6,163 research outputs found
Energy preserving model order reduction of the nonlinear Schr\"odinger equation
An energy preserving reduced order model is developed for two dimensional
nonlinear Schr\"odinger equation (NLSE) with plane wave solutions and with an
external potential. The NLSE is discretized in space by the symmetric interior
penalty discontinuous Galerkin (SIPG) method. The resulting system of
Hamiltonian ordinary differential equations are integrated in time by the
energy preserving average vector field (AVF) method. The mass and energy
preserving reduced order model (ROM) is constructed by proper orthogonal
decomposition (POD) Galerkin projection. The nonlinearities are computed for
the ROM efficiently by discrete empirical interpolation method (DEIM) and
dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and
mass are shown for the full order model (FOM) and for the ROM which ensures the
long term stability of the solutions. Numerical simulations illustrate the
preservation of the energy and mass in the reduced order model for the two
dimensional NLSE with and without the external potential. The POD-DMD makes a
remarkable improvement in computational speed-up over the POD-DEIM. Both
methods approximate accurately the FOM, whereas POD-DEIM is more accurate than
the POD-DMD
Tachyon Condensation on the Elliptic Curve
We use the framework of matrix factorizations to study topological B-type
D-branes on the cubic curve. Specifically, we elucidate how the brane RR
charges are encoded in the matrix factors, by analyzing their structure in
terms of sections of vector bundles in conjunction with equivariant R-symmetry.
One particular advantage of matrix factorizations is that explicit moduli
dependence is built in, thus giving us full control over the open-string moduli
space. It allows one to study phenomena like discontinuous jumps of the
cohomology over the moduli space, as well as formation of bound states at
threshold. One interesting aspect is that certain gauge symmetries inherent to
the matrix formulation lead to a non-trivial global structure of the moduli
space. We also investigate topological tachyon condensation, which enables us
to construct, in a systematic fashion, higher-dimensional matrix factorizations
out of smaller ones; this amounts to obtaining branes with higher RR charges as
composites of ones with minimal charges. As an application, we explicitly
construct all rank-two matrix factorizations.Comment: 69p, 6 figs, harvmac; v2: minor change
Phase transitions in 3D gravity and fractal dimension
We show that for three dimensional gravity with higher genus boundary
conditions, if the theory possesses a sufficiently light scalar, there is a
second order phase transition where the scalar field condenses. This three
dimensional version of the holographic superconducting phase transition occurs
even though the pure gravity solutions are locally AdS. This is in addition
to the first order Hawking-Page-like phase transitions between different
locally AdS handlebodies. This implies that the R\'enyi entropies of
holographic CFTs will undergo phase transitions as the R\'enyi parameter is
varied, as long as the theory possesses a scalar operator which is lighter than
a certain critical dimension. We show that this critical dimension has an
elegant mathematical interpretation as the Hausdorff dimension of the limit set
of a quotient group of AdS, and use this to compute it, analytically near
the boundary of moduli space and numerically in the interior of moduli space.
We compare this to a CFT computation generalizing recent work of Belin, Keller
and Zadeh, bounding the critical dimension using higher genus conformal blocks,
and find a surprisingly good match
Capturing the phase diagram of (2+1)-dimensional CDT using a balls-in-boxes model
We study the phase diagram of a one-dimensional balls-in-boxes (BIB) model
that has been proposed as an effective model for the spatial-volume dynamics of
(2+1)-dimensional causal dynamical triangulations (CDT). The latter is a
statistical model of random geometries and a candidate for a nonperturbative
formulation of quantum gravity, and it is known to have an interesting phase
diagram, in particular including a phase of extended geometry with classical
properties. Our results corroborate a previous analysis suggesting that a
particular type of potential is needed in the BIB model in order to reproduce
the droplet condensation typical of the extended phase of CDT. Since such a
potential can be obtained by a minisuperspace reduction of a (2+1)-dimensional
gravity theory of the Ho\v{r}ava-Lifshitz type, our result strengthens the link
between CDT and Ho\v{r}ava-Lifshitz gravity.Comment: 21 pages, 7 figure
Boson condensation and instability in the tensor network representation of string-net states
The tensor network representation of many-body quantum states, given by local
tensors, provides a promising numerical tool for the study of strongly
correlated topological phases in two dimension. However, tensor network
representations may be vulnerable to instabilities caused by small
perturbations of the local tensor, especially when the local tensor is not
injective. For example, the topological order in tensor network representations
of the toric code ground state has been shown to be unstable under certain
small variations of the local tensor, if these small variations do not obey a
local symmetry of the tensor. In this paper, we ask the questions of
whether other types of topological orders suffer from similar kinds of
instability and if so, what is the underlying physical mechanism and whether we
can protect the order by enforcing certain symmetries on the tensor. We answer
these questions by showing that the tensor network representation of all
string-net models are indeed unstable, but the matrix product operator (MPO)
symmetries of the local tensor can help to protect the order. We find that,
`stand-alone' variations that break the MPO symmetries lead to instability
because they induce the condensation of bosonic quasi-particles and destroy the
topological order in the system. Therefore, such variations must be forbidden
for the encoded topological order to be reliably extracted from the local
tensor. On the other hand, if a tensor network based variational algorithm is
used to simulate the phase transition due to boson condensation, then such
variation directions must be allowed in order to access the continuous phase
transition process correctly.Comment: 44 pages, 85 figures, comments welcom
Numerical study of Bose-Einstein condensation in the Kaniadakis-Quarati model for bosons
Kaniadakis and Quarati (1994) proposed a Fokker--Planck equation with
quadratic drift as a PDE model for the dynamics of bosons in the spatially
homogeneous setting. It is an open question whether this equation has solutions
exhibiting condensates in finite time. The main analytical challenge lies in
the continuation of exploding solutions beyond their first blow-up time while
having a linear diffusion term. We present a thoroughly validated time-implicit
numerical scheme capable of simulating solutions for arbitrarily large time,
and thus enabling a numerical study of the condensation process in the
Kaniadakis--Quarati model. We show strong numerical evidence that above the
critical mass rotationally symmetric solutions of the Kaniadakis--Quarati model
in 3D form a condensate in finite time and converge in entropy to the unique
minimiser of the natural entropy functional at an exponential rate. Our
simulations further indicate that the spatial blow-up profile near the origin
follows a universal power law and that transient condensates can occur for
sufficiently concentrated initial data.Comment: To appear in Kinet. Relat. Model
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