Preprint 19/99

Preprint 19/99 in the DFG-Schwerpunktprogramm "Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme":

W.-J. Beyn, J. Lorenz: Stability of Traveling Waves: Dichotomies and Eigenvalue Conditions on Finite Intervals: We present a continuation method for low-dimensional invariant subspaces of a parameterized family of large and sparse matrices. Such matrices typically occur when linearizing about branches of steady states in reaction-diffusion equations. Our continuation method provides bases of the invariant subspaces depending smoothly on the parameter. From these we can compute the corresponding eigenvalues efficiently. The predictor and the corrector step are reduced to solving bordered matrix equations of Sylvester type. For these equations we develop a bordered version of the Bartels-Stewart algorithm. The numerical techniques are used to study the stability problem for traveling waves in two examples: the Ginzburg-Landau and the FitzHugh-Nagumo system. In these cases there always exists a simple or multiple eigenvalue zero while the remaining eigenvalues determine the stability. We demonstrate the difficulties of separating these critical eigenvalues from clusters of eigenvalues that are generated by the essential spectrum of the continuous problem.
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Thorsten Pampel, 1999-07-30