Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain)
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Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain) solutions of partial differential equations Jonathan A. Sherratt
Received: 22 March 2012 / Accepted: 2 July 2012 / Published online: 15 August 2012 © Springer Science+Business Media, LLC 2012
Abstract A variety of numerical methods are available for determining the stability of a given solution of a partial differential equation. However for a family of solutions, calculation of boundaries in parameter space between stable and unstable solutions remains a major challenge. This paper describes an algorithm for the calculation of such stability boundaries, for the case of periodic travelling wave solutions of spatially extended local dynamical systems. The algorithm is based on numerical continuation of the spectrum. It is implemented in a fully automated way by the software package wavetrain, and two examples of its use are presented. One example is the Klausmeier model for banded vegetation in semi-arid environments, for which the change in stability is of Eckhaus (sideband) type; the other is the two-component Oregonator model for the photosensitive Belousov–Zhabotinskii reaction, for which the change in stability is of Hopf type. Keywords Numerical continuation · Periodic traveling wave · Wavetrain · Auto · Eckhaus · Hopf · Spectral stability Mathematics Subject Classifications (2010) 65P99 · 35P05
Communicated by: A. Zhou. J. A. Sherratt (B) Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK e-mail: [email protected]
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J.A. Sherratt
1 Introduction The stability of spatiotemporal solutions of partial differential equations (pdes) is fundamental to their mathematical role and to their relevance in applications. There are various different notions of stability and throughout this paper I consider spectral stability, defined formally in Section 2 below. In some special cases (e.g. [1, 2]) this can be determined analytically. Otherwise one must rely on numerical calculation, and a number of numerical methods are available. For wave fronts and pulses, the most established method uses shooting to solve the eigenfunction equation (e.g. [3–7], http://www.amsta.leeds.ac.uk/˜jitse/software.html); the mismatch of solutions obtained by shooting from plus and minus infinity is characterised by the Evans function [8–10]. An alternative method due to Deconinck and coworkers [11, 12] is based on truncating the Fourier–Floquet expansion of the eigenfunction equation (“Hill’s Method”). The method is particularly well suited to spatially periodic solutions, though it can be applied more generally, and a “black box” software package implementing the method is available at http://www.amath.washington.edu/hill/spectruw.html. A third method is due to Rademacher et al. [13], and involves numerical continuation of the spectrum using the (imaginary) spatial eigenvalue as a continuation parameter. This has been used successfully for both pulse wa
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