Testing the Instanton Approach to the Large Amplification Limit of a Diffraction-Amplification Problem

Abstract

The validity of the instanton analysis approach is tested numerically in the case of the diffraction-amplification problem ${\partial }_z\psi -\frac{i}{2m}{\partial }_{x^2}^2\psi =g|S{|}^2\psi$ for $\ln U\gg 1$, where $U=|\psi (0,L){|}^2$. Here, $S(x,z)$ is a complex Gaussian random field, $z$ and $x$ respectively are the axial and transverse coordinates, with $0\le z\le L$, and both $m\ne 0$ and $g>0$ are real parameters. We consider a class of $S$, called the `one-max class', for which we devise a specific biased sampling procedure. As an application, $p(U)$, the probability distribution of $U$, is obtained down to values less than $10^{-2270}$ in the far right tail. We find that the agreement of our numerical results with the instanton analysis predictions in Mounaix (2023 {\it J. Phys. A: Math. Theor.} {\bf 56} 305001) is remarkable. Both the predicted algebraic tail of $p(U)$ and concentration of the realizations of $S$ onto the leading instanton are clearly confirmed, which validates the instanton analysis numerically in the large $\ln U$ limit for $S$ in the one-max class.

23 pages, 9 figures, submitted to J. Phys. A: Math. Theor

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