  The nonlinear evolution of a unstable electrostatic wave is considered for a
multi-species Vlasov plasma. From the singularity structure of the associated
amplitude expansions, the asymptotic features of the electric field and
distribution functions are determined in the limit of weak instability, i.e.
$\gamma\to 0^+$ where $\gamma$ is the linear growth rate. The asymptotic
electric field is monochromatic at the wavelength of the linear mode with a
nonlinear time dependence. The structure of the distibutions outside the
resonant region is given by the linear eigenfunction but in the resonant region
the distribution is nonlinear. The details depend on whether the ions are fixed
or mobile; in either case the physical picture corresponds to the single wave
model originally proposed by O"Neil, Winfrey, and Malmberg for the interaction
of a cold weak beam with a plasma of fixed ions.
