S. Yu. Dobrokhotov, G. N. Makrakis, V. E. Nazaikinskii, “Maslov's canonical operator, Hörmander's formula, and localization of the Berry–Balazs solution in the theory of wave beams”, TMF, 180:2 (2014), 162–188; Theoret. and Math. Phys., 180:2 (2014), 894

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Teoreticheskaya i Matematicheskaya Fizika, 2014, Volume 180, Number 2, Pages 162–188
DOI: https://doi.org/10.4213/tmf8683 (Mi tmf8683)

This article is cited in 12 scientific papers (total in 12 papers)

Maslov's canonical operator, Hörmander's formula, and localization of the Berry–Balazs solution in the theory of wave beams

S. Yu. Dobrokhotov^ab, G. N. Makrakis^cd, V. E. Nazaikinskii^ba

^a Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Oblast, Russia
^b Institute for Problems in Mechanics, RAS, Moscow, Russia
^c Institute of Applied and Computational Mathematics, Foundation for Research and Technology-Hellas, Heraklion, Crete, Greece
^d Department of Applied Mathematics, University of Crete, Heraklion, Crete, Greece

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DOI: https://doi.org/10.4213/tmf8683

Abstract: For three-dimensional Schrödinger equations, we study how to localize exact solutions represented as the product of an Airy function (Berry–Balazs solutions) and a Bessel function and known as Airy–Bessel beams in the paraxial approximation in optics. For this, we represent such solutions in the form of Maslov's canonical operator acting on compactly supported functions on special Lagrangian manifolds. We then use a result due to Hörmander, which permits using the formula for the commutation of a pseudodifferential operator with Maslov's canonical operator to “move” the compactly supported amplitudes outside the canonical operator and thus obtain effective formulas preserving the structure based on the Airy and Bessel functions. We discuss the influence of dispersion effects on the obtained solutions.

Keywords: Schrödinger equation, paraxial approximation, Airy–Bessel beam, localization, Maslov's canonical operator.

Received: 25.03.2014

English version:
Theoretical and Mathematical Physics, 2014, Volume 180, Issue 2, Pages 894–916
DOI: https://doi.org/10.1007/s11232-014-0187-5

Bibliographic databases:

Document Type: Article

Language: Russian

Citation: S. Yu. Dobrokhotov, G. N. Makrakis, V. E. Nazaikinskii, “Maslov's canonical operator, Hörmander's formula, and localization of the Berry–Balazs solution in the theory of wave beams”, TMF, 180:2 (2014), 162–188; Theoret. and Math. Phys., 180:2 (2014), 894–916

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This publication is cited in the following 12 articles:

S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “Asymptotics of the Localized Bessel Beams and Lagrangian Manifolds”, Radiotekhnika i elektronika, 68:6 (2023), 527
S. Yu. Dobrokhotov, S. A. Sergeev, “Asymptotics of the Solution of the Cauchy Problem with Localized Initial Conditions for a Wave Type Equation with Time Dispersion. I. Basic Structures”, Russ. J. Math. Phys., 29:2 (2022), 149
S. Yu. Dobrokhotov, D. S. Minenkov, V. E. Nazaikinskii, “Representations of Bessel functions via the Maslov canonical operator”, Theoret. and Math. Phys., 208:2 (2021), 1018–1037
S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. I. Shafarevich, “Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations”, Russian Math. Surveys, 76:5 (2021), 745–819
A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “Nonstandard Liouville tori and caustics in asymptotics in the form of Airy and Bessel functions for two-dimensional standing coastal waves”, St. Petersburg Math. J., 33:2 (2022), 185–205
S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Lagrangian Manifolds and Efficient Short-Wave Asymptotics in a Neighborhood of a Caustic Cusp”, Math. Notes, 108:3 (2020), 318–338
Dobrokhotov S.Yu. Nazaikinskii V.E. Tolchennikov A.A., “Uniform Formulas For the Asymptotic Solution of a Linear Pseudodifferential Equation Describing Water Waves Generated By a Localized Source”, Russ. J. Math. Phys., 27:2 (2020), 185–191
A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “Uniform asymptotic solution in the form of an Airy function for semiclassical bound states in one-dimensional and radially symmetric problems”, Theoret. and Math. Phys., 201:3 (2019), 1742–1770
S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Efficient formulas for the Maslov canonical operator near a simple caustic”, Russ. J. Math. Phys., 25:4 (2018), 545–552
S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. I. Shafarevich, “New integral representations of the Maslov canonical operator in singular charts”, Izv. Math., 81:2 (2017), 286–328
S. A. Sergeev, “Asymptotic Solutions of the One-Dimensional Linearized Korteweg–de Vries Equation with Localized Initial Data”, Math. Notes, 102:3 (2017), 403–416
Dobrokhotov S.Yu., Nazaikinskii V.E., Shafarevich A.I., “Maslov's canonical operator in arbitrary coordinates on the Lagrangian manifold”, Dokl. Math., 93:1 (2016), 99–102

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

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