Computing stability of multi-dimensional travelling waves

TL;DR

This paper introduces a numerical Evans function shooting method for computing the stability spectrum of multi-dimensional travelling waves in parabolic systems, demonstrated on a cubic autocatalysis model.

Contribution

It develops a novel shooting approach using Fourier projection and Riccati equations to efficiently compute stability spectra of multi-dimensional travelling fronts.

Findings

The method accurately computes eigenvalues for stability analysis.

Comparison shows the shooting approach outperforms continuous orthogonalization.

Standard projection methods are less efficient for this problem.

Abstract

We present a numerical method for computing the pure-point spectrum associated with the linear stability of multi-dimensional travelling fronts to parabolic nonlinear systems. Our method is based on the Evans function shooting approach. Transverse to the direction of propagation we project the spectral equations onto a finite Fourier basis. This generates a large, linear, one-dimensional system of equations for the longitudinal Fourier coefficients. We construct the stable and unstable solution subspaces associated with the longitudinal far-field zero boundary conditions, retaining only the information required for matching, by integrating the Riccati equations associated with the underlying Grassmannian manifolds. The Evans function is then the matching condition measuring the linear dependence of the stable and unstable subspaces and thus determines eigenvalues. As a model application, we study the stability of two-dimensional wrinkled front solutions to a cubic autocatalysis model system. We compare our shooting approach with the continuous orthogonalization method of Humpherys and Zumbrun. We then also compare these with standard projection methods that directly project the spectral problem onto a finite multi-dimensional basis satisfying the boundary conditions.