Stability and Freezing of Nonlinear Waves in First Order Hyperbolic PDEs |
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Authors: | Jens Rottmann-Matthes |
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Institution: | 1. Bielefeld University, P.O. Box 100131, 33501, Bielefeld, Germany
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Abstract: | It is a well-known problem to derive nonlinear stability of a traveling wave from the spectral stability of a linearization.
In this paper we prove such a result for a large class of hyperbolic systems. To cope with the unknown asymptotic phase, the
problem is reformulated as a partial differential algebraic equation for which asymptotic stability becomes usual Lyapunov
stability. The stability proof is then based on linear estimates from (Rottmann-Matthes, J Dyn Diff Equat 23:365–393, 2011) and a careful analysis of the nonlinear terms. Moreover, we show that the freezing method (Beyn and Thümmler, SIAM J Appl
Dyn Syst 3:85–116, 2004; Rowley et al. Nonlinearity 16:1257–1275, 2003) is well-suited for the long time simulation and numerical approximation of the asymptotic behavior. The theory is illustrated
by numerical examples, including a hyperbolic version of the Hodgkin–Huxley equations. |
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