Nonlinear Instability in a Semiclassical Problem

TL;DR

This paper demonstrates that a nonlinear evolution can blow up rapidly despite the linearized operator being spectrally stable, highlighting the importance of pseudospectrum analysis in semiclassical problems.

Contribution

It constructs examples showing nonlinear blow-up occurs even when the linearized operator's spectrum indicates stability, emphasizing pseudospectrum's role.

Findings

Nonlinear blow-up can occur despite spectral stability.

Pseudospectrum growth explains instability.

Numerical illustrations support theoretical results.

Abstract

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.