A diagrammatic derivation of (convective) pattern equations

In complicated bifurcation problems where more than one instability can arise at onset, reasonably sound derivations of the equations that govern the amplitudes of the nearly marginal modes have been developed when the spectrum of the modes is discrete. The basis of these derivations lies in the center manifold theorem of dynamical systems theory. But when the spectrum of the modes is continuous and we no longer have that theorem to fall back on, there is nevertheless an equation (the Swift–Hohenberg equation) that well describes the patterns seen in Rayleigh–Bénard convection. Indeed, several ‘derivations’ of the S–H equation have been offered and here we describe how to obtain the S–H equation using Bogoliubov’s method. We suggest that this procedure clarifies and simplifies (though it does not make rigorous) the derivation of the S–H equation. Looking ahead to the derivation of pattern equations for more complicated problems with continuous spectra, we also describe a diagrammatic procedure that, once mastered, is useful in performing the complicated perturbative developments that are needed in such derivations. Here we illustrate the proposed combination of the ideas of Bogoliubov and Feynman for the standard form of the Rayleigh–Bénard convection problem. The resulting pattern equation is nonlocal but it reduces without approximation to the 1-D Swift–Hohenberg equation in the case of 2-D convection. Like the S–H equation, the nonlocal version admits a Lyapunov functional and we briefly indicate its utility in pattern selection both for the Swift–Hohenberg equation and its nonlocal extension. We conclude by describing the kinds of problems for which we intend the combined method but reserve the exhibition of the required calculations for a future festschrift.