Efficient Asymptotics in Problems on the Propagation of Waves Generated by Localized Sources in Linear Multidimensional
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L DIFFERENTIAL EQUATIONS
Efficient Asymptotics in Problems on the Propagation of Waves Generated by Localized Sources in Linear Multidimensional Inhomogeneous and Dispersive Media S. Yu. Dobrokhotova,* and V. E. Nazaikinskiia,** a
Ishlinsky Institute for Problems in Mechanics RAS, Moscow, 119526 Russia *e-mail: [email protected] **e-mail: [email protected] Received February 15, 2020; revised February 15, 2020; accepted April 9, 2020
Abstract—The Cauchy problem with localized initial conditions is considered for a large class of evolution equations that includes the Schrödinger and Dirac equations, Maxwell equations, linearized fluid dynamics equations, equations of the linear theory of surface water waves, equations of elasticity theory, acoustics equations, and many others. A general approach to the construction of efficient asymptotic formulas in such problems is discussed. Keywords: evolution equation, Cauchy problem, localized initial conditions, semiclassical asymptotics, WKB method, Maslov’s canonical operator, efficient formulas DOI: 10.1134/S0965542520080060
1. INTRODUCTION In this paper, we discuss a general approach to constructing efficient global asymptotics of the solution of the Cauchy problem with localized initial conditions for linear evolution differential and pseudodifferential equations and wave type systems with a small parameter multiplying the derivative (which may or may not coincide with the parameter representing the size of the neighborhood containing the initial condition). This class of equations includes the Schrödinger and Dirac equations, Maxwell equations, linearized fluid dynamics equations, equations of the linear theory of surface water waves, equations of elasticity theory, acoustics equations, and many others. When speaking of efficiency, we mean asymptotic formulas that can be relatively easily and computationally efficiently implemented in modern computer algebra systems, such as Wolfram Mathematica or MatLab. The global theory of semiclassical asymptotics based on Maslov’s canonical operator [1] (also see [2, 3]) typically does not yield efficient formulas in its standard version. This is not surprising, because this theory was developed more than 50 years ago, long before these computer algebra systems appeared. Therefore, there is a natural desire to adapt these constructs to the modern tools of mathematical investigations. In recent years, the authors of this paper, together with some colleagues, have made considerable progress in this direction, the key role being played by new representations, introduced in [4], of Maslov’s canonical operator in singular charts. The main practical result is that efficient asymptotics in these problems, at least in the leading term, can be constructed by combining the new formulas obtained in [4] with the well-known schemes described in [1–3] and by applying the resulting technique to special Lagrangian manifolds with singularities that were not covered by the “old” formulas. 2. EQUATIONS AND INITIAL CONDITIONS Consider
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