Research Project:
A steady or traveling wave is a solution to a time-dependent problem translates at a fixed velocity without altering its shape. Examples include surface and internal waves in the ocean, ignition fronts in combustion theory, and even stripe patterns in animal fur. More generally, one can find special solutions that evolve not by translation, but according to the action of more complicated symmetry groups. 
A number of tools now exist for constructing waves in a neighborhood of a known explicit solution, but often the most interesting solutions — both mathematically and physically — lie outside the perturbative regime. Classical global bifurcation tools for constructing these "large" solutions rely strongly on compactness properties that typically fail in the case of unbounded domains.

At ESI, the team will develop new global-bifurcation-theoretic techniques adapted to special solutions evolving via non-compact symmetry groups, and apply this theory to open problems in a variety of physical contexts.

Project Team: Robin Ming Chen (U of Pittsburgh), Samuel Walsh (U of Missouri), Miles H. Wheeler (U Vienna)