Asymptotic behavior for second order lattice dynamical systems

We consider the existence of the global attractor for a second order lattice dynamical systems

Existence and Multiplicity of Traveling Waves in a Lattice Dynamical System

AbstractThis work proves the existence and multiplicity results of monotonic traveling wave solutions for some lattice differential equations by using the monotone iteration method. Our results include the model of cellular neural networks (CNN). In addition to the monotonic traveling wave solutions, non-monotonic and oscillating traveling wave solutions in the delay type of CNN are also obtained

Asymptotic Behaviour of a Logistic Lattice System

In this paper we study the asymptotic behaviour of solutions of a lattice dynamical system of a logistic type. Namely, we study a system of in nite ordinary di erential equations which can be obtained after the spatial discretization of a logistic equation with di usion. We prove that a global attractor exists in suitable weighted spaces of sequences

Nonlinearity, topology and PT symmetry in Photonic Lattices

Since over a decade, there is an ever-growing interest of scientific community towards the hot topics of so-called parity-time (PT ) symmetry and topological phases of matter, historically originating from non-Hermitian extensions of Quantum Mechanics and phase transitions without symmetry breaking in Condensed Matter Physics, respectively. Recent technological advancements in Photonics allowed one to study and fruitfully develop some of the most peculiar aspects of PT symmetry and topological matter on both theoretical and experimental levels. PT -symmetric photonic structures, with a judicious tailoring of gain and loss bulk regions, became a new paradigm in controlling the ow of light in unconventional manner, thereby paving the way to novel applications in laser physics, synthetic optical materials, optical sensing and so on. Likewise, fundamental ideas of topology rapidly emerged in the field of Photonics and brought about new possibilities for harnessing light, such as robust backscattering-free transport and Thouless pumping, to name a few. Owing to universality of the topological principles, a wide range of experimental platforms became feasible, including waveguides, metamaterials, optical crystals, optomechanics, silicon-based photonics, cavities and circuit QED. Most of the aspects of PT symmetry and topology in Photonics are very well understood in the linear regime, where light particles, photons, do not interact with each other. In contrast, up to date, they remain hardly explored in nonlinear optical regimes, characterized by self-interaction and self-localization of light in nonlinear media. In that regard, the aim of this thesis is to extend those powerful ideas further on in the direction of nonlinear light, in order to eventually discover and experimentally observe novel phenomena and interplays between nonlinearity, PT symmetry and topology. For that, we study both experimentally and theoretically 1D and 2D Discrete Photonic Lattices with synthetic dimensions, mimicking the celebrated Discrete Quantum Walks and experimentally based on the extremely versatile and interferometrically robust technique, called time-multiplexing. The set-ups essentially consist of optical fiber loops, mutually coupled via passive or active in-fiber beam splitters. In particular, we discover and experimentally observe novel and fascinating aspects of non-Hermitian discrete solitons in PT -symmetric environments and topological chiral edge states under the action of optical Kerr nonlinearity.Bereits seit über einem Jahrzehnt besteht ein stetig wachsendes Interesse der Wissenschaftsgemeinde an den hochaktuellen Themen von sogenannter Parität-Zeit (PT ) Symmetrie und topologischen Phasen der Materie, die historisch aus nicht-hermitischen Erweiterungen der Quantenmechanik bzw. Phasenübergängen ohne Symmetrieeinbruch in der Physik der kondensierten Materie stammen. Jüngste technologische Fortschritte im Forschungsfeld der Photonik ermöglichten es, einige der besondersten Aspekte der PT Symmetrie und der topologischen Materie sowohl theoretisch als auch experimentell zu untersuchen und weiter zu entwickeln. PT -symmetrische, photonische Strukturen mit einer gezielten Anpassung von Verstärkungs- und Verlustbereichen wurden zu einem neuen Paradigma für die unkonventionelle Steuerung des Lichtusses, und ebneten damit den Weg für neuartige Anwendungen in der Laserphysik, der synthetischen optischen Materialien, Lichtsensoren und so weiter. Ebenso entstanden im Bereich der Photonik schnell grundlegende Topologie-Ideen, die neue Möglichkeiten der Lichtnutzung eröffneten, wie zum Beispiel einen robusten, rückstreuungsfreien Transport und ein Thouless Pumping, um nur Einige zu nennen. Aufgrund der Universalität der topologischen Prinzipien wurde eine Vielzahl von experimentellen optischen Plattformen realisierbar, darunter optischeWellenleiter, Metamaterialien, optische Kristalle, Optomechanik, Photonik auf Siliziumbasis, optische Resonatoren und Schaltkreis-QED. Die meisten Aspekte der PT Symmetrie und Topologie sind im linearen Bereich, in dem Lichtteilchen, Photonen, nicht miteinander interagieren, sehr gut erforscht. Im Gegensatz dazu sind sie in nichtlinearen optischen Systemen, die durch Selbstinteraktion und Selbstlokalisierung von Licht in nichtlinearen Medien gekennzeichnet sind, bislang kaum erforscht. In dieser Hinsicht ist das Ziel dieser Arbeit, diese einussreichen Ideen in Richtung nichtlineares Licht weiter auszudehnen, um schließlich neue Phänomene und Wechselwirkungen zwischen Nichtlinearität, PT Symmetrie und Topologie zu entdecken und experimentell zu beobachten. Dazu untersuchen wir sowohl experimentell als auch theoretisch diskrete, photonische 1D- und 2D-Gitter mit synthetischen Dimensionen, die die berühmten diskreten Quantenwanderungen imitieren und auf der extrem vielseitigen und interferometrisch robusten experimentellen Technik namens Zeitmultiplex basieren. Die experimentellen Aufbauten bestehen im Wesentlichen aus Lichtleiterschleifen, die über passive oder aktive Lichtleiterstrahlteiler miteinander gekoppelt sind. Unsere besonderen Aktivitäten betreffen die Entdeckung und experimentelle Beobachtung neuartiger und faszinierender Aspekte nicht-hermitischer diskreter Solitonen in PT -symmetrischen Umgebungen und topologischen chiralen Randzuständen unter der Einwirkung optischer Kerr-Nichtlinearität

Attractors for Stochastic Lattice Dynamical Systems with a Multiplicative Noise

In this paper, we consider a stochastic lattice di®erential equation with di®usive nearest neighbor interaction, a dissipative nonlinear reaction term, and a multiplicative white noise at each node. We prove the existence of a compact global random attractor which pulled back attracts tempered random bounded sets

On Differential Equations with Delay in Banach Spaces and Attractors for Retarded Lattice Dynamical Systems

In this paper we first prove a rather general theorem about existence of solutions for an abstract differential equation in a Banach space by assuming that the nonlinear term is in some sense weakly continuous. We then apply this result to a lattice dynamical system with delay, proving also the existence of a global compact attractor for such system

Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities

In this paper we study the asymptotic behavior of solutions of a first-order stochastic lattice dynamical system with a multiplicative noise. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true. Using the theory of multi-valued random dynamical systems we prove the existence of a random compact global attracto

Colloquium: Ice rule and emergent frustration in particle ice and beyond

Geometric frustration and the ice rule are two concepts that are intimately connected and widespread across condensed matter. The first refers to the inability of a system to satisfy competing interactions in the presence of spatial constraints. The second, in its more general sense, represents a prescription for the minimization of the topological charges in a constrained system. Both can lead to manifolds of high susceptibility and non-trivial, constrained disorder where exotic behaviors can appear and even be designed deliberately. In this Colloquium, we describe the emergence of geometric frustration and the ice rule in soft condensed matter. This Review excludes the extensive developments of mathematical physics within the field of geometric frustration, but rather focuses on systems of confined micro- or mesoscopic particles that emerge as a novel paradigm exhibiting spin degrees of freedom. In such systems, geometric frustration can be engineered artificially by controlling the spatial topology and geometry of the lattice, the position of the individual particle units, or their relative filling fraction. These capabilities enable the creation of novel and exotic phases of matter, and also potentially lead towards technological applications related to memory and logic devices that are based on the motion of topological defects. We review the rapid progress in theory and experiments and discuss the intimate physical connections with other frustrated systems at different length scales

Advances in honeycomb layered oxides: Part II -- Theoretical advances in the characterisation of honeycomb layered oxides with optimised lattices of cations

The quest for a successful condensed matter theory that incorporates diffusion of cations, whose trajectories are restricted to a honeycomb/hexagonal pattern prevalent in honeycomb layered materials is ongoing, with the recent progress discussed herein focusing on symmetries, topological aspects and phase transition descriptions of the theory. Such a theory is expected to differ both qualitatively and quantitatively from 2D electron theory on static carbon lattices, by virtue of the dynamical nature of diffusing cations within lattices in honeycomb layered materials. Herein, we have focused on recent theoretical progress in the characterisation of pnictogen- and chalcogen-based honeycomb layered oxides with emphasis on hexagonal/honeycomb lattices of cations. Particularly, we discuss the link between Liouville conformal field theory to expected experimental results characterising the optimal nature of the honeycomb/hexagonal lattices in congruent sphere packing problems. The diffusion and topological aspects are captured by an idealised model, which successfully incorporates the duality between the theory of cations and their vacancies. Moreover, the rather intriguing experimental result that a wide class of silver-based layered materials form stable Ag bilayers, each comprising a pair of triangular sub-lattices, suggests a bifurcation mechanism for the Ag triangular sub-lattices, which ultimately requires conformal symmetry breaking within the context of the idealised model, resulting in a cation monolayer-bilayer phase transition. Other relevant experimental, theoretical and computational techniques applicable to the characterisation of honeycomb layered materials have been availed for completeness.Comment: 93 pages, 21 figures, 4 tables, title updated, table of contents adde

Airborne topological acoustics

Advances in topological acoustics are leading to potential development for noise attenuation, ultrasonic imaging, sound manipulation, and information delivering, etc. Recently, ideas and methodologies from condensed-matter physics, such as the quantum Hall effect (QHE), the quantum spin Hall effect (QSHE), and the quantum valley Hall effect (QVHE), combined with configurations of sonic crystals and metamaterials, have been investigated in manipulating acoustic transmissions in the form of one-way edge modes and defect-immune protected acoustics. However, many related studies are still in their infancy and mostly rely on bulky, noisy, overly complicated, untunable and narrow-band-effective facilities, and so it is highly desirable but challenging to design more practical topological acoustic systems, with backscattering immune, tunable, broadband and miniaturized topological acoustic properties. This thesis investigates novel modulation mechanisms, versatile configurable lattice structures, and microscale acoustic transmission mechanisms to solve the aforementioned airborne topological acoustic challenges. Starting with the rotating modified spiral springs configuration adjusting the inner radius without altering the external lattice structure, a gapless topologically protected acoustic flow-free resonator system based on the QVHE in reconfigurable sonic crystals is designed to realize backscattering immune, tunable and broadband functional acoustic applications. Then, based on the acoustic analogue of the QHE, to replace the generating mechanism of the noisy fan-induced airflow, a new method using heat-induced natural convection coupled with an acoustic circulator is proposed to realize robust nonreciprocal acoustic propagation. This strategy is more feasible because of its dynamic control and versatile topological structures in the absence of moving parts. To further promote the topological acoustics into a more practical stage, based on the QSHE, a temperature modulation scheme is designed to demonstrate that the Floquet topological insulators with thermal-induced impedance matching can realize robust topological acoustic propagation, which is especially useful for noiseless and miniaturized airborne acoustics. Thermal modulation enables miniaturized topological airborne acoustics to the millimeter scale or even smaller. Additionally, a theoretical model with a second-order slip boundary to iii describe acoustic wave propagation in micro- and nanochannels is proposed to investigate the miniaturized topological acoustic transmission mechanism. Based on the molecular-based direct simulation Monte Carlo (DSMC) method, this model provides an analytical solution beneficial for topological acoustics in ultrasonic or in miniaturized structures