A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, M. Rouleux, “Lagrangian manifolds and the construction of asymptotics for (pseudo)differential equations with localized right-hand sides”, TMF, 214:1 (2023), 3–29; Theoret. and Math. Phys., 214:1 (2023), 1

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Teoreticheskaya i Matematicheskaya Fizika, 2023, Volume 214, Number 1, Pages 3–29
DOI: https://doi.org/10.4213/tmf10367 (Mi tmf10367)

This article is cited in 8 scientific papers (total in 8 papers)

Lagrangian manifolds and the construction of asymptotics for (pseudo)differential equations with localized right-hand sides

A. Yu. Anikin^a, S. Yu. Dobrokhotov^a, V. E. Nazaikinskii^a, M. Rouleux^b

^a Ishlinsky Institute for Problems in Mechanics of the RAS, Moscow, Russia
^b Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

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DOI: https://doi.org/10.4213/tmf10367

Abstract: We develop a method for constructing semiclassical asymptotic solutions of inhomogeneous partial differential and pseudodifferential equations with localized right-hand sides. These problems are related to the asymptotics of Green's function for this type of operators, in particular, for the Helmholtz equation, which has been studied in numerous papers. The method is based on a constructive description of the corresponding Lagrangian manifolds and on the recently proposed new representations of the Maslov canonical operator in a neighborhood of Lagrangian singularities (caustics and caustic sets). The method underlies an analytic-numerical algorithm for constructing efficient asymptotic solutions to problems of the above-mentioned type in various fields of physics and continuum mechanics.

Keywords: equation with right-hand side, Lagrangian manifold, semiclassical asymptotics, canonical operator.

Funding agency	Grant number
Russian Foundation for Basic Research	17-51-150006 НЦНИ_а
Ministry of Education and Science of the Russian Federation	АААА-А20-120011690131-7
The work was supported in part by the RFBR–CNRS (grant No. 17-51-150006) and by the Government program (contract no. AAAA-A20-120011690131-7).

Received: 13.09.2022
Revised: 13.09.2022

English version:
Theoretical and Mathematical Physics, 2023, Volume 214, Issue 1, Pages 1–23
DOI: https://doi.org/10.1134/S0040577923010014

Bibliographic databases:

Document Type: Article

MSC: 81Q20, 53D12

Language: Russian

Citation: A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, M. Rouleux, “Lagrangian manifolds and the construction of asymptotics for (pseudo)differential equations with localized right-hand sides”, TMF, 214:1 (2023), 3–29; Theoret. and Math. Phys., 214:1 (2023), 1–23

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https://doi.org/10.4213/tmf10367

https://www.mathnet.ru/eng/tmf/v214/i1/p3

This publication is cited in the following 8 articles:

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”, Theoret. and Math. Phys., 218:1 (2024), 19–40
I. A. Nosikov, A. A. Tolchennikov, M. V. Klimenko, “Boundary Value Problem of Calculating Ray Characteristics of Ocean Waves Reflected from Coastline”, Comput. Math. and Math. Phys., 64:3 (2024), 497
S Yu Dobrokhotov, A I Klevin, V E Nazaikinskii, A A Tolchennikov, “Asymptotics of solutions to systems of (pseudo)differential equations with localized right-hand sides”, J. Phys.: Conf. Ser., 2817:1 (2024), 012024
I. A. Nosikov, A. A. Tolchennikov, M. V. Klimenko, “Boundary value problem of calculating ray characteristics of ocean waves reflected from coastline”, Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, 64:3 (2024), 534
S. Yu. Dobrokhotov, V. E. Nazaikinskii, “On the arguments of Jacobians in local expressions of the Maslov canonical operator”, Math. Notes, 116:6 (2024), 1264–1276
A. Yu. Anikin, A. I. Klevin, “Asymptotics of the Helmholtz equation solutions in a two-layer medium with a localized right-hand side”, Theoret. and Math. Phys., 216:1 (2023), 1036–1054
S. T. Gataullin, T. M. Gataullin, “To the Problem of a Point Source in an Inhomogeneous Medium”, Math. Notes, 114:6 (2023), 1212–1216
Ilya Bogaevskii, Michel Rouleux, 2023 Days on Diffraction (DD), 2023, 12

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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References:	54
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