Coupled complex Ginzburg-Landau equations for the weak electrolyte model of electroconvection

The recently introduced weak electrolyte model (WEM) has successfully explained many linear properties of electroconvection in planarly aligned nematic liquid crystals such as the crossover from a stationary to a Hopf bifurcation in the parameter range observed experimentally. Here we present the first weakly nonlinear analysis of the WEM providing the coefficients of the complex Ginzburg-Landau equation. Whereas the Hopf bifurcation is always supercritical, the stationary bifurcation is for typical materials subcritical, which appears to be in agreement with experiments. In the oblique-roll range the (complex) cross coupling coefficient between the two degenerate roll systems (``zig'' and ``zag'') are also calculated leading to the superposition of traveling rectangles as observed in recent experiments on the material I52.