640 research outputs found
Time--Splitting Schemes and Measure Source Terms for a Quasilinear Relaxing System
Several singular limits are investigated in the context of a
system arising for instance in the modeling of chromatographic processes. In
particular, we focus on the case where the relaxation term and a
projection operator are concentrated on a discrete lattice by means of Dirac
measures. This formulation allows to study more easily some time-splitting
numerical schemes
Ginzburg-Landau model with small pinning domains
We consider a Ginzburg-Landau type energy with a piecewise constant pinning
term in the potential . The function is different from
1 only on finitely many disjoint domains, called the {\it pinning domains}.
These pinning domains model small impurities in a homogeneous superconductor
and shrink to single points in the limit ; here, \v is the inverse of
the Ginzburg-Landau parameter. We study the energy minimization in a smooth
simply connected domain with Dirichlet boundary
condition on \d \O, with topological degree {\rm deg}_{\d \O} (g) = d
>0. Our main result is that, for small \v, minimizers have distinct
zeros (vortices) which are inside the pinning domains and they have a degree
equal to 1. The question of finding the locations of the pinning domains with
vortices is reduced to a discrete minimization problem for a finite-dimensional
functional of renormalized energy. We also find the position of the vortices
inside the pinning domains and show that, asymptotically, this position is
determined by {\it local renormalized energy} which does not depend on the
external boundary conditions.Comment: 39 page
Calibrated Surfaces and Supersymmetric Wilson Loops
We study the dual gravity description of supersymmetric Wilson loops whose
expectation value is unity. They are described by calibrated surfaces that end
on the boundary of anti de-Sitter space and are pseudo-holomorphic with respect
to an almost complex structure on an eight-dimensional slice of AdS_5 x S^5.
The regularized area of these surfaces vanishes, in agreement with field theory
non-renormalization theorems for the corresponding operators.Comment: 28 pages, 2 figure
Classification of unit-vector fields in convex polyhedra with tangent boundary conditions
A unit-vector field n on a convex three-dimensional polyhedron P is tangent
if, on the faces of P, n is tangent to the faces. A homotopy classification of
tangent unit-vector fields continuous away from the vertices of P is given. The
classification is determined by certain invariants, namely edge orientations
(values of n on the edges of P), kink numbers (relative winding numbers of n
between edges on the faces of P), and wrapping numbers (relative degrees of n
on surfaces separating the vertices of P), which are subject to certain sum
rules. Another invariant, the trapped area, is expressed in terms of these. One
motivation for this study comes from liquid crystal physics; tangent
unit-vector fields describe the orientation of liquid crystals in certain
polyhedral cells.Comment: 21 pages, 2 figure
Euler semigroup, Hardy-Sobolev and Gagliardo-Nirenberg type inequalities on homogeneous groups
26 page
Positive Least Energy Solutions and Phase Separation for Coupled Schrodinger Equations with Critical Exponent: Higher Dimensional Case
We study the following nonlinear Schr\"{o}dinger system which is related to
Bose-Einstein condensate: {displaymath} {cases}-\Delta u +\la_1 u = \mu_1
u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in
\Omega, -\Delta v +\la_2 v =\mu_2 v^{2^\ast-1}+\beta v^{\frac{2^\ast}{2}-1}
u^{\frac{2^\ast}{2}}, \quad x\in \om, u\ge 0, v\ge 0 \,\,\hbox{in \om},\quad
u=v=0 \,\,\hbox{on \partial\om}.{cases}{displaymath} Here \om\subset \R^N
is a smooth bounded domain, is the Sobolev critical
exponent, -\la_1(\om)0 and , where
\lambda_1(\om) is the first eigenvalue of with the Dirichlet
boundary condition. When \bb=0, this is just the well-known Brezis-Nirenberg
problem. The special case N=4 was studied by the authors in (Arch. Ration.
Mech. Anal. 205: 515-551, 2012). In this paper we consider {\it the higher
dimensional case }. It is interesting that we can prove the existence
of a positive least energy solution (u_\bb, v_\bb) {\it for any } (which can not hold in the special case N=4). We also study the limit
behavior of (u_\bb, v_\bb) as and phase separation is
expected. In particular, u_\bb-v_\bb will converge to {\it sign-changing
solutions} of the Brezis-Nirenberg problem, provided . In case
\la_1=\la_2, the classification of the least energy solutions is also
studied. It turns out that some quite different phenomena appear comparing to
the special case N=4.Comment: 48 pages. This is a revised version of arXiv:1209.2522v1 [math.AP
Excitation Thresholds for Nonlinear Localized Modes on Lattices
Breathers are spatially localized and time periodic solutions of extended
Hamiltonian dynamical systems. In this paper we study excitation thresholds for
(nonlinearly dynamically stable) ground state breather or standing wave
solutions for networks of coupled nonlinear oscillators and wave equations of
nonlinear Schr\"odinger (NLS) type. Excitation thresholds are rigorously
characterized by variational methods. The excitation threshold is related to
the optimal (best) constant in a class of discr ete interpolation inequalities
related to the Hamiltonian energy. We establish a precise connection among ,
the dimensionality of the lattice, , the degree of the nonlinearity
and the existence of an excitation threshold for discrete nonlinear
Schr\"odinger systems (DNLS).
We prove that if , then ground state standing waves exist if
and only if the total power is larger than some strictly positive threshold,
. This proves a conjecture of Flach, Kaldko& MacKay in
the context of DNLS. We also discuss upper and lower bounds for excitation
thresholds for ground states of coupled systems of NLS equations, which arise
in the modeling of pulse propagation in coupled arrays of optical fibers.Comment: To appear in Nonlinearit
Adiabatic limit and the slow motion of vortices in a Chern-Simons-Schr\"odinger system
We study a nonlinear system of partial differential equations in which a
complex field (the Higgs field) evolves according to a nonlinear Schroedinger
equation, coupled to an electromagnetic field whose time evolution is
determined by a Chern-Simons term in the action. In two space dimensions, the
Chern-Simons dynamics is a Galileo invariant evolution for A, which is an
interesting alternative to the Lorentz invariant Maxwell evolution, and is
finding increasing numbers of applications in two dimensional condensed matter
field theory. The system we study, introduced by Manton, is a special case (for
constant external magnetic field, and a point interaction) of the effective
field theory of Zhang, Hansson and Kivelson arising in studies of the
fractional quantum Hall effect. From the mathematical perspective the system is
a natural gauge invariant generalization of the nonlinear Schroedinger
equation, which is also Galileo invariant and admits a self-dual structure with
a resulting large space of topological solitons (the moduli space of self-dual
Ginzburg-Landau vortices). We prove a theorem describing the adiabatic
approximation of this system by a Hamiltonian system on the moduli space. The
approximation holds for values of the Higgs self-coupling constant close to the
self-dual (Bogomolny) value of 1. The viability of the approximation scheme
depends upon the fact that self-dual vortices form a symplectic submanifold of
the phase space (modulo gauge invariance). The theorem provides a rigorous
description of slow vortex dynamics in the near self-dual limit.Comment: Minor typos corrected, one reference added and DOI give
Quantifying Robotic Swarm Coverage
In the field of swarm robotics, the design and implementation of spatial
density control laws has received much attention, with less emphasis being
placed on performance evaluation. This work fills that gap by introducing an
error metric that provides a quantitative measure of coverage for use with any
control scheme. The proposed error metric is continuously sensitive to changes
in the swarm distribution, unlike commonly used discretization methods. We
analyze the theoretical and computational properties of the error metric and
propose two benchmarks to which error metric values can be compared. The first
uses the realizable extrema of the error metric to compute the relative error
of an observed swarm distribution. We also show that the error metric extrema
can be used to help choose the swarm size and effective radius of each robot
required to achieve a desired level of coverage. The second benchmark compares
the observed distribution of error metric values to the probability density
function of the error metric when robot positions are randomly sampled from the
target distribution. We demonstrate the utility of this benchmark in assessing
the performance of stochastic control algorithms. We prove that the error
metric obeys a central limit theorem, develop a streamlined method for
performing computations, and place the standard statistical tests used here on
a firm theoretical footing. We provide rigorous theoretical development,
computational methodologies, numerical examples, and MATLAB code for both
benchmarks.Comment: To appear in Springer series Lecture Notes in Electrical Engineering
(LNEE). This book contribution is an extension of our ICINCO 2018 conference
paper arXiv:1806.02488. 27 pages, 8 figures, 2 table
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