Instabilities of nonlinear waves on unbounded domains manifest themselves in different ways. An absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain. In contrast, convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time, perturbations decay locally at each given point in the unbounded domain: wave packets are convected towards 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 aforementioned (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 domains 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 different way: firstly, separated boundary conditions may generate additional isolated eigenvalues. Secondly, 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 difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain. In particular, it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave, but that convectively unstable waves can destabilize under domain truncation. The theoretical predictions are compared with numerical computations.
Bibliographical noteFunding Information:
B. Sandstede was partially supported by the NSF under grant DMS-9971703 and by the Ohio State University through a seed grant. A. Scheel was partially supported by the Mathematical Research Institute of the Ohio State University and by the DFG-SP Analysis, Ergodentheorie und effiziente Simulation dynamischer Systeme. We are very grateful to the referees for many comments and suggestions that helped to improve the presentation of the material.
- Absolute instability
- Convective instability
- Domain truncation
- Essential spectrum
- Travelling waves