Gaussian summation method (review)

Babich, V. M.; Popov, M. M.

doi:10.1007/bf01038632

Cited by 75 publications

(28 citation statements)

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“…The accuracy of the Gaussian beam superposition to approximate the original wave field is important, but determining the error of the Gaussian beam superposition is highly nontrivial, see the conclusion section of the review article by Babič and Popov [6]. In the past few years, some significant progress on estimates of the error has been made.…”

Section: Introductionmentioning

confidence: 99%

Error estimates of the Bloch band-based Gaussian beam superposition for the Schrödinger equation

Liu¹,

Pryporov²

2015

Spectral Theory and Partial Differential Equations

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Abstract. This work is concerned with asymptotic approximations of the semi-classical Schrödinger equation in periodic media using Gaussian beams. For the underlying equation, subject to a highly oscillatory initial data, a hybrid of the Gaussian beam approximation and homogenization leads to the Bloch eigenvalue problem and associated evolution equations for Gaussian beam components in each Bloch band. We formulate a superposition of Blochband based Gaussian beams to generate high frequency approximate solutions to the original wave field. For initial data of a sum of finite number of band eigen-functions, we prove that the first-order Gaussian beam superposition converges to the original wave field at a rate of ǫ 1/2 , with ǫ the semiclassically scaled constant, as long as the initial data for Gaussian beam components in each band are prepared with same order of error or smaller. For a natural choice of initial approximation, a rate of ǫ 1/2 of initial error is verified. IntroductionWe consider the semiclassically scaled Schrödinger equation with a periodic potential:subject to the two-scale initial condition:where Ψ(t, x) is a complex wave function, ε is the re-scaled Planck constant, V e (x)-smooth external potential, S 0 (x)-real-valued smooth function, g(x, y) = g(x, y + 2π)-smooth function, compactly supported in x, i.e., g(x, y) = 0, x ∈ K 0 , K 0 -is a bounded set. V (y) is periodic with respect to the crystal lattice Γ = (2πZ) d , it models the electronic potential generated by the lattice of atoms in the crystal [14].A typical application arises in solid state physics where (1.1) describes the quantum dynamics of Bloch electrons moving in a crystalline lattice (generated by the ionic cores) [39]. The asymptotics of (1.1) as ε → 0+ is a well-studied two-scale problem in the physics and mathematics literature [8,16,20,41,23,13,33,1,14]. On the other hand, the computational challenge because of the small parameter ε has prompted a search for asymptotic model based numerical methods, see e.g., [29,37].1991 Mathematics Subject Classification. 35A21, 35A35, 35Q40.

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Section: Introductionmentioning

confidence: 99%

Error estimates of the Bloch band-based Gaussian beam superposition for the Schrödinger equation

Liu¹,

Pryporov²

2015

Spectral Theory and Partial Differential Equations

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“…This method implies: (i) decomposition of the primary wave field into Gaussian beams, (ii) description of the every individual Gaussian beam by ODEs of Babich's approach, of DRT method or of CGO method, and (iii) successive summation of all the terms, noticeably contributing in the wave field at the observation point. Further development of this approach was performed primary by efforts of St. Petersburg and Prague scientific schools (Červený et al, 1982;Červený, 1983, 1985Klimeš, 1984aKlimeš, ,1989aBabich and Popov, 1990). Other papers on the method of Gaussian beams summation, related predominantly to geophysics were outlined in the books by Červený (2001) andPopov (2002).…”

Section: 3 C O M P L E X -V a L U E D D Y N A M I C R A Y T R A Cmentioning

confidence: 99%

Gaussian beams in inhomogeneous media: A review

Kravtsov

Berczyński

2007

Stud Geophys Geod

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The paper outlines the most important results of the paraxial complex geometrical optics (CGO) in respect to Gaussian beams diffraction in the smooth inhomogeneous media and discusses interrelations between CGO and other asymptotic methods, which reduce the problem of Gaussian beam diffraction to the solution of ordinary differential equations, namely: (i) Babich's method, which deals with the abridged parabolic equation and describes diffraction of the Gaussian beams; (ii) complex form of the dynamic ray tracing method, which generalizes paraxial ray approximation on Gaussian beams and (iii) paraxial WKB approximation by Pereverzev, which gives the results, quite close to those of Babich's method. For Gaussian beams all the methods under consideration lead to the similar ordinary differential equations, which are complex-valued nonlinear Riccati equation and related system of complex-valued linear equations of paraxial ray approximation. It is pointed out that Babich's method provides diffraction substantiation both for the paraxial CGO and for complex-valued dynamic ray tracing method. It is emphasized also that the latter two methods are conceptually equivalent to each other, operate with the equivalent equations and in fact are twins, though they differ by names.The paper illustrates abilities of the paraxial CGO method by two available analytical solutions: Gaussian beam diffraction in the homogeneous and in the lens-like media, and by the numerical example: Gaussian beam reflection from a plane-layered medium.

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“…The standard approach to surmounting this difficulty is to use an approximate model for the wave propagation that converges to the exact model as the frequency increases. Examples of such asymptotic high frequency methods include geometric optics [5], Gaussian beam methods (GBs) [1,3], wavefront tracking methods [29] and others. For a comprehensive review, we refer the reader to [11].…”

Section: Introductionmentioning

confidence: 99%