10 research outputs found
Dedication to Professor Michael Tribelsky
Professor Tribelsky's accomplishments are highly appreciated by the international community. The best indications of this are the high citation rates of his publications, and the numerous awards and titles he has received. He has made numerous fundamental contributions to an extremely broad area of physics and mathematics, including (but not limited to) quantum solid-state physics, various problems in light–matter interaction, liquid crystals, physical hydrodynamics, nonlinear waves, pattern formation in nonequilibrium systems and transition to chaos, bifurcation and probability theory, and even predictions of the dynamics of actual market prices. This book presents several extensions of his results, based on his inspiring publications
Solitons in nonlinear lattices
This article offers a comprehensive survey of results obtained for solitons
and complex nonlinear wave patterns supported by purely nonlinear lattices
(NLs), which represent a spatially periodic modulation of the local strength
and sign of the nonlinearity, and their combinations with linear lattices. A
majority of the results obtained, thus far, in this field and reviewed in this
article are theoretical. Nevertheless, relevant experimental settings are
surveyed too, with emphasis on perspectives for implementation of the
theoretical predictions in the experiment. Physical systems discussed in the
review belong to the realms of nonlinear optics (including artificial optical
media, such as photonic crystals, and plasmonics) and Bose-Einstein
condensation (BEC). The solitons are considered in one, two, and three
dimensions (1D, 2D, and 3D). Basic properties of the solitons presented in the
review are their existence, stability, and mobility. Although the field is
still far from completion, general conclusions can be drawn. In particular, a
novel fundamental property of 1D solitons, which does not occur in the absence
of NLs, is a finite threshold value of the soliton norm, necessary for their
existence. In multidimensional settings, the stability of solitons supported by
the spatial modulation of the nonlinearity is a truly challenging problem, for
the theoretical and experimental studies alike. In both the 1D and 2D cases,
the mechanism which creates solitons in NLs is principally different from its
counterpart in linear lattices, as the solitons are created directly, rather
than bifurcating from Bloch modes of linear lattices.Comment: 169 pages, 35 figures, a comprehensive survey of results on solitons
in purely nonlinear and mixed lattices, to appear in Reviews of Modern
Physic
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
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Excitations in superfluids of atoms and polaritons
This thesis is devoted to the study of excitations in atomic and polariton Bose-Einstein condensates (BEC). These two specimens are prime examples for equilibrium and non equilibrium BEC. The corresponding condensate wave function of each system satisfies a particular partial differential equation (PDE). These PDEs are discussed in the beginning of this thesis and justified in the context of the quantum many-body problem. For high occupation numbers and when neglecting quantum fluctuations the quantum field operator simplifies to a semiclassical wave. It turns out that the interparticle interactions can be simplified to a single parameter, the scattering length, which gives rise to an effective potential and introduces a nonlinearity to the PDE. In both cases, i.e. equilibrium and non equilibrium, the main model corresponding to the semiclassical wave is the Gross-Pitaevskii equation (GPE), which includes certain mathematical adaptions depending on the physical context of the consideration and the nature of particles/quasiparticles, such as additional complex pumping and growth terms or terms due to motion. In the course of this work I apply a variety of state-of-the-art analytical and numerical tools to gain information about these semiclassical waves. The analytical tools allow e.g. to determine the position of the maximum density of the condensate wave function or to find the critical velocities at which excitations are expected to be generated within the condensate. In addition to analytical considerations I approximate the GPE numerically. This allows to gain the condensate wave function explicitly and is often a convenient tool to study the emergence of excitations in BEC. It is in particular shown that the form of the possible excitations significantly depends on the dimensionality of the considered system. The generated excitations within the BEC include quantum vortices, quantum vortex rings or solitons. In addition multicomponent systems are considered, which enable more complex dynamical scenarios. Under certain conditions imposed on the condensate one obtains dark-bright soliton trains within the condensate wave function. This is shown numerically and analytical expressions are found as well. In the end of this thesis I present results as part of an collaborative effort with a group of experimenters. Here it is shown that the wave function due to a complex GPE fits well with experiments made on polariton condensates, statically and dynamically.F.P. acknowledges financial support by the UK Engineering and Physical Sciences Re- search Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis and a KAUST gran
Theoretical Concepts of Quantum Mechanics
Quantum theory as a scientific revolution profoundly influenced human thought about the universe and governed forces of nature. Perhaps the historical development of quantum mechanics mimics the history of human scientific struggles from their beginning. This book, which brought together an international community of invited authors, represents a rich account of foundation, scientific history of quantum mechanics, relativistic quantum mechanics and field theory, and different methods to solve the Schrodinger equation. We wish for this collected volume to become an important reference for students and researchers
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Elliptic partial differential equations from an elementary viewpoint
These notes are the outcome of some courses taught to undergraduate and
graduate students from the University of Western Australia, the Pontif\'{\i}cia
Universidade Cat\'olica do Rio de Janeiro, the Indian Institute of Technology
Gandhinagar and the Ukrainian Catholic University in 2021 and 2022