Diffusion Effects on the Breakdown of a Linear Amplifier Model Driven by the Square of a Gaussian Field

Abstract

We investigate solutions to the equation ∂ _t ℰ−$\mathcal{D}$ Δℰ=λS ²ℰ, where S(x, t) is a Gaussian stochastic field with covariance C(x−x′, t, t′), and x∈$\mathbb{R}$ ^d. It is shown that the coupling λ _cN (t) at which the N-th moment <ℰ^N(x, t)> diverges at time t, is always less or equal for $\mathcal{D}$>0 than for $\mathcal{D}$=0. Equality holds under some reasonable assumptions on C and, in this case, λ _cN (t)=Nλ _c (t) where λ _c (t) is the value of λ at which <exp[λ ∫^t ₀ S ²(0, s) ds]> diverges. The $\mathcal{D}$=0 case is solved for a class of S. The dependence of λ _cN (t) on d is analyzed. Similar behavior is conjectured when diffusion is replaced by diffraction, $\mathcal{D}$→i $\mathcal{D}$, the case of interest for backscattering instabilities in laser-plasma interaction.