Nonlinear eigenvalue methods for linear pointwise stability of nonlinear waves

TL;DR

This paper introduces an iterative inverse power method to accurately find spectral values like eigenvalues and resonance poles in nonlinear matrix pencils, enhancing stability analysis of nonlinear waves on one-dimensional domains.

Contribution

The paper presents a novel inverse power method for locating spectral values in nonlinear eigenvalue problems, avoiding determinant computations and improving robustness.

Findings

Method effectively finds eigenvalues and resonance poles.

Algorithm does not require prior knowledge of spectral features.

Applicable to stability analysis of nonlinear wave systems.

Abstract

We propose an iterative method to find pointwise growth exponential growth rates in linear problems posed on essentially one-dimensional domains. Such pointwise growth rates capture pointwise stability and instability in extended systems and arise as spectral values of a family of matrices that depends analytically on a spectral parameter, obtained via a scattering-type problem. Different from methods in the literature that rely on computing determinants of this nonlinear matrix pencil, we propose and analyze an inverse power method that allows one to locate robustly the closest spectral value to a given reference point in the complex plane. The method finds branch points, eigenvalues, and resonance poles without a priori knowledge.