Model of Parametric Excitation by an Imperfect Pump

Abstract

We examine the three-wave decay instability due to a monochromatic pump with a phase which undergoes a random walk with diffusion coefficient $D$. Analytic results are obtained for small and large $\frac{D}{{\ensuremath{\gamma}}_{0}}$ (where ${\ensuremath{\gamma}}_{0}$ is the growth rate for $D=0$), which demonstrate, respectively, a fractional reduction of $\ensuremath{\gamma}$ by $\frac{D}{{\ensuremath{\gamma}}_{0}}$ and a growth rate $\frac{{{\ensuremath{\gamma}}_{0}}^{2}}{D}$. Results are also presented for intermediate values of $\frac{D}{{\ensuremath{\gamma}}_{0}}$.