Statistical theory of cubic Langmuir turbulence

The cubic direct interaction approximation is applied to a truncated (in Fourier space) version of the cubically nonlinear Schroedinger equation model of Langmuir physics. The results are compared (in the three-mode case) to those for an ensemble of numerical solutions of the dynamical equations with 10,000 different sets of Gaussianly distributed initial conditions. In the undriven, undamped case, the statistical theory (but not the ensemble) evolves to a state of thermal equilibrium. In the driven, damped case, the statistical theory appears to evolve to a state close to that corresponding to one of the limit cycles of the dynamical equations.