26 research outputs found
Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schrödinger equations
We consider the semiclassical limit of nonlinear Schr\"odinger equations with
wavepacket initial data. We recover the Wigner measure of the problem, a
macroscopic phase-space density which controls the propagation of the physical
observables such as mass, energy and momentum. Wigner measures have been used
to create effective models for wave propagation in random media, quantum
molecular dynamics, mean field limits, and the propagation of electrons in
graphene. In nonlinear settings, the Vlasov-type equations obtained for the
Wigner measure are often ill-posed on the physically interesting spaces of
initial data. In this paper we are able to select the measure-valued solution
of the 1+1 dimensional Vlasov-Poisson equation which correctly captures the
semiclassical limit, thus finally resolving the non-uniqueness in the seminal
result of [Zhang, Zheng \& Mauser, Comm. Pure Appl. Math. (2002) 55,
doi:10.1002/cpa.3017]. The same approach is also applied to the
Vlasov-Dirac-Benney equation with small wavepacket initial data, extending
several known results
Phase Resolved Simulation of the Landau–Alber Stability Bifurcation
It has long been known that plane wave solutions of the cubic nonlinear Schrödinger Equation (NLS) are linearly unstable. This fact is widely known as modulation instability (MI), and sometimes referred to as Benjamin–Feir instability in the context of water waves. In 1978, I.E. Alber introduced a methodology to perform an analogous linear stability analysis around a sea state with a known power spectrum, instead of around a plane wave. This analysis applies to second moments, and yields a stability criterion for power spectra. Asymptotically, it predicts that sufficiently narrow and high-intensity spectra are unstable, while sufficiently broad and low-intensity spectra are stable, which is consistent with empirical observations. The bifurcation between unstable and stable behaviour has no counterpart in the classical MI (where all plane waves are unstable), and we call it Landau–Alber bifurcation because the stable regime has been shown to be a case of Landau damping. In this paper, we work with the realistic power spectra of ocean waves, and for the first time, we produce clear, direct evidence for an abrupt bifurcation as the spectrum becomes narrow/intense enough. A fundamental ingredient of this work was to look directly at the nonlinear evolution of small, localised inhomogeneities, and whether these can grow dramatically. Indeed, one of the issues affecting previous investigations of this bifurcation seem to have been that they mostly looked for the indirect evidence of instability, such as an increase in overall extreme events. It is also found that a sufficiently large computational domain is crucial for the bifurcation to manifest
On a selection principle for multivalued semiclassical flows
We study the semiclassical behaviour of solutions of a Schr ̈odinger equation with a scalar po- tential displaying a conical singularity. When a pure state interacts strongly with the singularity of the flow, there are several possible classical evolutions, and it is not known whether the semiclassical limit cor- responds to one of them. Based on recent results, we propose that one of the classical evolutions captures the semiclassical dynamics; moreover, we propose a selection principle for the straightforward calculation of the regularized semiclassical asymptotics. We proceed to investigate numerically the validity of the proposed scheme, by employing a solver based on a posteriori error control for the Schr ̈odinger equation. Thus, for the problems we study, we generate rigorous upper bounds for the error in our asymptotic approximation. For 1-dimensional problems without interference, we obtain compelling agreement between the regularized asymptotics and the full solution. In problems with interference, there is a quantum effect that seems to survive in the classical limit. We discuss the scope of applicability of the proposed regularization approach, and formulate a precise conjecture
Modelling of Ocean Waves with the Alber Equation:Application to Non-Parametric Spectra and Generalisation to Crossing Seas
The Alber equation is a phase-averaged second-moment model for the statistics
of a sea state, which recently has been attracting renewed attention. We extend
it in two ways: firstly, we derive a generalized Alber system starting from a
system of nonlinear Schr\"odinger equations, which contains the classical Alber
equation as a special case but can also describe crossing seas, i.e. two
wavesystems with different wavenumbers crossing. (These can be two completely
independent wavenumbers, i.e. in general different directions and different
moduli.) We also derive the associated 2-dimensional scalar instability
condition. This is the first time that a modulation instability condition
applicable to crossing seas has been systematically derived for general
spectra. Secondly, we use the classical Alber equation and its associated
instability condition to quantify how close a given non-parametric spectrum is
to being modulationally unstable. We apply this to a dataset of 100
non-parametric spectra provided by the Norwegian Meteorological Institute, and
find the vast majority of realistic spectra turn out to be stable, but three
extreme sea states are found to be unstable (out of 20 sea states chosen for
their severity). Moreover, we introduce a novel "proximity to instability"
(PTI) metric, inspired by the stability analysis. This is seen to correlate
strongly with the steepness and Benjamin-Feir Index (BFI) for the sea states in
our dataset (>85% Spearman rank correlation). Furthermore, upon comparing with
phase-resolved broadband Monte Carlo simulations, the kurtosis and probability
of rogue waves for each sea state are also seen to correlate well with its PTI
(>85% Spearman rank correlation)
On the XFEL Schroedinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging
We analyse a nonlinear Schr\"odinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree-Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an X-ray Free Electron Laser (XFEL). We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding solutions of a Schr\"odinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential
Numerical simulations of X-rays Free Electron Lasers (XFEL)
We study a nonlinear Schr\"odinger equation which arises as an effective
single particle model in X-ray Free Electron Lasers (XFEL). This equation
appears as a first-principles model for the beam-matter interactions that would
take place in an XFEL molecular imaging experiment in \cite{frat1}. Since XFEL
is more powerful by several orders of magnitude than more conventional lasers,
the systematic investigation of many of the standard assumptions and
approximations has attracted increased attention.
In this model the electrons move under a rapidly oscillating electromagnetic
field, and the convergence of the problem to an effective time-averaged one is
examined. We use an operator splitting pseudo-spectral method to investigate
numerically the behaviour of the model versus its time-averaged version in
complex situations, namely the energy subcritical/mass supercritical case, and
in the presence of a periodic lattice.
We find the time averaged model to be an effective approximation, even close
to blowup, for fast enough oscillations of the external field. This work
extends previous analytical results for simpler cases \cite{xfel1}.Comment: 14 page
Strong and weak semiclassical limits for some rough Hamiltonians
We present several results concerning the semiclassical limit of the time
dependent Schr\"odinger equation with potentials whose regularity doesn't
guarantee the uniqueness of the underlying classical flow. Different topologies
for the limit are considered and the situation where two bicharateristics can
be obtained out of the same initial point is emphasized
A novel, structure-preserving, second-order-in-time relaxation scheme for Schrödinger-Poisson systems
We introduce a new second order in time relaxation-type scheme for
approximating solutions of the Schr\"odinger-Poisson system. More specifically,
we use the Crank-Nicolson scheme as a time stepping mechanism, whilst the
nonlinearity is handled by means of a relaxation approach in the spirit of
\cite{Besse, KK} for the nonlinear Schr\"odinger equation. For the spatial
discretisation we use the standard conforming finite element scheme. The
resulting scheme is explicit with respect to the nonlinearity, satisfies
discrete versions of the system's conservation laws, and is seen to be second
order in time. We conclude by presenting some numerical experiments, including
an example from cosmology, that demonstrate the effectiveness and robustness of
the new scheme.Comment: 17pages, 10 figure
Coarse-scale representations and smoothed Wigner transforms
Smoothed Wigner transforms have been used in signal processing, as a
regularized version of the Wigner transform, and have been proposed as an
alternative to it in the homogenization and / or semiclassical limits of wave
equations. We derive explicit, closed formulations for the coarse-scale
representation of the action of pseudodifferential operators. The resulting
``smoothed operators'' are in general of infinite order. The formulation of an
appropriate framework, resembling the Gelfand-Shilov spaces, is necessary.
Similarly we treat the ``smoothed Wigner calculus''. In particular this allows
us to reformulate any linear equation, as well as certain nonlinear ones (e.g.
Hartree and cubic non-linear Schr\"odinger), as coarse-scale phase-space
equations (e.g. smoothed Vlasov), with spatial and spectral resolutions
controlled by two free parameters. Finally, it is seen that the smoothed Wigner
calculus can be approximated, uniformly on phase-space, by differential
operators in the semiclassical regime. This improves the respective
weak-topology approximation result for the Wigner calculus.Comment: 58 pages, plain Te