Nonlinear Lattice Research Articles

Chiral Bulk Solitons in Photonic Graphene with Decorated Boundaries

AbstractA chiral bulk soliton in a nonlinear photonic lattice with decorated boundaries, presenting a novel approach to manipulate photonic transport without extensive bulk modifications is proposed. Unlike traditional methods that rely on topological edge and corner modes, this strategy leverages the robust chiral propagation of bulk modes. By introducing nonlinearity into the system, a stable bulk soliton, akin to the topological valley Hall effects is found. The chiral bulk soliton exhibits remarkable stability; the energy does not decay even after a long‐distance propagation; and the corresponding Fourier spectrum confirms the absence of inter‐valley scattering indicating a valley‐locking property. The findings not only contribute to the fundamental understanding of nonlinear photonic systems but also hold significant practical implications for the design and optimization of photonic devices.

Nonlinear elasticity tailoring and failure mode manipulation of functionally graded honeycombs under large deformation

Design of lattice metamaterials with tailored mechanical properties at a relatively higher (macro) length scale by architecting lower (micro) length scale geometric and material configurations leads to achieving unprecedented mechanical properties for fulfilling advanced multi-functional structural demands. In the design space of innovative microstructural configurations, we propose a novel class of lattice metamaterials with cell walls made of optimally designed functionally graded intrinsic materials. Under different modes of remotely applied mechanical stresses, two different intuitive architectures of functional gradations for the intrinsic cell wall materials are proposed. The large-deformation nonlinear lattices result in broadband modulation of effective stiffness, and an unprecedented manipulation capability of failure modes and corresponding strengths covering ductile and brittle types depending on architected material gradation. For estimating the nonlinear elasticity and microstructural stresses as a measure of failure mode for the proposed functionally graded lattices undergoing large deformation, a multi-scale mechanics-based semi-analytical framework is developed. Geometrically nonlinear functionally graded beams with generalized material gradation, integrated with unit cell architectures, are analyzed through iterative variational energy principle-based Ritz approach. Based on the developed physically insightful computational framework, effective nonlinear elastic properties and failure modes of the functionally graded honeycomb lattices are tailored as a function of the intrinsic material gradation at the lower length scale. The proposed novel class of lattices with optimally designed functionally graded intrinsic materials and coupled unit cell architectures would open up innovative avenues for designing advanced multi-functional engineering structures and mechanical systems.

Pseudo grid-based physics-informed convolutional-recurrent network solving the integrable nonlinear lattice equations

Traditional discrete learning methods involve discretizing continuous equations using difference schemes, necessitating considerations of stability and convergence. Integrable nonlinear lattice equations possess a profound mathematical structure that enables them to revert to continuous integrable equations in the continuous limit, particularly retaining integrable properties such as conservation laws, Hamiltonian structure, and multiple soliton solutions. The pseudo grid-based physics-informed convolutional-recurrent network (PG-PhyCRNet) is proposed to investigate the localized wave solutions of integrable lattice equations, which significantly enhances the model’s extrapolation capability to lattice points beyond the temporal domain. We conduct a comparative analysis of PG-PhyCRNet with and without pseudo grid by investigating the multi-soliton solutions and rational solitons of the Toda lattice and self-dual network equation. The results indicate that the PG-PhyCRNet excels in capturing long-term evolution and enhances the model’s extrapolation capability for solitons, particularly those with steep waveforms and high wave speeds. Finally, the robustness of the PG-PhyCRNet method and its effect on the prediction of solutions in different scenarios are confirmed through repeated experiments involving pseudo grid partitioning.

Rogue and solitary waves of a system of coupled nonlinear Schrödinger equations in a left-handed transmission line with second-neighbors coupling

The effects of the second-order neighbor interaction have been effective on the coherent localized waves and modulation instability spectrum. A coupled nonlinear Schrödinger equation is derived in the nonlinear left-handed electrical lattice, and the linear stability is used to formulate the modulation instability spectrum expression. Unstable modes are displayed to show symmetric lobes and increasing bandwidths under the influence of the neighbor coupling. The Benjamin-Feir instability has prospered in the network, and for weak perturbed wave numbers, the unstable modes increase. The numerical simulation is used to develop localized waves, including rogue waves with one crest and two humps, Akhmediev breathers, and other modulated structures. We notice that, for strong values of the perturbed wave number, an exponential growth of the continuous wave arises to confirm the fact that the nonlinear left-handed electrical lattice with coupling strength is opened to coherent localized waves. The long-lived nature of the obtained structures has also been demonstrated for specific times of propagation.

A privacy encryption scheme for medical images based on nonlinear dynamic coupled map lattices and protein sequence

A privacy encryption scheme for medical images based on nonlinear dynamic coupled map lattices and protein sequence

Nonlinear lattices from the physics of ecosystems: The Lefever–Lejeune nonlinear lattice in ℤ2

We argue that the spatial discretization of the strongly nonlinear Lefever–Lejeune partial differential equation defines a nonlinear lattice that is physically relevant in the context of the nonlinear physics of ecosystems, modelling the dynamics of vegetation densities in dry lands. We study the system in the lattice , which is especially relevant because of its natural dimension for the emergence of pattern formation. Theoretical results identify parametric regimes for the system that distinguish between extinction and potential convergence to nontrivial states. Importantly, we analytically identify conditions for Turing instability, detecting thresholds on the discretization parameter for the manifestation of this mechanism. Numerical simulations reveal the sharpness of the analytical conditions for instability and illustrate the rich potential for pattern formation even in the strongly discrete regime, emphasizing the importance of the interplay between higher dimensionality and discreteness.

Effect of squared, cubic and quartic on-site potentials on heat conduction in a nonlinear silicon lattice

This paper explains the process of heat conduction in a nonlinear silicon lattice governed by squared, cubic and quartic on-site potentials along with squared and quartic interaction potentials. The tanh method is employed to construct solutions to the resulting nonlinear equations. The results show the transfer of heat in the form of solitary waves which are presented graphically.

Delocalization and higher-order topology in a nonlinear elastic lattice

Topological elastic waves provide novel and robust ways for manipulating mechanical energy transfer and information transmission, with potential applications in vibration control, analog computation, and more. Recently discovered higher-order topological insulators (HOTIs) with multidimensional and hierarchical edge states can further expand the capabilities of topological elastic waves. However, the effects of nonlinearity on elastic HOTIs remain elusive. In this paper, we propose a nonlinear elastic higher-order topological Kagome lattice. After briefly reviewing its linear properties, we explore the effects of nonlinearity on the higher-order band topology and topological states. To do this, we have developed a method to calculate approximate nonlinear modes in order to identify the bulk polarization and probe the higher-order topological phase in the nonlinear lattice. We find that nonlinearity induces unusual delocalization of topological corner states, band crossing, and higher-order topological phase transition. The delocalization reveals that intracell hardening nonlinearity leads to direct delocalization of topological corner states while intracell softening nonlinearity first enhances and then reduces localization. The nonlinear higher-order topological phase is amplitude dependent, and we demonstrate a transition from a trivial to a non-trivial phase, enabling amplitude induced topological corner and edge states. Additionally, this phase transition corresponds to the closing and reopening of the bandgap, accompanied by an unusual band crossing. By examining the band topology before and after the band crossing, we find that the bulk polarization becomes quantized with respect to amplitude and can predict higher-order topological phases in nonlinear lattices. The obtained results are expected to be beneficial for the development of tunable and robust elastic wave devices.

Inverse scattering transform for a nonlinear lattice equation under non-vanishing boundary conditions

Inverse scattering transform for a nonlinear lattice equation under non-vanishing boundary conditions

Influence of the relative stiffness of second-neighbor interactions on chaotic discrete breathers in a square lattice

It is known that the modulational instability of a delocalized nonlinear vibrational mode (DNVM) with frequency outside the phonon band can lead to spontaneous energy localization in a nonlinear lattice on chaotic discrete breathers (CDBs). Considering a β-FPUT square lattice with nearest and next-nearest interactions, the appearance of CDBs is analyzed for different stiffnesses of the first- and second-nearest interactions, k1 and k2, keeping the density of the lattice unchanged. There are two DNVMs in the square lattice with frequencies above the phonon spectrum and both are studied. The appearance of CDBs in the lattice is monitored by calculating the time evolution of the energy localization parameter L and the maximum energy of the particles emax. For solid state physics and materials science, the important range of stiffness parameters is k2<k1, since the stiffness of chemical bonds typically decreases with the distance between atoms. It is found that CDBs form in the square lattice when k2>k1/4. This means that they can form in crystals if the stiffness of the second-neighbor bonds is smaller than that of the first-neighbor bonds, but not too small. Depending on the relation between the anharmonicity parameters of the first- and second-neighbor bonds CDBs can have different polarization.

Darboux’s Theorem, Lie series and the standardization of the Salerno and Ablowitz–Ladik models

In the framework of nonlinear Hamiltonian lattices, we revisit the proof of Moser-Darboux’s Theorem, in order to present a general scheme for its constructive applicability to Hamiltonian models with non-standard symplectic structures. We take as a guiding example the Salerno and Ablowitz–Ladik (AL) models: we justify the form of a well-known change of coordinates which is adapted to the Gauge symmetry, by showing that it comes out in a natural way within the general strategy outlined in the proof. Moreover, the full or truncated Lie-series technique in the extended phase-space is used to transform the Salerno model, at leading orders in the Darboux coordinates: thus the dNLS Hamiltonian turns out to be a normal form of the Salerno and AL models; as a byproduct we also get estimates of the dynamics of these models by means of dNLS one. We also stress that, once it is cast into the perturbative approach, the method allows to deal with the cases where the explicit trasformation is not known, or even worse it is not writable in terms of elementary functions.

Strong–strong beam–beam simulations with lattices of circular e+e- colliders

Strong–strong beam–beam simulations for circular colliders are very time-consuming. Thus far, such simulations have predominantly utilized a linear transfer matrix to represent the full nonlinear lattice. Otherwise, when nonlinear lattices were included, beam–beam interactions were usually studied by using weak–strong or quasi-strong–strong numerical models. The next-generation circular colliders, including FCC-ee at CERN, CEPC in China, and the Super Tau-Charm Factories in China and Russia, will leverage the crab waist collision scheme with a large Piwinski angle. This scheme has already been employed in the two currently operational colliders, DAFNE and SuperKEKB. Accurate predictions of the performance of such colliders call for strong–strong beam–beam simulations, taking into account the interplay of the beam–beam interaction with lattice nonlinearities, impedances, and other collective effects. To answer this call, a novel code APES-T that allows integrating the direct strong–strong beam–beam simulations with element-by-element nonlinear tracking has been developed. To overcome the limitations in computing time, APES-T employs parallelization techniques. Specifically, the beam–beam module is parallelized using MPI programming, while the element-by-element tracking modules are parallelized using GPU programming. In this paper, we present the development of the APES-T code, its benchmark against the SAD code, and its application to the CEPC in investigating the interplay between beam–beam interaction, beamstrahlung, and lattices.

Ordered Mesoporous Ceria and Cerium Gadolinium Oxide Prepared by Vacuum-Assisted Nanocasting.

Four ceria-based mesoporous oxide materials were prepared using a new vacuum impregnation (VI) templating method developed by the authors, namely, vacuum-assisted nanocasting (VAN). Two hard templates (SBA-15 and KIT-6) were employed, and products with compositions CeO2 and Ce0.9Gd0.1O1.9 (CGO) were made with each. The desired fluorite phase and composition were confirmed by powder XRD and EDS. The product structures were characterised by XRD, TEM, gas physisorption and SAXS. All products contained ordered mesoporous material in high yields. The specific surface areas (SSAs) and pore volumes of the products were determined to be high and the pore size and pore spacings related well to the templates from which the materials were synthesised. The TEM studies confirmed that the samples had a 3D pore structure, with this being the negative of the original template. The target materials were not only produced in high yields, but also displayed a porous single-crystal morphology with non-linear lattice planes. The highest SSA values and pore volumes were reported for materials impregnated using the KIT-6 template and with the CGO composition. The results suggest that VAN is an excellent and reproducible method for producing ordered mesoporous cerias and has considerable potential for wider application. All the mesoporous products showed dramatically increased reducibility in TPR experiments compared with a high-SSA nanoparticulate ceria reference. This is very promising for their potential applications in oxidation catalysts and in fuel cell components.

Exponential localization of odd, even, and multi-pulse discrete breathers in Fermi–Pasta–Ulam–Tsingou lattices

Discrete breathers are spatially localized periodic solutions in nonlinear lattices. The existence of odd and even symmetric single-pulse and multi-pulse discrete breathers has been proved in the one-dimensional Fermi–Pasta–Ulam–Tsingou lattices with even interaction potentials [Yoshimura and Doi, J. Differ. Equations 298, 560–608 (2021)]. We prove that those discrete breathers are exponentially localized in space.

Propagation of fundamental gap solitons in the fractional Schrödinger equation with combined linear and nonlinear optical lattices

The propagation of two-dimensional fundamental gap solitons is investigated in the framework of fractional Schrödinger equation with combined linear and nonlinear optical lattices. It is revealed that solitons exist in the first gap and can be stable in a large region. With a phase tilt, the fundamental gap solitons undergo oscillation during the propagation. And the oscillation is enhanced with the increase of the tilt. In addition, propagation of fundamental gap solitons are also studied by longitudinal modulation of Lévy index, nonlinear strength, and period of nonlinear lattices, respectively. With the sudden variation of Lévy index, the fundamental gap solitons undergo oscillation and become unstable, whereas the gradual variation generates the stable soliton. However, both sudden and gradual decreases of nonlinear strength can lead to the distortion of the soliton due to the weak nonlinearity. Interestingly, the small longitudinal modulation of the period of the nonlinear lattice has little effect on the propagation of fundamental gap soliton, while the effect of the large variation of the period of the nonlinear lattice is apparent.

Heat transport in an angular-momentum-conserving lattice.

It is expected that the energy-diffusion propagator in a one-dimensional nonlinear lattice with three conserved quantities: energy, momentum, and stretch, consists of a central heat mode and two sound modes. The heat mode follows a Lévy distribution. Consequently, the heat diffusion is super, i.e., the second moment of the diffusion propagator diverges as t^{β} with β>1; and the heat conduction is anomalous, i.e., the heat conductivity is size dependent and diverges with size N by N^{α}, with α>0. In this paper, we study a one-dimensional lattice with two-dimensional transverse motions, in which the total angular momentum also conserves. More importantly, the diffusion of this conserved quantity is ballistic. Surprisingly, the above pictures and the values of the mentioned power exponents keep unchanged. The universality of the scalings is then further extended. On the other hand, the detailed strengths of heat transports are largely enhanced. Such a counterintuitive finding can be explained by the change of the phonon mean-free path of the lattices.

Scattering of Bright Solitons Due to Defects in Binary Bose‐Einstein Condensates with Nonlinear Optical Lattices

AbstractSpatially overlapped coupled matter‐wave bright solitons are considered in binary Bose‐Einstein condensates with nonlinear optical lattices(NOLs) and study the properties of scattering patterns due to a localized defect. It is shown that spatially overlapped solitons become separated and exhibit different scattering patterns with the variation of inter‐species interaction. The NOL interplays with the defect and tends to influence the scattering properties. However, the effect of NOL varies with strength of inter‐component interaction. It is found that, if this interaction is weak then the scattering center gets shifted outside the defect. However, for a stronger inter‐component interaction, the scattering takes place from the defect and leads to different scattering patterns.

Breathers for nonlinear energy management in acoustic waveguides

Recent investigations of nonlinear metamaterials have revealed interesting wave propagation phenomena with no counterpart in linear systems. One of the appealing phenomena is traveling discrete breathers. Breathers are traveling oscillatory wavepackets with spatially localized envelopes and energy-dependent speeds. These nonlinear waves were reported in lattices with purely (or nearly purely) nonlinear neighbor interactions (possessing zero speed of sound as defined in the classical acoustics sense, and, hence, designated as sonic vacua). However, the effect of local resonators on breather propagation is elusive to date. In this work, we aim to fill this gap by studying 1D grounded strongly nonlinear lattices with linear local resonators (SNLRMs) under impulsive force excitation. Numerical simulations have demonstrated that SNLRMs support the generation of new families of breathers. These families resulted from the opening of a new optical propagation zone (PZ) in the spectrum, which is associated with the linear dynamics of the local resonators. These families are characterized by the local resonator parameters and can support multiple fast frequencies or a single frequency that belongs to either the optical or acoustical PZs. Further aspects of the nonlinear acoustics are also demonstrated using the complexification averaging method. The newly reported breathers open the venue to employ SNLRMs in passive nonlinear energy management applications. Examples are presented through acoustics waveguides and frequency filtering.

Bright solitons in a spin-orbit-coupled dipolar Bose-Einstein condensate trapped within a double-lattice.

By effectively controlling the dipole-dipole interaction, we investigate the characteristics of the ground state of bright solitons in a spin-orbit coupled dipolar Bose-Einstein condensate. The dipolar atoms are trapped within a double-lattice which consists of a linear and a nonlinear lattice. We derive the motion equations of the different spin components, taking the controlling mechanisms of the dipole-dipole interaction into account. An analytical expression of dipole-dipole interaction is derived. By adjusting the dipole polarization angle, the dipole interaction can be adjusted from attraction to repulsion. On this basis, we study the generation and manipulation of the bright solitons using both the analytical variational method and numerical imaginary time evolution. The stability of the bright solitons is also analyzed and we map out the stability phase diagram. By adjusting the long-range dipole-dipole interaction, one can achieve manipulation of bright solitons in all aspects, including the existence, width, nodes, and stability. Considering the complexity of our system, our results will have enormous potential applications in quantum simulation of complex systems.

Active Regulation of Elastic Waves in a Type of Two-Dimensional Periodic Structures With Piezoelectric Springs

Abstract Wave propagations exhibit direction and frequency selectivity in two-dimensional (2D) periodic structures, which provides possibilities to regulate wave dispersion and bandgap properties. Most of current researches focus on regulations of 1D waves, and there are few works about active regulations of 2D waves, especially in the structures with strong nonlinearities that have remarkable influences on dispersions. In this work, two types of 2D periodic nonlinear lattice structures with piezoelectric springs, which include a monatomic and a diatomic structure, are designed to implement controllable dispersion and propagation direction of 2D waves. Considering the strong nonlinearities caused by the cubic spring, dynamic models of the wave propagations in the two kinds of periodic structures are established, and an improved incremental harmonic balance (IHB) method is developed to implement efficient and accurate calculations of the 2D wave propagation. Influences of active and structural parameters on dispersion and bandgap properties are comprehensively studied, and the regulation ability of the piezoelectric springs is demonstrated where the proportional voltage constant is the active control parameter with particle displacements as the feedback. Results also show that a piezoelectric modulated bandgap and a critical wave vector region are created by positive and negative proportional constants, respectively, which indicate that the structures can be used to filter a wide range of low-frequency long-wavelength noises and waves at particular directions. The properties predicted by the improved IHB method are verified by numerical experiments.