Testing the instanton approach to the large amplification limit of a diffraction–amplification problem.
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| Titel: | Testing the instanton approach to the large amplification limit of a diffraction–amplification problem. |
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| Autoren: | Mounaix, Philippe1 (AUTHOR) philippe.mounaix@polytechnique.edu |
| Quelle: | Journal of Physics A: Mathematical & Theoretical. 12/4/2024, Vol. 57 Issue 48, p1-18. 18p. |
| Schlagwörter: | *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] |
| Datenbank: | Academic Search Index |
Nájsť tento článok vo Web of Science | 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] |
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| ISSN: | 17518113 |
| DOI: | 10.1088/1751-8121/ad8f08 |