  We consider the simplest instabilities involving multiple unstable
electrostatic plasma waves corresponding to four-dimensional systems of mode
amplitude equations. In each case the coupled amplitude equations are derived
up to third order terms. The nonlinear coefficients are singular in the limit
in which the linear growth rates vanish together. These singularities are
analyzed using techniques developed in previous studies of a single unstable
wave. In addition to the singularities familiar from the one mode problem,
there are new singularities in coefficients coupling the modes. The new
singularities are most severe when the two waves have the same linear phase
velocity and satisfy the spatial resonance condition $k_2=2k_1$. As a result
the short wave mode saturates at a dramatically smaller amplitude than that
predicted for the weak growth rate regime on the basis of single mode theory.
In contrast the long wave mode retains the single mode scaling. If these
resonance conditions are not satisfied both modes retain their single mode
scaling and saturate at comparable amplitudes.
