Oscillating two-stream instability of a plasma wave in a plasma channel

A long wavelength Langmuir wave (w(0), (k) over right arrow(0)), propagating through a parabolic plasma density channel, can decay into a low-frequency mode (w, (k) over right arrow) and two short wavelength Langmuir wave sidebands (w(1,2), (k) over right arrow(1,2)), via two-stream instability where w(1,2) = w -/+ w(0) and (k) over right arrow(1,2) = (k) over right arrow -/+ (k) over right arrow(0) . Depending on the mode number n, the growth rate maximizes in the range gamma(max) similar or equal to 0.1 w(pi) - 0.4w(pi) for the range of k from 0.1(w(pi)/c(s)) to 0.2(w(pi)/c(s)) for upsilon(0) similar to Vth where upsilon(0) and upsilon(th) are the oscillatory and thermal velocities of electrons, w(pi) is the ion plasma frequency on the axis, and c(s) is the sound speed, The growth rate increases with the width a of the plasma density channel, It decreases with the mode number, The instability may be relevant to laser based charged particle accelerators.