The derivation and application of convective pattern equations

In this report, we review an elegant technique, known as the Bogoliubov method, for deriving amplitude equations in pattern forming systems, through detailed solution to the classical problem of Rayleigh-Bénard convection, with both free and rigid upper and lower boundaries. The computation is facilitated by the use of a newly proposed diagrammatic technique. The resulting equation is a nonlocal pattern equation, which reduces to the 1D Swift-Hohenberg equation for 2D convection. We show that the nonlocal pattern equation is variational by finding a Lyapunov functional. Subsequently, we formulate a few properties of steady state solutions of general variational PDE, and test their utility in pattern prediction for models including the Swift-Hohenberg equation. Finally, we describe normal form theory for Hopf bifurcation in the context of pattern equations, and point out possible extensions of the diagrammatic technique.