We derive and analyze the amplitude equation for roll patterns in the generalized Active Model B (AMB) with chemical reactions in $d=1$. We start from the generalized $\mathrm{AMB}+$, which differs from the original $\mathrm{AMB}+$ introduced by Tjhung et al. [Phys. Rev. X 8, 031080 (2018)] through the inclusion of an additional quadratic term $g{\varphi }^2$ in the equilibrium part of the current. The model incorporates a rotation-free active current with strength $\lambda$ and a rotational current with strength $\xi$. In $d=1$, generalized $\mathrm{AMB}+$ reduces to generalized AMB with an effective rotation-free active current of strength ${\lambda }_{\mathrm{eff}}=\lambda -\frac{\xi }{2}$, while the rotational current is absent. The inclusion of a chemical reaction with rate $\mathrm{\Gamma }$ removes the conservation constraint and introduces a preferred wave number 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 derive 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 of 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. Furthermore, we derive the condition for the 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 $g$.

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• Bifurcations
• Pattern formation
• Living matter & active matter
• Bifurcation analysis