View in EDS

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

Saved in:
Bibliographic Details
Title: Testing the instanton approach to the large amplification limit of a diffraction–amplification problem.
Authors: Mounaix, Philippe
Source: Journal of Physics A: Mathematical & Theoretical; 12/4/2024, Vol. 57 Issue 48, p1-18, 18p
Subject Terms: STOCHASTIC partial differential equations, DISTRIBUTION (Probability theory), STOCHASTIC analysis, SAMPLING (Process), RANDOM fields
Abstract: 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. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Physics A: Mathematical & Theoretical is the property of IOP Publishing and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Complementary Index
Description
Abstract: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<sup>−2270</sup> 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. [ABSTRACT FROM AUTHOR]
ISSN:17518113
DOI:10.1088/1751-8121/ad8f08