Abstract
A model Helmholtz equation with a localized right-hand side is considered. When writing asymptotics of a solution satisfying the limit absorption principle, a Lagrangian surface naturally appears that has a logarithmic singularity at one point. Because of this singularity, the solution is localized not only in a neighborhood of the projection of the Lagrangian surface onto the coordinate space but also in a neighborhood of a certain ray “escaping” from the Lagrangian surface and going into the region forbidden in the classical approximation.
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Acknowledgments
The authors thank V. E. Nazaikinskii and A. I. Shafarevich for the useful remarks.
Funding
This work is supported by the Russian Science Foundation (grant No. 21-11-00341).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2024, Vol. 218, pp. 23–47 https://doi.org/10.4213/tmf10553.
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Bogaevskii, I.A., Dobrokhotov, S.Y. & Tolchennikov, A.A. Arnold Lagrangian singularity in the asymptotics of the solution of a model two-dimensional Helmholtz equation with a localized right-hand side. Theor Math Phys 218, 19–40 (2024). https://doi.org/10.1134/S0040577924010021
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DOI: https://doi.org/10.1134/S0040577924010021