The Maslov canonical operator on a pair of Lagrangian manifolds and asymptotic solutions of stationary equations with localized right-hand sides
- Authors: Anikin A.Y.1, Dobrokhotov S.Y.1, Nazaikinskii V.E.1,2, Rouleux M.3
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Affiliations:
- Ishlinsky Institute for Problems in Mechanics
- Moscow Institute of Physics and Technology (State University)
- Aix Marseille Univ
- Issue: Vol 96, No 1 (2017)
- Pages: 406-410
- Section: Mathematical Physics
- URL: https://journals.rcsi.science/1064-5624/article/view/225344
- DOI: https://doi.org/10.1134/S1064562417040275
- ID: 225344
Cite item
Abstract
The problem of constructing the asymptotics of the Green function for the Helmholtz operator h2Δ + n2(x), x ∈ Rn, with a small positive parameter h and smooth n2(x) has been studied by many authors; see, e.g., [1, 2, 4]. In the case of variable coefficients, the asymptotics was constructed by matching the asymptotics of the Green function for the equation with frozen coefficients and a WKB-type asymptotics or, in a more general situation, the Maslov canonical operator. The paper presents a different method for evaluating the Green function, which does not suppose the knowledge of the exact Green function for the operator with frozen variables. This approach applies to a larger class of operators, even when the right-hand side is a smooth localized function rather than a δ-function. In particular, the method works for the linearized water wave equations.
About the authors
A. Yu. Anikin
Ishlinsky Institute for Problems in Mechanics
Author for correspondence.
Email: anikin83@inbox.ru
Russian Federation, Moscow, 119526
S. Yu. Dobrokhotov
Ishlinsky Institute for Problems in Mechanics
Email: anikin83@inbox.ru
Russian Federation, Moscow, 119526
V. E. Nazaikinskii
Ishlinsky Institute for Problems in Mechanics; Moscow Institute of Physics and Technology (State University)
Email: anikin83@inbox.ru
Russian Federation, Moscow, 119526; Dolgoprudnyi, Moscow oblast, 141700
M. Rouleux
Aix Marseille Univ
Email: anikin83@inbox.ru
France, Marseille