical simulations are bounded. We are interested in the eeects that the truncation of the unbounded to a large but bounded domain has on the stability properties of a nonlinear wave. These eeects depend upon the boundary conditions that are imposed on the bounded domain. We compare the spectra of the relevant operators on the unbounded and the bounded domain for two classes of boundary conditions. It is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately. Spectra for separated boundary conditions behave in quite a diierent way: rst, separated boundary conditions may generate additional eigenvalues. Second, the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain. Instead, the so-called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights. We interpret the diierence between the absolute and the essential spectrum in terms of convective behavior of the linearization about the wave. It is proved that stability of the absolute spectrum implies pointwise stability of the wave but that pointwise stable waves can destabilize under domain truncation. The theoretical predictions are compared with numerical computations.