Absolute and convective instabilities of waves on unbounded and large bounded domains

Instabilities of nonlinear waves on unbounded domains manifest themselves i n different ways. An absolute instability occurs if the amplitude of locali zed wave packets grows in time at each fixed point in the domain. In contra st, convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time, perturbations decay locall y at each given point in the unbounded domain: wave packets are convected t owards infinity In experiments as well as in numerical simulations, bounded domains are often more relevant. We are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the afore mentioned (in)stability properties of a wave. These effects depend upon the boundary conditions that are imposed on the bounded domain. We compare the spectra of the linearized evolution operators on unbounded and bounded dom ains for two classes of boundary conditions. It is proved that periodic bou ndary conditions reproduce the point and essential spectrum on the unbounde d domain accurately. Spectra for separated boundary conditions behave in qu ite a different way: firstly, separated boundary conditions may generate ad ditional isolated eigenvalues. Secondly, the essential spectrum on the unbo unded domain is in general not approximated by the spectrum on the bounded domain. Instead, the so-called absolute spectrum is approximated that corre sponds to the essential spectrum on the unbounded domain calculated with ce rtain optimally chosen exponential weights. We interpret the difference bet ween the absolute and the essential spectrum in terms of the convective beh avior of the wave on the unbounded domain. In particular, it is demonstrate d that the stability of the absolute spectrum implies convective instabilit y of the wave, but that convectively unstable waves can destabilize under d omain truncation. The theoretical predictions are compared with numerical c omputations. (C) 2000 Elsevier Science B.V. All rights reserved.