A reaction dispersion system and Raman interactions

We consider the problem of amplification of an optical signal wave with an optical pump wave when both are propagating in the fundamental mode of a single mode optical waveguide. We introduce a system of Ginzburg-Landau type and study the radiation loss due to the nonlinear interaction between the signal and the pump waves. The linear dynamics are dispersive, while nonlinearity governs the transfer of energy from the pump wave to the signal wave. The strength of the effect is shown to depend on a dimensionless parameter, which is given by the ratio of the diffraction length and amplification length. If this parameter is small, then the radiation loss is small. This result is established by (i) verifying the absence of resonant terms that can potentially drive the growth of radiative components and (ii) then by estimating the oscillatory (nonresonant) terms by proving the relevant PDE a priori estimates. These estimates require appropriate bounds on the solutions of the PDE, whose only conserved integral is the L ² norm. However, the special structure of the nonlinear term, dictated by the physics of the Raman effect, implies a weak space-time bound involving the signal and pump intensities. This bound and L ² conservation are used together with Strichartz (space-time) estimates for the Schrödinger equation to obtain control of stronger classical norms of the signal and pump fields.