ar X iv : 0 80 5 . 17 06 v 2 [ m at h . D S ] 1 D ec 2 00 8 COMPUTING STABILITY OF MULTI - DIMENSIONAL TRAVELLING WAVES

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 eigen-values. 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. 1. Introduction. Our goal in this paper is to present a practical extension of the Evans function shooting method to computing the stability of multi-dimensional travelling wave solutions to parabolic nonlinear systems. The problem is thus to solve the linear spectral equations for small perturbations of the travelling wave. Hence we need to determine the values of the spectral parameter for which solutions match longitudinal far-field zero boundary conditions and, in more than one spatial dimension, the transverse boundary conditions. These are the eigenvalues. There are two main approaches to solving such eigenvalue problems. • Projection: Project the spectral equations onto a finite dimensional basis, which by construction satisfies the boundary conditions. Then solve the resulting matrix eigenvalue problem. • Shooting: Start with a specific value of the spectral parameter and the correct boundary conditions at one end. Shoot/integrate towards the far end. Examine how close this solution is to satisfying the boundary conditions at the far end. Readjust the spectral parameter accordingly. Repeat the shooting procedure until the solution matches the far boundary conditions. The projection method can be applied to travelling waves of any dimension. Its main disadvantage is that to increase accuracy or include adaptation is costly and