User:Dwight Barkley/Proposed/Symmetry breaking in reaction-diffusion systems

• Dwight Barkley, Mathematics, University of Warwick, UK

Reaction-diffusion systems often possess symmetries. Specifically, the equations describing a reaction-diffusion system often are left unchanged by certain groups of transformations, such as reflection, translation or rotation. Symmetry breaking refers to the situation in which solutions to the equations have less symmetry than the equations themselves.

Figure 1: Two diffusively coupled stirred tank reactors form a simple left-right symmetric reaction-diffusion system. States of the system may be symmetric or asymmetric. The asymmetric states have less symmetry than the system and therefore are said to have broken symmetry.

For example, in the simplest case a reaction-diffusion system might be left-right symmetric as in Figure 1. States of the system may, however, fail to be left-right symmetric. In this example, the broken-symmetry states will come in pairs, related to one another by reflection.

Symmetric breaking is important in the study of reaction-diffusion equations because solutions often arise through symmetry-breaking bifurcations. As a result the behavior of solutions is strongly dictated by the nature of the symmetry breaking and is independent of details of the underlying reaction-diffusion system. These properties of symmetry breaking follow from considerations of equivariant bifurcation theory and equivariant dynamical systems

It is frequently observed that in systems with several symmetries, not all symmetries break at once as a control parameter of the system is varied. For example, in a system with both translation and reflection symmetry, often one observes that as a control parameter is varied translation symmetry breaks first. Further variation in the control parameter lead to breaking of reflection symmetry.

Reaction-diffusion Equations in Multiple Dimensions

Many examples possible. The most interesting perhaps in Euclidian symmetry breaking.