Swift-Hohenberg roll-amplitude saturation: A^2_saturation = 4r/3 (Lyapunov functional minimum)

Swift-Hohenberg roll-amplitude saturation framework. Setup: Swift-Hohenberg equation u_t = r u - (1 + d_xx)^2 u - u^3 (Swift-Hohenberg 1977 PRA 15, 319) is the simplest PDE exhibiting spatially periodic roll patterns near onset. Take u(x,…