Article

Advances in Computational Mathematics

, Volume 39, Issue 1, pp 175-192

Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain) solutions of partial differential equations

  • Jonathan A. SherrattAffiliated withDepartment of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University Email author 

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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