Nonlinear Instability in a Semiclassical Problem

Details Nonlinear Instability in a Semiclassical Problem Nonlinear Instability in a Semiclassical Problem

2011-09-05

arxiv.org/abs/1109.0773v2

Mathematics

Analysis of PDEs

Scientific paper

We consider a nonlinear evolution problem with an asymptotic parameter and construct examples in which the linearized operator has spectrum uniformly bounded away from Re z >= 0 (that is, the problem is spectrally stable), yet the nonlinear evolution blows up in short times for arbitrarily small initial data. We interpret the results in terms of semiclassical pseudospectrum of the linearized operator: despite having the spectrum in Re z < -c < 0, the resolvent of the linearized operator grows very quickly in parts of the region Re z > 0. We also illustrate the results numerically.

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