Testing the instanton approach to the large amplification limit of a diffraction-amplification problem

The validity of the instanton analysis approach is tested numerically in the case of the diffraction-amplification problem ∂ z ψ − i 2 m ∂ x 2 2 ψ = g | S | 2 ψ for ln ⁡ U ≫ 1 , where U = | ψ ( 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 ⩽ z ⩽ L , and both m ≠ 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 J. Phys. A: Math. Theor. 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.