Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium

The authors describe the evolution in time and one spatial dimension (steady-state in two spatial dimensions) of coherent three-wave interactions in a lossless homogeneous medium. They obtain their results by a complementary use of inverse scattering methods and numerical integration of the equations. The nonlinear saturation by pump depletion of instabilities which are absolute in the pump reference frame is found to involve only soliton transfer, thus allowing a complete solution in terms of an $N$-soliton formula. Threshold formulas for explosive instabilities, and conditions for optimal energy transfer in two-pump interactions, are presented. Nonlinear collisions of wave packets are found to give rise to upconverted packets which are sharply spiked. A nonlinear modulation is described in stimulated backscatter.