This paper presents a systematic investigation of the existence, stability, and dynamics of localized modes—specifically fundamental gap solitons and nonlinear truncated Bloch waves (TBWs)—in a one-dimensional system with a combined linear lattice and a purely quintic nonlinear lattice. The model is governed by the nonlinear Schrödinger equation with a periodic linear potential and a spatially modulated quintic nonlinearity. Using numerical methods, including the Newton-conjugate gradient technique for stationary solutions, linear stability analysis, and direct propagation simulations, we demonstrate that the spatially periodic modulation of the quintic nonlinear term plays a crucial role in stabilizing fundamental gap solitons. Stable families emerge when specific commensurability conditions are satisfied between the linear and nonlinear lattice periods. Furthermore, we report the first discovery of stable TBWs in a quintic nonlinear setting, demonstrating that their formation and spatial period are intrinsically governed by the coupling between the linear and nonlinear lattices, independent of initial conditions. These multi-peak localized states exhibit high robustness during propagation when phase-matching conditions are satisfied. Our results underscore the critical role of quintic nonlinear lattice modulation in stabilizing diverse localized states and offer insights for designing advanced photonic devices based on higher-order nonlinear periodic media.

We demonstrate the epitaxial growth of single-phase (100) BaTiO3 films on (100) β-Ga2O3 substrates at substrate temperatures ranging from 600 to 700 °C using molecular-beam epitaxy. Characterization of a 47 nm thick BaTiO3 film by atomic force microscopy reveals a step-and-terrace morphology with unit-cell-high BaTiO3 steps and an rms surface roughness of 0.26 nm. Scanning transmission electron microscopy (STEM) images show that in some regions the β-Ga2O3 substrate terminates with a (100)A plane as it transitions to BaTiO3 and in other regions with a (100)B plane. The (100) BaTiO3 films are fully relaxed and consist of a mixture of two types of a-axis domains: a1 and a2. The orientation relationship determined by X-ray diffraction and confirmed by STEM is (100) BaTiO3 || (100) β-Ga2O3 and [011] BaTiO3||[010] β-Ga2O3. Despite the average linear lattice mismatch of 3.8%, BaTiO3 films with rocking curve full width at half maximum widths as narrow as 28 arc sec are achieved. From capacitance–voltage measurements on a metal–oxide–semiconductor capacitor structure with a-axis BaTiO3 as the oxide layer and Si-doped β-Ga2O3 as the semiconducting layer, we extract a dielectric constant of K11 = 670 for the BaTiO3 epitaxially integrated with (100) β-Ga2O3. We anticipate that this high-K epitaxial dielectric will be useful for electric-field management in β-Ga2O3-based device structures.

We study the stability of one-dimensional linear lattice Boltzmann schemes for scalar hy- perbolic equations with respect to boundary data. Our approach is based on the original raw algorithm on several unknowns, thereby avoiding the need for a transformation into an equivalent scalar for- mulation – a challenging process in presence of boundaries. To address different behaviors exhibited by the numerical scheme, we introduce appropriate notions of strong stability. They account for the potential absence of a continuous extension of the stable vector bundle associated with the bulk scheme on the unit circle for certain components. Rather than developing a general theory, complicated by the fact that discrete boundaries in lattice Boltzmann schemes are inherently characteristic, we focus on strong stability–instability for methods whose characteristic equations have stencils of breadth one to the left. In this context, we study three representative schemes. These are endowed with various boundary conditions drawn from the literature, and our theoretical results are supported by numerical simulations.

Our study focuses on the optimisation of the internal structure of unimodal and pentamodal metamaterials, modelled as three-dimensional linear elastic lattice structures. For optimisation, we represent the metamaterials with discrete truss models of their respective Periodic Unit Cells (PUCs), whose effective response is determined by the first-order numerical homogenisation. The optimisation is formulated as an inverse homogenisation problem with objective functions comprising a ratio of selected eigenvalues of the effective stiffness matrix, which allows us to dispense with the traditional volume constraint and solve the optimisation problem with a simple gradient method combined with the line search method. We demonstrate the efficacy of the formulation with a design of a unimodal material compliant in a chosen shear deformation mode and we also show that our formulation recovers the traditional pentamodal metafluid.

A relationship for LYM inequalities between Boolean lattices and linear lattices with applications

We demonstrate that a defocusing cubic nonlinearity with a bichromatic lattice can support various types of bright soliton families, including fundamental solitons and their clusters with an arbitrary number of peaks. We investigate the profiles, powers, amplitudes, stability domains, and propagation dynamics of these soliton families in both the first and second linear Bloch band gaps. Notably, the profiles of soliton families in the bichromatic lattice are different from the counterparts in the monochromatic lattice; specifically, additional humps appear adjacent to the central major peaks of the solitons. This phenomenon becomes more pronounced when the magnitude of the propagation constant becomes large. Introduction of the second lattice makes the gaps significantly wider and offers the possibility of controlling the shape of monochromatic solitons. The stability domains of such solitons are obtained by the method of linear stability analysis and are verified by direct numerical simulations. As the propagation constant increases, instabilities develop in the second band gap and close to the band edge. Curiously, the soliton families under low strengths of bichromatic lattice are always unstable in the second band gap, whereas the ones under high strengths are always stable.

Abstract This paper investigates statics and natural vibration of linear elastic cubic lattices, together with their continuum approximations. The lattice endowed with central and angular interactions, referred to as Gazis et al . ’s, is considered first: since the stiffness of each lattice phase must be positive, the equivalent macroscopic Poisson’s ratio must be lower than its central limit 1/4. A volumetric interaction based on a volume-dependent internal pressure is introduced as an additional non-central interaction for a complete calibration of the equivalent Poisson’s ratio up to its incompressibility limit 1/2. This volumetric interaction can also be classified as Fuchs-type, providing a potential energy that depends on the volume variation of each cell. The mixed differential-difference equations of the associated lattice derive from Hamilton’s principle applied to the discrete energies. The algebraic properties of the stiffness matrix of the discrete cell provide information on the positive definiteness of the potential energy, for each lattice with central and non-central interactions. The convergence of this finite lattice towards a linear elastic continuous right parallelepiped is shown in several static loading schemes. The discrete Lamé problem for the free vibration of this parallelepiped is solved for all the considered lattices. It is concluded that discrete and continuum elasticity can be connected by this cubic lattice within a complete range of elasticity parameters.

We present an asymptotic analysis of shell lattice metamaterials based on Ciarlet's shell theory, introducing a new metric— asymptotic directional stiffness (ADS)—to quantify how the geometry of the middle surface governs the effective stiffness. We prove a convergence theorem that rigorously characterizes ADS and establishes its upper bound, along with necessary and sufficient condition for achieving it. As a key result, our theory provides the first rigorous explanation for the high bulk modulus observed in Triply Periodic Minimal Surfaces (TPMS)-based shell lattices. To optimize ADS on general periodic surfaces, we propose a triangular-mesh-based discretization and shape optimization framework. Numerical experiments validate the theoretical findings and demonstrate the effectiveness of the optimization under various design objectives. Our implementation is available at https://github.com/lavenklau/minisurf.

Autonomous quantum memories are a way to passively protect quantum information using engineered dissipation that creates an “always-on'' decoder. We analyze Markovian autonomous decoders that can be implemented with a wide range of qubit and bosonic error-correcting codes, and derive several upper bounds and a lower bound on the logical error rate in terms of correction and noise rates. These bounds suggest that, in general, there is always a correction rate, possibly size-dependent, above which autonomous memories exhibit arbitrarily long coherence times. For any given autonomous memory, size dependence of this correction rate is difficult to rule out: we point to common scenarios where autonomous decoders that stochastically implement active error correction must operate at rates that grow with code size. For codes with a threshold, we show that it is possible to achieve faster-than-polynomial decay of the logical error rate with code size by using superlogarithmic scaling of the correction rate. We illustrate our results with several examples. One example is an exactly solvable global dissipative toric code model that can achieve an effective logical error rate that decreases exponentially with the linear lattice size, provided that the recovery rate grows proportionally with the linear lattice size.

The statistical mechanics of structured particles with arbitrary size and shape adsorbed onto discrete lattices presents a longstanding theoretical challenge, mainly due to complex spatial correlations and entropic effects that emerge at finite densities. Even for simplified systems such as hard-core linear k-mers, exact solutions remain limited to low-dimensional or highly constrained cases. In this review, we summarize the main theoretical approaches developed by our research group over the past three decades to describe adsorption phenomena involving linear k-mers-also known as multisite occupancy adsorption-on regular lattices. We examine modern approximations such as an extension to two dimensions of the exact thermodynamic functions obtained in one dimension, the Fractional Statistical Theory of Adsorption based on Haldane's fractional statistics, and the so-called Occupation Balance based on expansion of the reciprocal of the fugacity, and hybrid approaches such as the semi-empirical model obtained by combining exact one-dimensional calculations and the Guggenheim-DiMarzio approach. For interacting systems, statistical thermodynamics is explored within generalized Bragg-Williams and quasi-chemical frameworks. Particular focus is given to the recently proposed Multiple Exclusion statistics, which capture the correlated exclusion effects inherent to non-monomeric particles. Applications to monolayer and multilayer adsorption are analyzed, with relevance to hydrocarbon separation technologies. Finally, computational strategies, including advanced Monte Carlo techniques, are reviewed in the context of high-density regimes. This work provides a unified framework for understanding entropic and cooperative effects in lattice-adsorbed polyatomic systems and highlights promising directions for future theoretical and computational research.

Background. The development of many elements of modern communication systems is increasingly based on the use of various types of dielectric resonators (DR). The theory of coupled oscillations of resonators is the basis for further calculations and optimisation of the scattering matrices of electromagnetic waves on various devices. When calculating devices built on a large number of resonators, direct numerical methods are often not effective. They usually require the use of powerful computers, therefore, the calculation of elements on a large number of DR is impossible without building analytical models of complex structures based on electrodynamic modelling. Objective. The study aims to find analytical expressions for the frequencies and distributions of electromagnetic fields of natural oscillations of lattices, consisting of a large number of various types of dielectric resonators for use in various devices of optical communication systems. To solve this problem, a linear system of equations, which relates the complex amplitudes and frequencies of the resonators, obtained earlier from the perturbation theory, was used. Methods. To find analytical expressions, methods of matrix theory are used. In this case, both known methods of calculating the determinants of tri-diagonal and circulant matrices are used, as well as their modifications related to the calculations of more complex matrices, which, after transformations, are reduced to much simpler formulas. The final result is the receipt of new general analytical formulas for describing coupled oscillations of lattices consisting of a large number of dielectric resonators of various types. Results. Coupled oscillations of one-dimensional linear lattices of two types of dielectric resonators are considered. New analytical expressions for complex frequencies and amplitudes of resonators, as well as Q-factor expressions without restrictions on their number, are obtained. A new model of natural oscillations of two-dimensional lattices, consisting of dielectric resonators of two different types, is constructed. General analytical solutions are found for the frequencies and amplitudes of coupled oscillations for two types of two-dimensional lattices with different arrangements of resonators. Analytical solutions are found for the amplitudes and frequencies of coupled oscillations of two axially symmetric ring lattices with different types of resonators, which are characterised by different placement symmetry in free space. The obtained general analytical expressions for the frequencies of coupled oscillations are compared with the results of calculations obtained numerically, by solving linear systems of equations. A very good agreement between the solutions obtained by the two methods is demonstrated. Conclusions. The developed theory is the basis for the design of many devices of the optical wavelength range, which are built on the basis of the use of a large number of dielectric resonators of various types. The obtained new analytical expressions for calculating coupled oscillations of dielectric resonators allow building new more efficient models of scattering for optimization of various optical communication devices.

Abstract We study the existence and stability of dark-gap solitons in linear lattice and nonlinear lattices. The results indicate that the combination of linear and nonlinear lattices gives dark-gap solitons unique properties. The linear lattice can stabilize dark-gap solitons, while the nonlinear lattice reduces the stability of dark-gap solitons. On the basis of numerical analysis, we investigate the effects of lattice depth, chemical potential, nonlinear lattice amplitude, and nonlinear lattice period on the soliton in mixed lattices with the same and different periods. The stability of dark-gap soliton is studied carefully by means of real-time evolution and linear stability analysis. Dark-gap solitons can exist stably in the band gap, but the solitons formed by the mixed lattices are slightly different when the period is the same or different.

Nonlinear lattice models can support “discrete breather” excitations that stay localized in space for all time. By contrast, the localized Wannier states of linear lattice models are dynamically unstable. Nevertheless, symmetric and exponentially localized Wannier states are a central tool in the classification of band structures with crystalline symmetries. Moreover, the quantized transport observed in nonlinear Thouless pumps relies on the fact that—at least in a specific model—discrete breathers recover Wannier states in the limit of vanishing nonlinearity. Motivated by these observations, we investigate the correspondence between nonlinear breathers and exponentially localized Wannier states for a family of discrete nonlinear Schrödinger equations with crystalline symmetries. We develop a formalism to analytically predict the breathers' spectrum, center of mass and symmetry data, and apply this to nonlinear generalizations of the Su-Schrieffer-Heeger chain and the breathing kagome lattice.

Hefei Advanced Light Facility (HALF) is a VUV and soft X-ray diffraction-limited storage ring, which uses a modified hybrid multi-bend achromat (HMBA) lattice. In a HMBA lattice, many nonlinear effects are cancelled by the -I transformation between sextupoles. But for amplitude-dependent tune shifts (ADTS) and high-order chromaticities, generally they cannot be controlled well, which will limit the dynamic aperture and momentum aperture. To further improve the performance of the HALF lattice, linear parameters and nonlinear dynamics are optimized in this paper. First, linear lattice, ADTS and second-order chromaticity are simultaneously optimized with magnet strengths varied in large ranges to better explore good solutions. In this optimization, lattice solutions with both low emittances and relatively good nonlinear dynamics are efficiently obtained. Then, starting from the good solution region generated above, direct optimization of dynamic aperture and Touschek lifetime based on particle tracking is performed. After this second optimization, the lattices with larger DAs and longer lifetimes are obtained for HALF. This method can also be used for optimizing other HMBA lattices.

Dark gap soliton families in coupled nonlinear Schrödinger equations with linear lattices

A. Haghany and M. Vedadi, as well as M. K. Patel, explored the relationship between a semi-projective and retractable module and its endomorphism ring. In this work, we study the lattice-theoretic counterparts of these results. To this end, we consider the category of linear modular lattices. Specifically, we show a relation between a retractable and semi-projective complete modular lattice and its monoid of endomorphisms.

Much is understood about 1-dimensional spin chains in terms of entanglement properties, physical phases, and integrability. However, the Lie algebraic properties of the Hamiltonians describing these systems remain largely unexplored. In this work, we provide a classification of all Lie algebras generated by the terms of 2-local spin chain Hamiltonians, or so-called dynamical Lie algebras, on 1-dimensional linear and circular lattice structures. We find 17 unique dynamical Lie algebras. Our classification includes some well-known models such as the transverse-field Ising model and the Heisenberg chain, and we also find more exotic classes of Hamiltonians that appear new. In addition to the closed and open spin chains, we consider systems with a fully connected topology, which may be relevant for quantum machine learning approaches. We discuss the practical implications of our work in the context of variational quantum computing, quantum control and the spin chain literature.

We demonstrate the existence of two types of dark gap solitary waves-the dark gap solitons and the dark gap soliton clusters-in Bose-Einstein condensates trapped in a bichromatic optical superlattice with cubic-quintic nonlinearities. The background of these dark soliton families is different from the one in a common monochromatic linear lattice; namely, the background in our model is composed of two types of Gaussian-like pulses, whereas in the monochromatic linear lattice, it is composed of only one type of Gaussian-like pulses. Such a special background of dark soliton families is convenient for the manipulation of solitons by the parameters of bichromatic and chemical potentials. The dark soliton families in the first, second, and third bandgap in our model are studied. Their stability is assessed by the linear-stability analysis, and stable as well as unstable propagation of these gap solitons are displayed. The profiles, stability, and perturbed evolution of both types of dark soliton families are distinctly presented in this work.

Convex expectations for countable-state uncertain processes with càdlàg sample paths

The linear lattice of Riemann μ \mu -integrable functions on a completely regular space with a bounded positive Radon measure μ \mu is viewed as an extension of the linear lattice of bounded continuous functions. To characterize this Riemann extension, the new functional analysis category of c c -latlineals with refinements ( ≡ c r \equiv cr -latlineals) is introduced. On this basis, the procedure of c r cr -completion of certain order-boundary type is defined. The μ \mu -Riemann extension is shown to be the result of applying this procedure to the c r μ cr_{\mu } -latlineal of bounded continuous functions.