Asymptotic treatment of the three-wave nonlinear parametric interaction

A novel integral representation suitable for the study of asymptotic (L/L₀ ≫ 1, L being the plasma length and L₀ the basic gain length) stability of the inhomogeneous three-wave interaction is formulated and applied to the special case of uniform gain and ramp wavenumber mismatch. The absolute instability condition is κ L₀ < π /8 (κ being the mismatch gradient), with the wave conversion performed in a zone of width 4/κ L₀ located at the boundary of wave incidence.