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

## TL;DR

This paper extends the study of Turing patterns to conservation laws, deriving instability conditions, identifying bifurcating periodic solutions, and numerically analyzing their stability, thus confirming the existence of stable periodic waves in such systems.

## Contribution

It introduces the first conditions for Turing instability in conservation laws and demonstrates the emergence and stability of periodic solutions through numerical continuation.

## Key findings

- Stable periodic waves can arise from supercritical Turing bifurcations.
- Secondary bifurcations can produce stable waves from sub-critical bifurcations.
- Numerical analysis suggests stable periodic solutions are possible in conservation laws.

## Abstract

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.