  Turing patterns on unbounded domains have been widely studied in systems of
reaction-diffusion equations. However, up to now, they have not been studied
for systems of conservation laws. Here, we (i) derive conditions for Turing
instability in conservation laws and (ii) use these conditions to find families
of periodic solutions bifurcating from uniform states, numerically continuing
these families into the large-amplitude regime. For the examples studied,
numerical stability analysis suggests that stable periodic waves can emerge
either from supercritical Turing bifurcations or, via secondary bifurcation as
amplitude is increased, from sub-critical Turing bifurcations. This answers in
the affirmative a question of Oh-Zumbrun whether stable periodic solutions of
conservation laws can occur. Determination of a full small-amplitude stability
diagram-- specifically, determination of rigorous Eckhaus-type stability
conditions-- remains an interesting open problem.
