Predicting pattern formation in coupled reaction-diffusion systems

Coupled reaction-diffusion equations are known to exhibit a wealth of multiple coexisting stationary solution patterns as the characteristic length of the system grows. We describe and implement a technique which allows us, by studying only stationary solution branches at small lengths, to predict the complex structure of steady-state bifurcations occuring at large system lengths without actually computing this structure. This technique is applicable to arbitrary isothermal or nonisothermal systems of coupled reaction-diffusion equations or no-flux boundary conditions. To illustrate it we use a standard literature example (the Brussellator), where, by computing only the first two solution branches, we accurately predict the steady-state bifurcation structure reported up to much larger system lengths. This technique also provides a compact way of describing and comparing stationary pattern formation in a large class of systems, extending beyond coupled reaction-diffusion equations. We demonstrate this by comparing stationary pattern formation for our test problem (the Brussellator), with the formation of complex surface wave patterns in thin liquid film flow.