An amplitude equation for the conserved-Hopf bifurcation—Derivation, analysis, and assessment

We employ weakly nonlinear theory to derive an amplitude equation for the conserved-Hopf instability, i.e., a generic large-scale oscillatory instability for systems with two conservation laws. The resulting equation represents in the conserved case the equivalent of the complex Ginzburg-Landau equation obtained in the nonconserved case as an amplitude equation for the standard Hopf bifurcation. Considering first the case of a relatively simple symmetric two-component Cahn-Hilliard model with purely nonreciprocal coupling, we derive the nonlinear nonlocal amplitude equation with real coefficients and show that its bifurcation diagram and time evolution well agree with the results for the full model. The solutions of the amplitude equation and their stability are analytically obtained, thereby showing that in such oscillatory phase separation, the suppression of coarsening is universal. Second, we lift the two restrictions and obtain the amplitude equation in the generic case. It has complex coefficients and also shows very good agreement with the full model as exemplified for some transient dynamics that converges to traveling wave states.