This article is cited in 17 scientific papers (total in 17 papers)
Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations
Abstract:
We say that the initial data in the Cauchy problem are localized if they are given by functions concentrated in a neighbourhood of a submanifold of positive codimension, and the size of this neighbourhood depends on a small parameter and tends to zero together with the parameter. Although the solutions of linear differential and pseudodifferential equations with localized initial data constitute a relatively narrow subclass of the set of all solutions, they are very important from the point of view of physical applications. Such solutions, which arise in many branches of mathematical physics, describe the propagation of perturbations of various natural phenomena (tsunami waves caused by an underwater earthquake, electromagnetic waves emitted by antennas, etc.), and there is extensive literature devoted to such solutions (including the study of their asymptotic behaviour). It is natural to say that an asymptotics is efficient when it makes it possible to examine the problem quickly enough with relatively few computations. The notion of efficiency depends on the available computational tools and has changed significantly with the advent of \textsf{Wolfram Mathematica}, \textsf{Matlab}, and similar computing systems, which provide fundamentally new possibilities for the operational implementation and visualization of mathematical constructions, but which also impose new requirements on the construction of the asymptotics. We give an overview of modern methods for constructing efficient asymptotics in problems with localized initial data. The class of equations and systems under consideration includes the Schrödinger and Dirac equations, the Maxwell equations, the linearized gasdynamic and hydrodynamic equations, the equations of the linear theory of surface water waves, the equations of the theory of elasticity, the acoustic equations, and so on.
Bibliography: 109 titles.
Citation:
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
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\jour Russian Math. Surveys
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This publication is cited in the following 17 articles:
S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Metod osredneniya dlya zadach o kvaziklassicheskikh asimptotikakh”, Funktsionalnye prostranstva. Differentsialnye operatory. Problemy
matematicheskogo obrazovaniya, SMFN, 70, no. 1, Rossiiskii universitet druzhby narodov, M., 2024, 53–76
V.E. Nazaikinskii, “Semiclassical Asymptotics on Stratified Manifolds”, Russ. J. Math. Phys., 31:2 (2024), 299
S.Yu. Dobrokhotov, E.S. Smirnova, “Asymptotics of the Solution of the Initial Boundary Value Problem for the One-Dimensional Klein–Gordon Equation with Variable Coefficients”, Russ. J. Math. Phys., 31:2 (2024), 187
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
V.E. Nazaikinskii, “On the Phase Spaces for a Class of Boundary-Degenerate Equations”, Russ. J. Math. Phys., 31:4 (2024), 713
A. T. Fomenko, A. I. Shafarevich, V. A. Kibkalo, “Glavnye napravleniya i dostizheniya kafedry differentsialnoi geometrii i prilozhenii na sovremennom etape”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2024, no. 6, 27–37
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”, Theoret. and Math. Phys., 214:1 (2023), 1–23
S. Yu. Dobrokhotov, S. B. Levin, A. A. Tolchennikov, “Keplerian orbits and global asymptotic solution in the form of an Airy function for the scattering problem on a repulsive Coulomb potential”, Russian Math. Surveys, 78:4 (2023), 788–790
E. S. Smirnova, “Asymptotics of the Solution of an Initial–Boundary Value Problem for the One-Dimensional Klein–Gordon Equation on the Half-Line”, Math. Notes, 114:4 (2023), 608–618
T. A. Suslina, “Operator-theoretic approach to the homogenization of Schrödinger-type equations with periodic coefficients”, Russian Math. Surveys, 78:6 (2023), 1023–1154
D. S. Minenkov, S. A. Sergeev, “Asymptotics of the whispering gallery-type in the eigenproblem for the Laplacian in a domain of revolution diffeomorphic to a solid torus”, Russ. J. Math. Phys., 30:4 (2023), 599
V. L. Chernyshev, V. E. Nazaikinskii, A. V. Tsvetkova, “Lattice equations and semiclassical asymptotics”, Russ. J. Math. Phys., 30:2 (2023), 152
V. E. Nazaikinskii, “Canonical Operator on Punctured Lagrangian Manifolds and Commutation with Pseudodifferential Operators: Local Theory”, Math. Notes, 112:5 (2022), 709–725
S. Yu. Dobrokhotov, A. A. Tolchennikov, “Keplerian trajectories and an asymptotic solution of the Schrödinger equation with repulsive Coulomb potential and localized right-hand side”, Russ. J. Math. Phys., 29:4 (2022), 456–466
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), 149–169
S. A. Sergeev, “Asymptotic solution of the Cauchy problem with localized initial data for a wave equation with small dispersion effects”, Differ. Equ.
58, no. 10, 2022, 1376–1395
V. E. Nazaikinskii, A. Yu. Savin, “On semiclassical asymptotics for nonlocal equations”, Russ. J. Math. Phys., 29:4 (2022), 568–575
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