Speed of travelling waves in spatially quasi-periodic media

We consider travelling waves in spatially quasi-periodic media. We first define the notion of travelling waves and discuss their uniquness and stability properties in the case of bistable nonlinearities. Much of the discussions go in parallel to the case of periodic spatial inhomogeneity, but in the quasi-periodic case, some difficulties related to the small divisor problem also occur. We then study travelling waves in a periodically or quasi-periodically racheted cylinder. Our goal is to estimate their avarage speed near the homogenization limit. Surprisingly, the limit speed of the travelling wave depends only on the maximal opening angle of the rachet, despite its complex behavior near the boundary.