Schrödinger Equation Driven by the Square of a Gaussian Field: Instanton Analysis in the Large Amplification Limit

Abstract

We study the tail of $p(U)$, the probability distribution of $U=|\psi (0,L){|}^2$, for $\ln U\gg 1$, $\psi (x,z)$ being the solution to ${\partial }_z\psi -\frac{i}{2m}{\nabla }_{\perp }^2\psi =g|S{|}^2\psi$, where $S(x,z)$ 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 perform the first instanton analysis of the corresponding Martin-Siggia-Rose action, from which it is found that the realizations of $S$ concentrate onto long filamentary instantons, as $\ln U\to +\infty$. The tail of $p(U)$ is deduced from the statistics of the instantons. The value of $g$ above which $⟨U⟩$ diverges coincides with the one obtained by the completely different approach developed in Mounaix et al. 2006 {\it Commun. Math. Phys.} {\bf 264}~741. Numerical simulations clearly show a statistical bias of $S$ towards the instanton for the largest sampled values of $\ln U$. The high maxima -- or `hot spots' -- of $|S(x,z){|}^2$ for the biased realizations of $S$ tend to cluster in the instanton region.

arXiv comment41 pages, 10 figures, submitted to J. Phys. A: Math. Theor

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Cited by in corpus (2)

• Testing the Instanton Approach to the Large Amplification Limit of a Diffraction-Amplification Problem
• From the Rose-DuBois Ansatz of Hot Spot Fields to the Instanton Solution: a Pedestrian Presentation