Wavelength selection for periodic travelling waves: an unsolved problem
by Lukas Eigentler and Mattia Sensi.
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Many models for spatio-temporal patterns admit periodic travelling wave solutions (PTWs) and predict that PTWs can undergo wavelength changes as model parameters are varied; for example leading to ecological theory that dryland vegetation patterns can adaptively change their wavelength to adapt to climate change. While PTW destabilisation is well-understood thanks to Busse balloon theory, we lack understanding of what determines wavelength selection during PTW-to-PTW transitions. In this unsolved problems article, we review the state of the art in predicting PTW destabilisation and knowledge of wavelength selection principles in special cases. We present new numerical data, hoping to stimulate new research into this open problem.
Busse balloon for a model of dryland vegetation patterns showing jumps between wavelength contours.
