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Recent developments on wave instability. (English) Zbl 0870.76030

Aston, Philip J. (ed.), Nonlinear mathematics and its applications. Proceedings of the 4th spring school on applied nonlinear mathematics, Surrey, GB, April 3-7, 1995. Cambridge: Cambridge University Press. 205-217 (1996).
We review recent developments of methods for proving sufficient conditions for the exitence of unstable eigenvalues; a problem which is tractable and leads to new results on wave dynamics in open systems.
First, a model problem is considered in more detail and, with a combination of Floquet theory and a geometric analysis of the basic state, a criterion for the existence of unstable eigenvalues is presented. Although the model problem is fairly simple, the formulation and proof are a prototype for more complex wave problems. The formulation is abstracted and then extended to the instability problem for spatially quasiperiodic waves. Here there are interesting connections with the geometry of invariant tori. Stability of quasiperiodic waves is especially difficult to analyse since Floquet’s theorem is not, in general, applicable.
The instability problem for a class of dispersive wave equations is formulated, and it is shown that the geometric instability criterion for periodic travelling waves is similar to that for stationary quasiperiodic waves. Moreover, there are interesting connections with the Whitham modulation theory; the instability criterion provides a rigorous proof of the instability predicted by the Whitham modulation equation. Finally, a sketch of some other recent developments is presented.
For the entire collection see [Zbl 0845.00057].

MSC:

76E99 Hydrodynamic stability
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
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