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 ${\partial }_t\mathcal{E}-\mathcal{D}\Delta \mathcal{E}=\lambda S^2\mathcal{E}$, where $S(x,t)$ is a Gaussian stochastic field with covariance $C(x-x^{\prime },t,t^{\prime })$, and $x\in {\mathbb{R}}^d$. It is shown that the coupling ${\lambda }_{cN}(t)$ at which the $N$-th moment $<{\mathcal{E}}^N(x,t)>$ 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, ${\lambda }_{cN}(t)=N{\lambda }_c(t)$ where ${\lambda }_c(t)$ is the value of $\lambda$ at which $<\exp [\lambda {\int }_0^t{S}^2(0,s)ds]>$ diverges. The $\mathcal{D}=0$ case is solved for a class of $S$. The dependence of ${\lambda }_{cN}(t)$ on $d$ is analyzed. Similar behavior is conjectured when diffusion is replaced by diffraction, $\mathcal{D}\to i\mathcal{D}$, the case of interest for backscattering instabilities in laser-plasma interaction.

19 pages, in LaTeX, e-mail addresses: [email protected], [email protected], [email protected], [email protected]

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