Langmuir wave self-focusing versus decay instability
  Electron trapping in a finite amplitude Langmuir wave (LW) leads to a frequency shift, \Delta\omega_{TP} < 0, and reduced Landau damping. These may lead to modulational instability. Its growth rate and damping threshold, due to escape of trapped electrons at rate \nu, are calculated for the first time in the short wavelength regime. If the background plasma is in thermal equilibrium, it is shown that this trapped particle modulational instability (TPMI) is not possible when k \lambda_D > 0.46, while for 0.33 < k \lambda_D < 0.46, TPMI requires that the fluctuation wavevector have a component perpendicular to k, the LW wavevector, with \lambda_D the electron Debye length. Its nonlinear evolution leads to self-focusing. Comparison is made with a re-evaluated LW ion acoustic decay instability (LDI): compared to classical estimates, the new LDI threshold is lowered by primary LW \Delta\omega_{TP} since frequency matching leads to wavenumber and hence damping reduction of the daughter LW. For parameters estimates relevant to a recent stimulated Raman scatter experiment (Kline et al., submitted to PRL), the LDI and TPMI thresholds cross in the range 0.28 < k \lambda_D < 0.34, consistent with the observed LDI regime change. However, if \nu exceeds a critical value, estimated to be order 1% of the electron plasma frequency, then TPMI is not possible at any wavenumber. 
Harvey. A. Rose