Schrödinger equation driven by the square of a Gaussian field: instanton analysis in the large amplification limit

We study the tail of p(U), the probability distribution of U = | ψ ( 0 , L ) | 2 , for ln U > > 1 , ψ ( x , z ) being the solution to ∂ z ψ − i 2 m ∇ ⊥ 2 ψ = g | S | 2 ψ , where S ( x , z ) is a complex Gaussian random field, z and x respectively are the axial and transverse coordinates, with 0 ⩽ z ⩽ L , and both m ≠ 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 → + ∞ . 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 Commun. Math. Phys. 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.