Numerical continuation methods for studying periodic travelling wave (wavetrain) solutions of partial differential equations. (English) Zbl 1244.65155
Summary: Periodic travelling waves (wavetrains) are an important solution type for many partial differential equations. In this paper I review the use of numerical continuation for studying these solutions. I discuss the calculation of the form and stability of a given periodic travelling wave, and the calculation of boundaries in a two-dimensional parameter plane for wave existence and stability. I also describe the automated implementation of these numerical continuation procedures via the software package wavetrain (http://www.ma.hw.ac.uk/wavetrain). I conclude by discussing ongoing work on numerical continuation methods for determining the absolute stability of periodic travelling waves.
MSC:
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 35G20 | Nonlinear higher-order PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
numerical continuation; periodic travelling wave; wavetrain; numerical examples; stability; software package wavetrainReferences:
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