A class of pattern{forming models

A general class of nonlinear evolution equations is described, which support stable spatially oscillatory steady solutions. These equations are composed of an indefinite self-adjoint linear operator acting on the solution plus a nonlinear function, a typical example of the latter being a double-well potential. Thus a Lyapunov functional exists. The linear operator contains a parameter ρ which could be interpreted as a measure of the pattern-forming tendency for the equation. Examples in this class of equations are an integrodifferential equation studied by Goldstein, Muraki and Petrich and others in an activator-inhibitor context, and a class of fourth order parabolic PDE’s appearing in the literature in various physical connections and investigated mathematically by Coleman, Leizarowitz, Marcus, Mizel, Peletier, and Troy. The former example reduces to the real Ginzburg-Landau equation when ρ = 0. The most complete results, including threshold results for the appearance of globally minimizing patterns and many other properties of the patterns themselves, are given for complex-valued solutions in one space variable. A complete linear stability analysis for all such sinusoidal solutions is also given; it extends the set of stable solutions considerably beyond the global minimizers.