Linear Lattice Research Articles

One-dimensional gap solitons in quintic and cubic–quintic fractional nonlinear Schrödinger equations with a periodically modulated linear potential

Competing nonlinearities, such as the cubic (Kerr) and quintic nonlinear terms whose strengths are of opposite signs (the coefficients in front of the nonlinearities), exist in various physical media (in particular, in optical and matter-wave media). A benign competition between self-focusing cubic and self-defocusing quintic nonlinear nonlinearities (known as cubic–quintic model) plays an important role in creating and stabilizing the self-trapping of D-dimensional localized structures, in the contexts of standard nonlinear Schrödinger equation. We incorporate an external periodic potential (linear lattice) into this model and extend it to the space-fractional scenario that begins to surface in very recent years—the nonlinear fractional Schrödinger equation (NLFSE), therefore obtaining the cubic–quintic or the purely quintic NLFSE, and investigate the propagation and stability properties of self-trapped modes therein. Two types of one-dimensional localized gap modes are found, including the fundamental and dipole-mode gap solitons. Employing the techniques based on the linear-stability analysis and direct numerical simulations, we get the stability regions of all the localized modes; and particularly, the anti-Vakhitov–Kolokolov criterion applies for the stable portions of soliton families generated in the frameworks of quintic-only nonlinearity and competing cubic–quintic nonlinear terms.

Asymptotic homogenization models for pantographic lattices with variable order rotational resistance at pivots

In this paper, different asymptotic orders of torsional resistance at the pivots for linear pantographic lattices are introduced within an asymptotic homogenization of the discrete micro-beam model associated with the summation of internal forces and moments at the pivots. The small parameter is defined by the ratio of the square root of a unit cell size for the lattice and the problem domain size. The objective is to systematically identify leading order continuum field equations and to define homogenized elasticity tensors and their properties for increasing rotational torque stiffness. To measure the effect of the torsional spring stiffness on the deformation of the lattice, the torsional resistance of a beam element between pivots which compares the torque stiffness to its bending moment stiffness is introduced as a power of the small parameter. Consistent with previous models for pantographic sheets with zero (perfect) or light torsional resistance, an asymptotic expansion with the small scaling parameter shows that the leading order continuum field equations are characterized by second gradient elasticity theory. The homogenized elasticity coefficients are determined in terms of the fiber characteristics and the torque stiffness. A shear modulus is calculated proportional to the torque stiffness. By increasing the torsional resistance to order one, the resulting field equations move to first-gradient classical elasticity with a shear modulus varying as a homographic function in the torsional resistance. By increasing torsional stiffness to higher orders, the shear modulus is obtained for the limiting case of rigid connections. Numerical results are presented for standard elongation and shear bias tests. Strain energies from the first gradient and second gradient terms present for the models with different orders of torsional resistance are computed to demonstrate the zero and infinite limiting cases, and the need to move from a second-gradient continuum model with low-order torsional resistance as scaled with the small parameter, to a first-gradient model for high-orders of torsional resistance.

Investigation of the lattice defects density and optical characteristics for the anatase phase of titanium dioxide nanocrystalline films

The spin-coating sol-gel method was used and the nanocrystallines of titanium dioxide were deposited on the ordinary glass slide substrates. The films were annealed in air at temperature range between 350 and \(550\,\,^{\circ }\hbox {C}\) and the scanning electron microscopy images of them were analyzed to study the morphology as well as the thickness of films in terms of shape and size distribution behavior of embedded nanoparticles. The behavior of distribution functions was applied; the X-ray diffraction patterns were simulated and compared to the measured \(\hbox {TiO}_2\) peak profiles, using the advanced X-ray diffraction analysis. The volume weighted average crystallite size and the density of linear lattice defects were investigated as a function of annealing temperature. The transmittance spectra of our \(\hbox {TiO}_2\) films were taken by spectrophotometer and the film thicknesses as well as the optical energy gap were calculated. It is found that the film thickness of our samples is in the range of 343–397 nm which is in agreement with the SEM-cross section results. It is also seen that with the elevation of annealing temperature, the energy gap of films decreases. Finally, the correlation between the optical band gap and microstructure parameters (i.e., crystallite size, dislocation density as well as the strain along the z direction) and optical characteristics was studied.

Resilience for stochastic systems interacting via a quasi-degenerate network.

A stochastic reaction-diffusion model is studied on a networked support. In each patch of the network, two species are assumed to interact following a non-normal reaction scheme. When the interaction unit is replicated on a directed linear lattice, noise gets amplified via a self-consistent process, which we trace back to the degenerate spectrum of the embedding support. The same phenomenon holds when the system is bound to explore a quasidegenerate network. In this case, the eigenvalues of the Laplacian operator, which governs species diffusion, accumulate over a limited portion of the complex plane. The larger the network, the more pronounced the amplification. Beyond a critical network size, a system deemed deterministically stable, hence resilient, can develop seemingly regular patterns in the concentration amount. Non-normality and quasidegenerate networks may, therefore, amplify the inherent stochasticity and so contribute to altering the perception of resilience, as quantified via conventional deterministic methods.

Avoiding Brooms, Forks, and Butterflies in the Linear Lattices

Let $n$ be a positive integer, $q$ a power of a prime, and $\mathcal{L}_n(q)$ the poset of subspaces of an $n$-dimensional vector space over a field with $q$ elements. This poset is a normalized matching poset and the set of subspaces of dimension $\left\lfloor n/2 \right\rfloor$ or those of dimension $\left\lceil n/2 \right\rceil$ are the only maximum-sized anti-chains in this poset. Strengthening this well-known and celebrated result, we show that, except in the case of $\mathcal{L}_3(2)$, these same collections of subspaces are the only maximum-sized families in $\mathcal{L}_n(q)$ that avoid both a $\wedge$ and a $\vee$ as a subposet. We generalize some of the results to brooms and forks, and we also show that the union of the set of subspaces of dimension $k$ and $k+1$, for $k = \left\lfloor n/2 \right\rfloor$ or $k = \left\lceil n/2 \right\rceil-1$, are the only maximum-sized families in $\mathcal{L}_n(q)$ that avoid a butterfly (definitions below).

Some isomorphic big lattices and some properties of lattice preradicals

According to Albu and Iosif, [2, Definition 1.1] a lattice preradical is a subfunctor of the identity functor on the category [Formula: see text] of linear modular lattices, whose objects are the complete modular lattices and whose morphisms are linear morphisms. In this paper, we describe some big lattices which are isomorphic to the big lattice of lattice preradicals and we study the four classical operations that occur in the lattice of preradicals of modules over a ring [Formula: see text], namely, the join, the meet, the product and the coproduct. We show that some results about the lattice of module preradicals can be extended to the lattice of lattice preradicals. In particular, we show the existence of the equalizer, the annihilator, the coequalizer and the totalizer for a lattice preradical [Formula: see text], as well as some of their properties.

Hydroelastic damping of low aspect ratio cantilevered plates

This study considers the hydroelastic damping of cantilevered flat plates undergoing free vibration in flowing water. Of particular interest are plates with aspect ratios (defined as span length divided by chord length) less than one. Experimental trials involving plates of aspect ratios from 0.3 to 0.7 are conducted in a high-speed water tunnel. The tests involve flow velocities well below those corresponding to hydroelastic instability. Additionally, an unsteady, linear vortex lattice model is coupled to a structural dynamic plate model to predict flow-induced damping as a function of flow speed. The numerical model has been previously used with air, but never with water as the fluid medium. The model is able to predict the hydroelastic damping associated with the first several plate modes with less than 30% error in most cases. While the hydroelastic damping remains linear in the experimental flow regime, at higher flow speeds more complicated hydroelastic damping behavior is predicted. Numerical studies involving mass ratio, aspect ratio, and reduced velocity are conducted to predict hydroelastic damping across a wide parameter space. All else being equal, chordwise bending modes are found to exhibit two to three times greater hydroelastic damping than spanwise bending modes. There is also an inverse relationship between mass ratio and hydroelastic damping. Further, for plates with low aspect ratios and at fixed reduced velocities, hydroelastic damping increases with increasing aspect ratio.

Self-induced topological transition in phononic crystals by nonlinearity management

A new design paradigm of topology has recently emerged to manipulate the flow of phonons. At its heart lies a topological transition to a nontrivial state with exotic properties. This framework has been limited to linear lattice dynamics so far. Here we show a topological transition in a nonlinear regime and its implication in emerging nonlinear solutions. We employ nonlinearity management such that the system consists of masses connected with two types of nonlinear springs, "stiffening" and "softening" types, alternating along the length. We show, analytically and numerically, that the lattice makes a topological transition simply by changing the excitation amplitude and invoking nonlinear dynamics. Consequently, we witness the emergence of a new family of finite-frequency edge modes, not observed in linear phononic systems. We also report the existence of kink solitons at the topological transition point. These correspond to heteroclinic orbits that form a closed curve in the phase portrait separating the two topologically-distinct regimes. These findings suggest that nonlinearity can be used as a strategic tuning knob to alter topological characteristics of phononic crystals. These also provide fresh perspectives towards understanding a new family of nonlinear solutions in light of topology.

Quantum variational principle and quantum multiform structure: The case of quadratic Lagrangians

A modern notion of integrability is that of multidimensional consistency (MDC), which classically implies the coexistence of (commuting) dynamical flows in several independent variables for one and the same dependent variable. This property holds for both continuous dynamical systems as well as for discrete ones defined in discrete space-time. Possibly the simplest example in the discrete case is that of a linear quadrilateral lattice equation, which can be viewed as a linearised version of the well-known lattice potential Korteweg-de Vries (KdV) equation. In spite of the linearity, the MDC property is non-trivial in terms of the parameters of the system. The Lagrangian aspects of such equations, and their nonlinear analogues, has led to the notion of Lagrangian multiform structures, where the Lagrangians are no longer scalar functions (or volume forms) but genuine p-forms in a multidimensional space of independent variables. The variational principle involves variations not only with respect to the field variables, but also with respect to the geometry in the space of independent variables. In this paper we consider a quantum analogue of this new variational principle by means of quantum propagators (or equivalently Feynman path integrals). In the case of quadratic Lagrangians these can be evaluated in terms of Gaussian integrals. We study also periodic reductions of the lattice leading to discrete multi-time dynamical commuting mappings, the simplest example of which is the discrete harmonic oscillator, which surprisingly reveals a rich integrable structure behind it. On the basis of this study we propose a new quantum variational principle in terms of multiform path integrals.

Perfect localization on flat-band binary one-dimensional photonic lattices

The existence of flat bands is generally thought to be physically possible only for dimensions larger than one. However, by exciting a system with different orthogonal states this condition can be reformulated. In this work, we demonstrate that a one-dimensional binary lattice supports always a trivial flat band, which is formed by isolated single-site vertical dipolar states. These flat band modes correspond to the highest localized modes for any discrete system, without the need of any aditional mechanism like, e.g., disorder or nonlinearity. By fulfilling a specific relation between lattice parameters, an extra flat band can be excited as well, with modes composed by fundamental and dipolar states that occupy only three lattice sites. Additionally, by inspecting the lattice edges, we are able to construct analytical Shockley surface modes, which can be compact or present staggered or unstaggered tails. We believe that our proposed model could be a good candidate for observing transport and localization phenomena on a simple one-dimensional linear photonic lattice.

Certain aspects of the acoustics of a strongly nonlinear discrete lattice

The aim of this paper is to analytically investigate certain aspects of the nonlinear acoustics, especially traveling breathers, of a one-dimensional, non-dissipative, essentially nonlinear lattice. The considered semi-infinite lattice is formed by coupling linearly grounded single-degree-of-freedom oscillators by means of essentially nonlinear stiffnesses with purely cubic force–displacement characteristics and is subject to an impulsive excitation at its free boundary. In the limit of small energy, the complexification-averaging method is employed to analytically prove the existence of a nonlinear, energy-dependent propagation zone (PZ) and two complementary attenuation zones (AZs) in frequency–energy domain of the corresponding infinite lattice (i.e., the one without a free boundary). Excitations within the PZ yield waves that propagate in the far field of the lattice, whereas excitations within either one of the AZs generate exponentially decaying, spatially localized, standing wave close to the excitation site. Therefore, unlike in the linear case nonlinear lattices possess energy-tunable filtering properties. After determining the region in the frequency–energy domain where wave (and energy) propagation within the lattice is allowed, the excitation of traveling breathers in the semi-infinite lattice is studied. It is analytically and numerically shown that such breathers can exist only close to, but outside of the upper boundary of the PZ of the lattice in frequency–energy domain. Moreover, due to the localized nature of the traveling breather, each oscillator experiences a continuous and smooth change in its total energy as the breather passes. It is analytically shown and explained that this smooth energy transition results in a similar form of transition in the instantaneous frequency of each oscillator.

First-passage properties of mortal random walks: Ballistic behavior, effective reduction of dimensionality, and scaling functions for hierarchical graphs.

We consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites. The configuration comprising the graph, the starting point of the walk, and the locations of the trap sites is taken to be exactly self-similar as one goes from one generation of the family to the next. Under these circumstances, the total probability that the walker hits a trap is determined exactly as a function of the single-step survival probability q of the mortal walker. On the nth generation graph of the family, this probability is shown to be given by the nth iterate of a certain scaling function or map q→f(q). The properties of the map then determine, in each case, the behavior of the trapping probability, the mean time to trapping, the temporal scaling factor governing the random walk dimension on the graph, and other related properties. The formalism is illustrated for the cases of a linear hierarchical lattice and the Sierpinski graphs in two and three Euclidean dimensions. We find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint. The relevance of this finding for experiments involving travel times of particles in diffusion-decay systems is discussed.

Wigner entropy production and heat transport in linear quantum lattices

When a quantum system is coupled to several heat baths at different temperatures, it eventually reaches a non-equilibrium steady state featuring stationary internal heat currents. These currents imply that entropy is continually being produced in the system at a constant rate. In this paper we apply phase-space techniques to the calculation of the Wigner entropy production on general linear networks of harmonic nodes. Working in the ubiquitous limit of weak internal coupling and weak dissipation, we obtain simple closed-form expressions for the entropic contribution of each individual quasi-probability current. Our analysis highlights the essential role played by the internal unitary interactions (node-node couplings) in maintaining a non-equilibrium steady-state and hence a finite entropy production rate. We also apply this formalism to the paradigmatic problem of energy transfer through a chain of oscillators subject to self-consistent internal baths that can be used to tune the transport from ballistic to diffusive. We find that the entropy production scales with different power law behaviors in the ballistic and diffusive regimes, hence allowing us to quantify what is the "entropic cost of diffusivity."

Lattice Dynamical Systems Associated with a Fractional Laplacian

We derive optimal well-posedness results and explicit representations of solutions in terms of special functions for the linearized version of the equationfor some constant where denotes the Caputo fractional derivative in time of order β and denotes the discrete fractional Laplacian of order We also prove a comparison principle. A special case of this equation is the discrete Fisher- equation with and without delay. We show that if for every and the function is concave on f(x, s) is nonnegative for every and satisfiesfor some then the system has a nonnegative unique solution u satisfying for every and Our results include cubic nonlinearities and incorporate new results for the discrete Newell-Whitehead-Segel equation. We use Lévy stable processes as well as Mittag-Leffler, Wright and modified Bessel functions to describe the solutions of the linear lattice model, providing a useful framework for further study. For the nonlinear model, we use a generalization of the upper–lower solution method for reaction–diffusion equations in order to prove existence and uniqueness of solutions.

Ultra-High bit rate all-optical AND/OR logic gates based on photonic crystal with multi-wavelength simultaneous operation

ABSTRACTIn this paper, a novel two-dimensional photonic crystal based all-optical AND/OR logic gates are designed, simulated and optimized. The structure is built on a linear square lattice photonic crystal platform. A multi-wavelength operation, together with a simultaneous operation, is achieved at ultra-high bit rates. The concurrent operation is attained without altering the proposed design continuously, as stated in the literature. It provides simplicity because there is no auxiliary input required along with the absence of externally attached phase shift units. The enhancement process has been done to the rod radius. A magnificent representation tool is developed. The benefit of the mentioned tool lies in the data combination of different operating wavelengths, contrast ratio, and bit rate; which will establish an efficient optimization process. Each gate is enhanced independently, then an overall improvement has been done. As a result, the operation at 1.52 µm will provide a successful multi logic gate operation with ultra-high bit rates of 6.76 and 4.74 Tbit/s for AND and OR logic gates, respectively. The design has an acceptable size of (19.8 × 12.6 µm) and a contrast ratio of 9.74 dB and 17.95 dB for the designed AND and OR gate, respectively. The design is highly sensitive to the waveguide length to verify the gates on demand.

Tunable photoluminescence in Ba1-xSrxSi3O4N2: Eu2+/ Ce3+, Li+ solid solution phosphors induced by linear structural evolution

A series of oxynitride phosphors, Ba1-xSrxSi3O4N2: Eu2+/ Ce3+, Li+ (0 ≤ x ≤ 0.5) have been synthesized through a solid state reaction method at a high temperature. The influence of Sr2+-doping on the structure and photoluminescence properties of Ba1-xSrxSi3O4N2: Eu2+/ Ce3+, Li+ was studied systemically. All of the samples were found to retain their hexagonal crystal structures with the linear lattice shrinking upon the increase of Sr2+ concentration, indicating that Ba2+ is substituted by Sr2+ to form the intermediate solid-solution compositions of Ba1-xSrxSi3O4N2. It is noteworthy that red shifts of about 30 and 7 nm in the emission spectra of Eu2+ and Ce3+-activated samples were observed, respectively, through the Sr substitution at Ba sites from 0 to 0.5, originating from the increment of crystal field strength and Stokes shift. The thermal quenching stability, fluorescence decay behavior and CIE values were discussed and compared. Based on these results, the Ba1-xSrxSi3O4N2: Eu2+ phosphors are considered to be potential candidate phosphors applicable to n-UV LED for solid-state lighting.

A Molecular Dynamics Study on the Effects of Lattice Defects on the Phase Transformation from BCC to FCC Structures

Molecular dynamics simulations of the phase transformation from body- centered-cubic (bcc) to face-centered-cubic (fcc) structures were performed. A Morse-type function was applied, and the parameters were determined so that both fcc and bcc structures were stable for the perfectcrystal model. When the fcc structure was superior to the bcc structure, the bcc model transformed to fcc. Two mechanisms, based on the Bain and Nishiyama- Wasserman (NW) relationships, were considered. Then, point or linear lattice defects, i.e., randomly scattered or regularly aligned vacancies, were introduced. Consequently, bcc models tended to transform to an fcc structure, whereas fcc models remained stable. The transformation process was also investigated in detail. BCC-to-FCC transformation is often considered as a homogeneous process based on changes in the axis lengths, and such a process was observed for the perfectcrystal model. Conversely, for the defect models, local heterogeneous deformation patterns, including cylindrical domain and planar interface formation, were observed. These behaviors are considered to be related to plastic deformation during phase transformation, and the validity of the presented model for further investigation was confirmed.

Three-dimensional solitons in Bose-Einstein condensates with spin-orbit coupling and Bessel optical lattices

Applying a variational approach and numerical analysis to the system of Gross-Pitaevskii equations, we find three-dimensional (3D) stable solitons in binary atomic Bose-Einstein condensates with spin-orbit coupling (SOC) and out-of-phase linear and nonlinear Bessel optical lattices. We discuss the stability of 3D solitons by utilizing their norm and energy. The introduction of Bessel potentials makes the evolution and collisions of solitons more stable and improves their resistance to collapse. Depending on the strength of the intra- and intercomponent spatial modulation of the nonlinearity and SOC, we find stable solitons of the semivortex and mixed-mode structures. Furthermore, we show that the solitons are stable against small perturbations in propagation and collisions.

Making yttrium orthovanadate a better color emission host: Case study of hollow-like nanocrystals

Novel Eu3+-doped YVO4 hollow-like nanocrystals were constructed via initiating homogeneous precipitation followed by a hydrothermal treatment. The phase structure, microstructure, band gap, and fluorescence performance of the hollow-like nanocrystals were characterized by X-ray diffraction (XRD), transmission electron microscopy (TEM), Uv–Vis diffusion reflectance spectra (Uv–Vis), and photoluminescence spectroscopy (PL). The results showed that all prepared samples were well crystallized in tetragonal YVO4 structure regardless of the doping concentration of Eu3+ ions from 0 to 15 mol%. TEM showed that all samples are composed of hollow-like nanocrystals with average diameters of about 20 nm, exhibiting a large-scale uniform morphology. A linear lattice expansion was observed due to evenly distributed of Eu3+ ions in the YVO4 crystal lattice matrix. The bandgap energy Eg showed a red-shift below the doping level of 3 mol% of Eu3+ ions. More importantly, YVO4 nanocrystals with hollow-like structure have a higher quenching Eu3+ ions concentration up to 10 mol%, compared to 5 mol% of the bulk YVO4 host. Furthermore, the tunable color emission from blue to white to red can be achieved in hollow-like YVO4 nanocrystals by controlling the doping concentration of Eu3+ ions. The nanocrystals with hollow-like structure reported in this work are very important, which may pave a new route to better the yttrium orthovandate host and further realize color tunable emission for laser, optoelectronics, and biomedical applications.

Analytic study on optical solitons in parity-time-symmetric mixed linear and nonlinear modulation lattices with non-Kerr nonlinearities

This paper studies soliton solutions in parity-time-symmetric mixed linear and nonlinear optical lattices with Kerr and non-Kerr law nonlinearities. Four different kinds of rough mediums namely, parabolic law, power law, dual power law and polynomial law are considered in the context of non-Kerr media. With the help of the inverse engineering scheme, different types of soliton solutions are presented. The extracted outcomes show that exact bright and dark soliton solutions can exist for those different physical states.