Gaussian beam methods for the Schrӧdinger equation with periodic potentials and strictly hyperbolic systems

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2013-01-01
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Pryporov, Maksym
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Hailiang Liu
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Abstract

In this dissertation, we study Gaussian beam superposition methods for the computation of high frequency wavefields governed by two important time-dependent partial differential equations, the Shcr ӧdinger equation with periodic potentials and strictly hyperbolic systems, both subject to highly oscillatory initial conditions. Gaussian beams form a high frequency asymptotic model which is closely related to geometrical optics. However, unlike geometrical optics, there is no breakdown at caustics. The beam solution is concentrated near a single ray of geometrical optics. The superposition of first order Gaussian beams constitute our asymptotic solution to the underlying initial value problems. Based on the well-posedness result, we obtain optimal error estimates in terms of the high frequency parameter &epsilon. For the Schr ӧdinger equation, our error

estimate is obtained in L^2 norm, and for hyperbolic systems the energy norm is taken.

For the linear semi-classical Shcr ӧdinger equation in periodic media, the geometric optics ansatz together with homogenization leads to the Bloch eigenvalue problem. We provide Gaussian beam evolution equations for each Bloch band, following the idea in the earlier work by M. Dimassi J-C. Guillot and J. Ralston &rdquo Gaussian beam construction for adiabatic perturbations &ldquo published in the Journal of Mathematical Physics, Analysis and Geometry in 2006, [10]. Our contribution to the analysis of this problem is in obtaining error estimates of the Gaussian beam superposition. Using the superposition principle, we obtain high frequency approximate solutions to the original wavefield. When the initial data can be decomposed into a finite number of band eigen-functions and under regularity assumptions for Bloch bands and energy bands, we prove that thefirst-order Gaussian beam superposition converges to the original wavefield at a rate of 1/2, with 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 &epsilon^(1/2) of initial error is verified.

For the strictly hyperbolic systems we construct Gaussian beam approximations and study the accuracy of the Gaussian beam superposition. Under some regularity assumptions of data we show

error estimates between the exact solution and the Gaussian beam superposition in terms of the high frequency parameter . The main result is that the relative local error measured in energy norm in the beam approximations decay as ^(1/2) independent of dimension and presence of caustics, forfirst order beams. This result is shown to be valid when the gradient of the initial phase may vanish on a set of measure zero.

For the strictly hyperbolic systems we construct Gaussian beam approximations and study the accuracy of the Gaussian beam superposition. Under some regularity assumptions of data we show

error estimates between the exact solution and the Gaussian beam superposition in terms of the high frequency parameter . The main result is that the relative local error measured in energy norm in the beam approximations decay as ^(1/2) independent of dimension and presence of caustics, forfirst order beams. This result is shown to be valid when the gradient of the initial phase may vanish on a set of measure zero.

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Tue Jan 01 00:00:00 UTC 2013