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 Uspekhi Mat. Nauk, 2021, Volume 76, Issue 5(461), Pages 3–80 (Mi umn9973)

Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations

S. Yu. Dobrokhotova, V. E. Nazaikinskiia, A. I. Shafarevichbacd

a Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, Russia
b Lomonosov Moscow State University, Moscow, Russia
c Moscow Institute of Physics and Technology (National Research University), Moscow, Russia
d National Research Centre "Kurchatov Institute", Moscow, Russia

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.

Keywords: differential equations, semiclassical asymptotics, efficient asymptotics, Maslov canonical operator, Cauchy problem, localized initial conditions.

 Funding Agency Grant Number Russian Foundation for Basic Research 19-11-50111 This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 19-11-50111).

DOI: https://doi.org/10.4213/rm9973

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English version:
Russian Mathematical Surveys, 2021, 76:5, 745–819

Bibliographic databases:

UDC: 517.9
MSC: 81Q20, 35L15, 35L45, 35S10, 53D12

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”, Uspekhi Mat. Nauk, 76:5(461) (2021), 3–80; Russian Math. Surveys, 76:5 (2021), 745–819

Citation in format AMSBIB
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