Geodesic
Arnold Lagrangian singularity in the asymptotics of the solution of a model two-dimensional Helmholtz equation with a localized right-hand side
Teoretičeskaâ i matematičeskaâ fizika, Tome 218 (2024) no. 1, pp. 23-47

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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.
Keywords: semiclassical asymptotics, canonical Maslov operator, Lagrangian surfaces. 
@article{TMF_2024_218_1_a1,
     author = {I. A. Bogaevsky and S. Yu. Dobrokhotov and A. A. Tolchennikov},
     title = {Arnold {Lagrangian} singularity in the asymptotics of the solution of a model two-dimensional {Helmholtz} equation with a localized right-hand side},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {23--47},
     publisher = {mathdoc},
     volume = {218},
     number = {1},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2024_218_1_a1/}
}
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I. A. Bogaevsky; S. Yu. Dobrokhotov; A. A. Tolchennikov. Arnold Lagrangian singularity in the asymptotics of the solution of a model two-dimensional Helmholtz equation with a localized right-hand side. Teoretičeskaâ i matematičeskaâ fizika, Tome 218 (2024) no. 1, pp. 23-47. http://geodesic.mathdoc.fr/item/TMF_2024_218_1_a1/