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

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 $\mathit{\text{et}}\mathit{\text{al.}}$ [E. Tjhung $\mathit{\text{et}}\mathit{\text{al.}}$, Phys. Rev. X $\mathbf{\text{8}}$, 031080 (2018)] by the addition of a quadratic term, $g{\phi }^2$, in the expression for the equilibrium part of the current. Also, the model includes a rotation-free active current of strength $\lambda$ and a rotational current of strength $\xi$. The inclusion of a chemical reaction with rate $\mathrm{\Gamma }$ removes the conservation constraint and introduces a preferred wavenumber that governs the pattern formation below a critical reaction rate ${\mathrm{\Gamma }}_c$. 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 $g$, $\lambda$, and $\xi$, 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 $g\mathrm{\ne }0$, the transition between the supercritical and subcritical regimes depends sensitively on the model parameters. Further, we derive the condition for the $\mathit{\text{Eckhaus}}\mathit{\text{instability}}$ from the stability analysis of the amplitude equation as well as from the phase diffusion equation, and find that it is independent of $g$.