  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 the
equivalent in the conserved case of the complex Ginzburg-Landau equation
obtained in the nonconserved case as amplitude equation for the standard Hopf
bifurcation.
  Considering first the case of a relatively simple symmetric Cahn-Hilliard
model with purely nonreciprocal coupling, we derive the nonlinear nonlocal
amplitude equation and show that its bifurcation diagram and time evolution
well agree with results for the full model. The solutions of the amplitude
equation and their stability are obtained analytically thereby showing that in
oscillatory phase separation the suppression of coarsening is universal.
Second, we lift the restrictions and obtain the amplitude equation in a more
generic case, that also shows very good agreement with the full model as
exemplified for some transient dynamics that converges to traveling wave
states.
