Note on a diffraction-amplification problem

TL;DR

This paper analyzes a diffraction-amplification equation with stochastic input, showing how the coupling threshold for divergence depends on the diffusion coefficient, with implications for wave amplification in random media.

Contribution

It provides a mathematical analysis of the diffraction-amplification problem with stochastic fields, revealing the relationship between diffusion and amplification thresholds.

Findings

Coupling threshold decreases with increasing diffusion D.

Divergence of <|E|> occurs at lower coupling for D>0.

Threshold coupling  is always less or equal for D>0 than D=0.

Abstract

We investigate the solution of the equation \partial_t E(x,t)-iD\partial_x^2 E(x,t)= \lambda |S(x,t)|^2 E(x,t)$, for x in a circle and S(x,t) a Gaussian stochastic field with a covariance of a particular form. It is shown that the coupling \lambda_c at which <|E|> diverges for t>=1 (in suitable units), is always less or equal for D>0 than D=0.