Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves

• Published in: Physica. D 2018-03, Vol.367, p.11-18
• Main Authors: Barker, Blake, Jung, Soyeun, Zumbrun, Kevin
• Format: Article
• Language: English
• Subjects: Conservation laws; Stable periodic waves; Turing instability
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2017.12.003

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 subcritical 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. •Conditions for Turing instability in conservation laws are derived.•There exist no Turing-type instabilities in conservation laws for 2×2 systems.•Stable periodic waves in conservation laws are numerically observed.