In this paper, we begin with the nonlinear Schrödinger/Gross–Pitaevskii equation
(NLSE/GPE) for modeling Bose–Einstein condensation (BEC) and nonlinear optics as well …

We consider time-dependent (linear and nonlinear) Schrödinger equations in a
semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive …

In some applications, it is reasonable to assume that geodesics (rays) have a consistent
orientation so that the Helmholtz equation may be viewed as an evolution equation in one of …

Gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial
differential equations, concentrated on a single curve through the physical domain. They can …

We introduce a new multiscale Gaussian beam method for the numerical solution of the
wave equation with smooth variable coefficients. The first computational question addressed …

We propose the frozen Gaussian approximation for computation of high frequency wave
propagation. This method approximates the solution to the wave equation by an integral …

A rigorous method for simulations of quantum dynamics is introduced on the basis of
concatenation of semiclassical thawed Gaussian propagation steps. The time-evolving state …

We perform a time-frequency analysis of Fourier multipliers and, more generally,
pseudodifferential operators with symbols of Gevrey, analytic and ultra-analytic regularity. As …

We propose a new algorithm for solving the semiclassical time-dependent Schrödinger
equation. The algorithm is based on semiclassical wavepackets. The focus of the analysis is …

The Dirac equation is an important model in relativistic quantum mechanics. In the semi-
classical regime $\epsilon\ll1 $, even a spatially spectrally accurate time splitting method\cite …