38 research outputs found
Stability of systems of fractional-order differential equations with caputo derivatives
Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting from the well-known Matignon’s results on stability of single-order systems, for which a different proof is provided together with a clarification of a limit case, the investigation is moved towards multi-order systems as well. Due to the key role of the Mittag–Leffler function played in representing the solution of linear systems of FDEs, a detailed analysis of the asymptotic behavior of this function and of its derivatives is also proposed. Some numerical experiments are presented to illustrate the main results
Positive Solutions for Multi-Order Nonlinear Fractional Systems
In this paper, we study the existence of positive solutions for a class of multi-order systems of fractional differential equations with nonlocal conditions. The main tool used is Schauder fixed point theorem and upper and lower solutions method. The results obtained are illustrated by a numerical example
Phase transitions in a holographic s+p model with backreaction
In a previous paper (arXiv:1309.2204, JHEP 1311 (2013) 087), we present a
holographic s+p superconductor model with a scalar triplet charged under an
SU(2) gauge field in the bulk. We also study the competition and coexistence of
the s-wave and p-wave orders in the probe limit. In this work we continue to
study the model by considering the full back-reaction The model shows a rich
phase structure and various condensate behaviors such as the "n-type" and
"u-type" ones, which are also known as reentrant phase transitions in condensed
matter physics. The phase transitions to the p-wave phase or s+p coexisting
phase become first order in strong back-reaction cases. In these first order
phase transitions, the free energy curve always forms a swallow tail shape, in
which the unstable s+p solution can also play an important role. The phase
diagrams of this model are given in terms of the dimension of the scalar order
and the temperature in the cases of eight different values of the back reaction
parameter, which show that the region for the s+p coexisting phase is enlarged
with a small or medium back reaction parameter, but is reduced in the strong
back-reaction cases.Comment: 15 pages(two-column), 9 figure
Krein-like extensions and the lower boundedness problem for elliptic operators
For selfadjoint extensions tilde-A of a symmetric densely defined positive
operator A_min, the lower boundedness problem is the question of whether
tilde-A is lower bounded {\it if and only if} an associated operator T in
abstract boundary spaces is lower bounded. It holds when the Friedrichs
extension A_gamma has compact inverse (Grubb 1974, also Gorbachuk-Mikhailets
1976); this applies to elliptic operators A on bounded domains.
For exterior domains, A_gamma ^{-1} is not compact, and whereas the lower
bounds satisfy m(T)\ge m(tilde-A), the implication of lower boundedness from T
to tilde-A has only been known when m(T)>-m(A_gamma). We now show it for
general T.
The operator A_a corresponding to T=aI, generalizing the Krein-von Neumann
extension A_0, appears here; its possible lower boundedness for all real a is
decisive. We study this Krein-like extension, showing for bounded domains that
the discrete eigenvalues satisfy
N_+(t;A_a)=c_At^{n/2m}+O(t^{(n-1+varepsilon)/2m}) for t\to\infty .Comment: 35 pages, revised for misprints and accepted for publication in
Journal of Differential Equation
Final value problems for parabolic differential equations and their well-posedness
This article concerns the basic understanding of parabolic final value
problems, and a large class of such problems is proved to be well posed. The
clarification is obtained via explicit Hilbert spaces that characterise the
possible data, giving existence, uniqueness and stability of the corresponding
solutions. The data space is given as the graph normed domain of an unbounded
operator occurring naturally in the theory. It induces a new compatibility
condition, which relies on the fact, shown here, that analytic semigroups
always are invertible in the class of closed operators. The general set-up is
evolution equations for Lax--Milgram operators in spaces of vector
distributions. As a main example, the final value problem of the heat equation
on a smooth open set is treated, and non-zero Dirichlet data are shown to
require a non-trivial extension of the compatibility condition by addition of
an improper Bochner integral.Comment: 39 pages. Revised version, with minor improvements. Essentially
identical to the accepted version, which appeared in Axioms on 9 May 201
Multiferroicity in plastically deformed SrTiO
A major challenge in the development of quantum technologies is to induce
additional types of ferroic orders into materials that exhibit other useful
quantum properties. Various techniques have been applied to this end, such as
elastically straining, doping, or interfacing a compound with other materials.
Plastic deformation introduces permanent topological defects and large local
strains into a material, which can give rise to qualitatively new
functionality. Here we show via local magnetic imaging that plastic deformation
induces robust magnetism in the quantum paraelectric SrTiO3, in both conducting
and insulating samples. Our analysis indicates that the magnetic order is
localized along dislocation walls and coexists with polar order along the
walls. The magnetic signals can be switched on and off in a controllable manner
with external stress, which demonstrates that plastically deformed SrTiO3 is a
quantum multiferroic. These results establish plastic deformation as a
versatile platform for quantum materials engineering