In the fourth part of this paper devoted to the problem of the thermal self-focusing of an electromagnetic wave in an underdense plasma, we examine the spatial nonlinear development of the instability, assuming that a stationary state of the type described earlier has been established. Our treatment considers the interaction of the pump wave with the Stokes side-band waves and the density perturbation corresponding to the most unstable linear mode. We show that the system of equations describing their interaction possesses three invariants. These invariants are used to reduce the problem to a single differential equation for the axial behaviour of the amplitude of the density perturbation. The form of the invariants differs according as the inclination of the beam to the magnetic field is above or below the critical value defined earlier. Above the critical orientation, the amplitude of the density perturbation can be expressed algebraically in terms of the amplitudes of the pump and side-band waves. We show that this case is integrable in terms of the Jacobean elliptic function dn and we use the results to find the saturation level. Below the critical orientation, the response of the plasma becomes non-local, above a certain threshold power. The algebraic relation is then replaced by a differential equation which results in a significantly more complicated differential equation for the amplitude of the density perturbation. A particular solution of this equation is found for a special choice of initial conditions and from this the saturation level is found. The results of these calculations are applied to the problem of ionospheric modification by microwave transmissions from the proposed solar power satellite systems.
