On the nonlinear development of the filamentation of an electromagnetic wave in a plasma

The problem of the filamentation of an electromagnetic wave in a plasma has been formulated in terms of the nonlinear interaction between the initial plane electromagnetic wave, the Stokes, and anti-Stokes electromagnetic waves and a density perturbation. Since a perturbation procedure is used the analysis is limited to situations where the total pump wave energy density is a small fraction of the energy density in the undisturbed plasma. A crucial ingredient of the problem is the frequency mis-match between the interacting waves although only those waves are included which satisfy perfect k-matching. The pump wave is treated on the same footing as the other electromagnetic waves and these waves are treated as distinct throughout the interaction. When the pump amplitude is assumed to remain constant the equations yield the linear threshold and growth rate. By neglecting ion inertia, three coupled nonlinear equations (with cubic nonlinearities) are obtained for the initial, Stokes and anti-Stokes electromagnetic waves. These equations have been solved analytically under various assumptions