Spatiotemporal chaos in the nonlinear three wave interaction

The nonlinear three wave interaction (3WI) has applications to plasma physics, non-linear optics, and hydrodynamics. We consider one form of these equations, which describes the saturation of a linearly unstable parent wave by nonlinearly coupling to two damped daughter waves. The system exhibits spatiotemporal chaos (STC) in the long time, large system limit. The term STC specifically refers to the chaotic dynamics of coherent structures or spatial patterns. This is contrasted with fully developed turbulence where there is a cascade to small scales and with low dimensional chaos where spatial degrees of freedom are not involved. The conservative form of the 3WI is integrable by inverse scattering transforms (1ST) and has soliton solutions. We use perturbation theory about those solutions to gain some understanding of the behaviour