  The longitudinal dynamics of a resonantly driven Langmuir wave are analyzed
in the limit that the growth of the electrostatic wave is slow compared to the
bounce frequency. Using simple physical arguments, the nonlinear distribution
function is shown to be nearly gaussian in the canonical particle action, with
a slowly evolving mean and fixed variance. Self-consistency with the
electrostatic potential provide the basic properties of the nonlinear
distribution function including a frequency shift that agrees well with driven,
electrostatic particle simulations. This extends earlier work on nonlinear
Langmuir waves by Morales and O'Neil [G. J. Morales and T. M. O'Neil, Phys.
Rev. Lett. 28, 417 (1972)], and could form the basis of a reduced kinetic
treatment of Raman backscatter in a plasma.
