Weakly collisional steady state linear and nonlinear plasma waves

Landau’s collisionless result for a weakly damped plasma wave is precisely recovered in a weakly collisional, steady state plasma by treating the physics of the narrow collisional boundary layer associated with the resonant electrons. To recover Landau’s results, the collision frequency must be large enough that islands are unable to form and/or the wave amplitude must be small enough to allow linearization. However, the Landau treatment fails once the collision frequency becomes too weak and/or the wave amplitude too large. Remarkably, Landau’s weakly damped plasma wave results require collisions and are shown to be inappropriate in the collisionless limit for a nonlinear, finite amplitude, steady state wave!