Related papers: The Anisotropic Gaussian Semi-Classical Schr\"{o}d…
Superoscillations are a phenomenon in physics, where linear combinations of low-frequency plane waves interfere almost destructively in such a way that the resulting wave has a higher frequency than any of the individual waves. The…
We consider the case of a cubic nonlinear Schr\"{o}dinger equation with an additional chaotic potential, in the sense that such a potential produces chaotic dynamics in classical mechanics. We derive and describe an appropriate…
This work considers numerical methods for the time-dependent Schr\"{o}dinger equation of incommensurate systems. By using a plane wave method for spatial discretization, the incommensurate problem is lifted to a higher dimension that…
We study the propagation of singularities for semilinear Schrodinger equations with quadratic Hamiltonians, in particular for the semilinear harmonic oscillator. We show that the propagation still occurs along the flow the Hamiltonian flow,…
We build Gaussian wave packets for the linear Schr\"odinger equation and its finite difference space semi-discretization and illustrate the lack of uniform dispersive properties of the numerical solutions as established in Ignat, Zuazua,…
This paper introduces weighted finite difference methods for numerically solving dispersive evolution equations with solutions that are highly oscillatory in both space and time. We consider a semiclassically scaled cubic nonlinear…
We consider Schr\"odinger operators $H$ on $R^n$ with variable coefficients. Let $H_0=-\frac12\triangle$ be the free Schr\"odinger operator and we suppose $H$ is a "short-range" perturbation of $H_0$. Then, under the nontrapping condition,…
We study inelastic resonant scattering of a Gaussian wave packet with the parameters close to a zero of the complex scattering coefficient. We demonstrate, both theoretically and experimentally, that such near-zero scattering can result in…
The focussing anisotropic nonlinear Schr\"odinger equation \begin{align*} \mathrm{i} u_t-\partial_{xx} u + (-\partial_{yy})^s u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R} \times \mathbb{R}^2 \end{align*} is considered for $0<s<1$ and $p>2$.…
Using the Kantorovitch method in combination with a Gaussian ansatz, we derive the equations of motion for spatial, temporal and spatiotemporal optical propagation in a dispersive Kerr medium with a general transverse and spectral gain…
We study generalised magnetic Schroedinger operators of the form H(A,V)=h(P^A)+V, where h is an elliptic symbol, P^A is the generator of the magnetic translations, with A a vector potential defining a variable magnetic field B, and V is a…
Motivated by the phase space analysis of Schr\"odinger evolution operators, in this paper we investigate how metaplectic operators are approximately diagonalized along the corresponding symplectic flows by exponentially localized Gabor wave…
We construct a special class of semiclassical Fourier integral operators whose wave fronts are symplectic micromorphisms. These operators have very good properties: they form a category on which the wave front map becomes a functor into the…
Let $ H:=-\tfrac12\Delta+V$ be a one-dimensional continuum Schr\"odinger operator. Consider ${\hat H}:= H+\xi$, where $\xi$ is a translation invariant Gaussian noise. Under some assumptions on $\xi$, we prove that if $V$ is locally…
We consider Schr\"odinger equations with variable coefficients and the harmonic potential. We suppose the perturbation is short-range type in the sense of [Nakamura 2004]. We characterize the wave front set of the solutions to the equation…
We develop a gauge-invariant frozen Gaussian approximation (GIFGA) method for the linear Schr\"odinger equation (LSE) with periodic potentials in the semiclassical regime. The method generalizes the Herman-Kluk propagator for LSE to the…
Sea ice attenuates waves propagating from the open ocean. Here we model the evolution of energetic unidirectional random waves in the marginal ice zone with a nonlinear Schr\"{o}dinger equation, with a frequency dependent dissipative term…
We model the effects of cross-phase modulation in frequency (or wavelength) division multiplexed optical communications systems, using a Schr\"odinger equation with a spatially and temporally random potential. Green's functions for the…
We present an ab initio approach to solve the time-dependent Schr\"odinger equation to treat electron and photon impact multiple ionization of atoms or molecules. It combines the already known time scaled coordinate method with a new high…
The quantum optical problem of the propagation of electromagnetic waves in a nonlinear waveguide is related to the solutions of the classical nonstationary harmonic oscillator using the method of linear integrals of motion [ Malkin et.al.,…