56 research outputs found
Construction of non-PT-symmetric complex potentials with all-real spectra
We review recent work on the generalization of PT symmetry. We show that, in
addition to PT-symmetric complex potentials, there are also large classes of
non-PT-symmetric complex potentials which also feature all-real spectra. In
addition, some classes of these non-PT-symmetric potentials allow phase
transitions which do or do not go through exceptional points. These
non-PT-symmetric potentials are constructed by a variety of methods, such as
the symmetry and supersymmetry methods and the soliton theory. A generalization
of PT symmetry in multi-dimensions is also reviewed.Comment: 22 pages, 6 figure
Exactly solvable Wadati potentials in the PT-symmetric Gross-Pitaevskii equation
This note examines Gross-Pitaevskii equations with PT-symmetric potentials of
the Wadati type: . We formulate a recipe for the construction of
Wadati potentials supporting exact localised solutions. The general procedure
is exemplified by equations with attractive and repulsive cubic nonlinearity
bearing a variety of bright and dark solitons.Comment: To appear in Proceedings of the 15 Conference on Pseudo-Hermitian
Hamiltonians in Quantum Physics, May 18-23 2015, Palermo, Italy (Springer
Proceedings in Physics, 2016
Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates
Solitons are among the most distinguishing fundamental excitations in a wide
range of non-linear systems such as water in narrow channels, high speed
optical communication, molecular biology and astrophysics. Stabilized by a
balance between spreading and focusing, solitons are wavepackets, which share
some exceptional generic features like form-stability and particle-like
properties. Ultra-cold quantum gases represent very pure and well-controlled
non-linear systems, therefore offering unique possibilities to study soliton
dynamics. Here we report on the first observation of long-lived dark and
dark-bright solitons with lifetimes of up to several seconds as well as their
dynamics in highly stable optically trapped Rb Bose-Einstein
condensates. In particular, our detailed studies of dark and dark-bright
soliton oscillations reveal the particle-like nature of these collective
excitations for the first time. In addition, we discuss the collision between
these two types of solitary excitations in Bose-Einstein condensates.Comment: 9 pages, 4 figure
An instability criterion for nonlinear standing waves on nonzero backgrounds
A nonlinear Schr\"odinger equation with repulsive (defocusing) nonlinearity
is considered. As an example, a system with a spatially varying coefficient of
the nonlinear term is studied. The nonlinearity is chosen to be repelling
except on a finite interval. Localized standing wave solutions on a non-zero
background, e.g., dark solitons trapped by the inhomogeneity, are identified
and studied. A novel instability criterion for such states is established
through a topological argument. This allows instability to be determined
quickly in many cases by considering simple geometric properties of the
standing waves as viewed in the composite phase plane. Numerical calculations
accompany the analytical results.Comment: 20 pages, 11 figure
Scattering Theory and -Symmetry
We outline a global approach to scattering theory in one dimension that
allows for the description of a large class of scattering systems and their
-, -, and -symmetries. In
particular, we review various relevant concepts such as Jost solutions,
transfer and scattering matrices, reciprocity principle, unidirectional
reflection and invisibility, and spectral singularities. We discuss in some
detail the mathematical conditions that imply or forbid reciprocal
transmission, reciprocal reflection, and the presence of spectral singularities
and their time-reversal. We also derive generalized unitarity relations for
time-reversal-invariant and -symmetric scattering
systems, and explore the consequences of breaking them. The results reported
here apply to the scattering systems defined by a real or complex local
potential as well as those determined by energy-dependent potentials, nonlocal
potentials, and general point interactions.Comment: Slightly expanded revised version, 38 page
Optical Lattices: Theory
This chapter presents an overview of the properties of a Bose-Einstein
condensate (BEC) trapped in a periodic potential. This system has attracted a
wide interest in the last years, and a few excellent reviews of the field have
already appeared in the literature (see, for instance, [1-3] and references
therein). For this reason, and because of the huge amount of published results,
we do not pretend here to be comprehensive, but we will be content to provide a
flavor of the richness of this subject, together with some useful references.
On the other hand, there are good reasons for our effort. Probably, the most
significant is that BEC in periodic potentials is a truly interdisciplinary
problem, with obvious connections with electrons in crystal lattices, polarons
and photons in optical fibers. Moreover, the BEC experimentalists have reached
such a high level of accuracy to create in the lab, so to speak, paradigmatic
Hamiltonians, which were first introduced as idealized theoretical models to
study, among others, dynamical instabilities or quantum phase transitions.Comment: Chapter 13 in Part VIII: "Optical Lattices" of "Emergent Nonlinear
Phenomena in Bose-Einstein Condensates: Theory and Experiment," edited by P.
G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonzalez (Springer
Series on Atomic, Optical, and Plasma Physics, 2007) - pages 247-26
PT-Symmetric Dimer in a Generalized Model of Coupled Nonlinear Oscillators
Abstract In the present work, we explore the case of a general PT -symmetric dimer in the context of two both linearly and nonlinearly coupled cubic oscillators. To obtain an analytical handle on the system, we first explore the rotating wave approximation converting it into a discrete nonlinear Schrödinger type dimer. In the latter context, the stationary solutions and their stability are identified numerically but also wherever possible analytically. Solutions stemming from both symmetric and anti-symmetric special limits are identified. A number of special cases are explored regarding the ratio of coefficients of nonlinearity between oscillators over the intrinsic one of each oscillator. Finally, the considerations are extended to the original oscillator model, where periodic orbits and their stability are obtained. When the solutions are found to be unstable their dynamics is monitored by means of direct numerical simulations
Dynamics of generalized PT-symmetric dimers with time-periodic gainâloss
A parity-time (PT)-symmetric system with periodically varying-in-time gain and loss modeled by two coupled Schrödinger equations (dimer) is studied. It is shown that the problem can be reduced to a perturbed pendulum-like equation. This is done by finding two constants of motion. Firstly, a generalized problem using Melnikov-type analysis and topological degree arguments is studied for showing the existence of periodic (libration), shift- periodic (rotation), and chaotic solutions. Then these general results are applied to the PT-symmetric dimer. It is interestingly shown that if a sufficient condition is satisfied, then rotation modes, which do not exist in the dimer with constant gainâloss, will persist. An approximate threshold for PT-broken phase corresponding to the disappearance of bounded solutions is also presented. Numerical study is presented accompanying the analytical results
A Sheep in Wolf's Clothing or a Gift from Heaven? Left-Left Coalitions in Comparative Perspective
Communist, post-Communist and radical socialist political parties have recently been brought into the coalition equation in a number of democratic states. This article seeks to develop and apply an analytical framework suitable for understanding under what conditions this occurs. It concentrates primarily on coalition formation at the sub-state level between social democratic parties and competitors to their ideological left. The framework is applied in the German and Spanish cases. The article concludes that there do indeed appear to be optimal sets of conditions facilitating the creation of left-left coalitions
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