Numerical continuation of boundaries in parameter space between stable and unstable periodic travelling wave (wavetrain) solutions of partial differential equations. (English) Zbl 1271.65132
Summary: 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, the 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.
MSC:
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 35G61 | Initial-boundary value problems for systems of nonlinear higher-order PDEs |
| 35B35 | Stability in context of PDEs |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35K57 | Reaction-diffusion equations |
Keywords:
numerical continuation; periodic traveling wave; spectral stability; numerical examples; algorithm; Klausmeier model; banded vegetation in semi-arid environments; Eckhaus (sideband) type; Oregonator model; photosensitive Belousov-Zhabotinskii reaction; Hopf typeReferences:
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