 Quintic real Ginzburg-Landau equation

A cubic destabilizing term and quintic saturation describe a subcritical stationary pattern bifurcation. A plane wave $A=R{e}^{iQX}$ has squared amplitude $x=R^2$ satisfying $\mu -Q^2+\alpha x-x^2=0$. The nonzero branches meet at $x=\alpha /2$, $\mu =Q^2-{\alpha }^2/4$. The lower branch is unstable to amplitude perturbations; the upper branch can also undergo an Eckhaus instability of its phase.