On the potential lack of response in a model of second-harmonic generation. A computer-assisted proof
Abstract
This paper provides a rigorous computer-assisted proof of the existence of generalized transmission eigenvalues arising in nonlinear optics in the context of high-order harmonic generation, a result conjectured in Cakoni, Hovsepyan, Lassas, and Vogelius, SIAM J. Math. Anal. 57 (2025), 1370-1405. The analysis is carried out for a one-dimensional nonlinear medium, where the problem reduces to a coupled system of nonlinear homogeneous ordinary differential equations subject to nonstandard boundary conditions. These eigenvalues correspond to probing frequencies ω for which there exists a nontrivial incident ω-wave such that the second-harmonic field generated does not persist outside the compact support of the nonlinear medium, thereby rendering its nonlinear properties undetectable to an external observer. Building on earlier numerical evidence, we establish that, in the low-frequency regime, there exist generalized transmission eigenvalues whose associated eigenfunctions exhibit blow-up behavior as the frequency tends to zero. The proof combines analytical arguments with validated numerics, employing a Newton-Kantorovich framework together with interval arithmetic to rigorously control approximation errors. The algorithmic implementation underlying the computer-assisted proof is made available in the GitHub repository TransmissionEigenvalues.jl by Dominic Blanco (2026).
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BibTeX
@misc{ayala2026potential,
title = {{On the potential lack of response in a model of second-harmonic generation. A computer-assisted proof}},
author = {Miguel Ayala and Dominic Blanco and Fioralba Cakoni and Narek Hovsepyan and Michael S. Vogelius},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23217},
eprint = {2609.23217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23217},
}
RIS
TY - UNPB
TI - On the potential lack of response in a model of second-harmonic generation. A computer-assisted proof
AU - Miguel Ayala
AU - Dominic Blanco
AU - Fioralba Cakoni
AU - Narek Hovsepyan
AU - Michael S. Vogelius
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23217
UR - https://arxiv.org/abs/2609.23217
AB - This paper provides a rigorous computer-assisted proof of the existence of generalized transmission eigenvalues arising in nonlinear optics in the context of high-order harmonic generation, a result conjectured in Cakoni, Hovsepyan, Lassas, and Vogelius, SIAM J. Math. Anal. 57 (2025), 1370-1405. The analysis is carried out for a one-dimensional nonlinear medium, where the problem reduces to a coupled system of nonlinear homogeneous ordinary differential equations subject to nonstandard boundary conditions. These eigenvalues correspond to probing frequencies ω for which there exists a nontrivial incident ω-wave such that the second-harmonic field generated does not persist outside the compact support of the nonlinear medium, thereby rendering its nonlinear properties undetectable to an external observer. Building on earlier numerical evidence, we establish that, in the low-frequency regime, there exist generalized transmission eigenvalues whose associated eigenfunctions exhibit blow-up behavior as the frequency tends to zero. The proof combines analytical arguments with validated numerics, employing a Newton-Kantorovich framework together with interval arithmetic to rigorously control approximation errors. The algorithmic implementation underlying the computer-assisted proof is made available in the GitHub repository TransmissionEigenvalues.jl by Dominic Blanco (2026).
ER -