Derivation and Analysis of Amplitude Equation for Generalized AMB+ in Presence of Chemical Reaction

Abstract: We derive and analyze the amplitude equation for the roll patterns in case of generalized Active Model B+ (AMB+) in the presence of chemical reactions. The generalized AMB+ differs from the original AMB+ introduced by Tjhung \textit{et al.} [E. Tjhung \textit{et al.}, Phys. Rev. X \textbf{8}, 031080 (2018)] by the addition of a quadratic term, $gφ^2$, in the expression for the equilibrium part of the current. Also, the model includes a rotation-free active current of strength $λ$ and a rotational current of strength $ξ$. The inclusion of a chemical reaction with rate $Γ$ removes the conservation constraint and introduces a preferred wavenumber that governs the pattern formation below a critical reaction rate $Γ_c$. We argue for the analytical form of the amplitude equation based on symmetry considerations and explicitly derived it using multiscale analysis. By taking different limits of $g$, $λ$, and $ξ$, we recover amplitude equations for several well-known physical models as special cases and determine the nature transitions close to the onset of instability. We find that for $g = 0$, the transition is always supercritical, whereas for $g \ne 0$, the transition between the supercritical and subcritical regimes depends sensitively on the model parameters. Further, we derive the condition for the \textit{Eckhaus instability} from the stability analysis of the amplitude equation as well as from the phase diffusion equation, and find that it is independent of $g$.