| Abstract: |
| In pattern-forming systems, an unstable `trivial' state is often first invaded by stripe patterns, which are themselves unstable to hexagonal patterns that emerge through a secondary (slower) invasion front. Such cascading invasions naturally give rise to a non-steady (i.e., the fronts travel at different speeds) two-front superposition in which the connecting state is unstable.
In this talk, I present recent results on the linear convective stability of such a superposition of two travelling waves with unstable connecting state in a reaction-diffusion system. We find that convective stability holds for a strictly smaller range of propagation speeds than for the corresponding single-front waves, reflecting the long-range influence the fronts exert on one another. Since the superposition is time-dependent, classical techniques are not applicable to analyse this phenomenon. We therefore rely on numerical range estimates that imply time-uniform resolvent bounds. I will also discuss open challenges in the nonlinear analysis.
This is joint work with Louis Gar\`{e}naux (INRIA Saclay). |
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