Lattice Waves Research Articles

Existence of solitary waves in particle lattices with power-law forces

Existence of solitary waves in particle lattices with power-law forces

Solitary Waves and Kinks in FPU Lattices with Soft–Hard–Soft Trilinear Interactions

Solitary Waves and Kinks in FPU Lattices with Soft–Hard–Soft Trilinear Interactions

Nonlocal-in-time finite-velocity diffusion approach on a ring.

The ubiquitous telegrapher's equationis presented in the context of a non-local-in-time master equationon the lattice. From the exact solution of this transport equation, for different hopping models, the second moment in the infinite lattice and the time evolution of the probability in the ring have been analyzed as a function of the two characteristic timescales appearing in the memory kernel of the finite-velocity approach: the rate of energy loss and the timescale characterizing the jumping process in the lattice. We have demonstrated how these timescales characterize the constraint to find positive solutions, the time variation of entropy and therefore the approach to the disordered stationary state on the ring. This lattice model provides an analytic treatment. Thus, this result is relevant in the study of Shannon entropy, transport of information, and waves in lattices and sheds light on the functional role of the loss of energy in the finite-velocity diffusion dynamics.

Dynamics of discrete solitons in the fractional discrete nonlinear Schrödinger equationwith the quasi-Riesz derivative.

We elaborate a fractional discrete nonlinear Schrödinger (FDNLS) equationbased on an appropriately modified definition of the Riesz fractional derivative, which is characterized by its Lévy index (LI). This FDNLS equationrepresents a novel discrete system, in which the nearest-neighbor coupling is combined with long-range interactions, that decay as the inverse square of the separation between lattice sites. The system may be realized as an array of parallel quasi-one-dimensional Bose-Einstein condensates composed of atoms or small molecules carrying, respectively, a permanent magnetic or electric dipole moment. The dispersion relation (DR) for lattice waves and the corresponding propagation band in the system's linear spectrum are found in an exact form for all values of LI. The DR is consistent with the continuum limit, differing in the range of wave numbers. Formation of single-site and two-site discrete solitons is explored, starting from the anticontinuum limit and continuing the analysis in the numerical form up to the existence boundary of the discrete solitons. Stability of the solitons is identified in terms of eigenvalues for small perturbations, and verified in direct simulations. Mobility of the discrete solitons is considered too, by means of an estimate of the system's Peierls-Nabarro potential barrier, and with the help of direct simulations. Collisions between persistently moving discrete solitons are also studied.

Standing and traveling waves in a minimal nonlinearly dispersive lattice model

In the work of Colliander et al. (2020) a minimal lattice model was constructed describing the transfer of energy to high frequencies in the defocusing nonlinear Schrödinger equation. In the present work, we present a systematic study of the coherent structures, both standing and traveling, that arise in the context of this model. We find that the nonlinearly dispersive nature of the model is responsible for standing waves in the form of discrete compactons. On the other hand, analysis of the dynamical features of the simplest nontrivial variant of the model, namely the dimer case, yields both solutions where the intensity is trapped in a single site and solutions where the intensity moves between the two sites, which suggests the possibility of moving excitations in larger lattices. Such excitations are also suggested by the dynamical evolution associated with modulational instability. Our numerical computations confirm this expectation, and we systematically construct such traveling states as exact solutions in lattices of varying size, as well as explore their stability. A remarkable feature of these traveling lattice waves is that they are of “antidark” type, i.e., they are mounted on top of a non-vanishing background. These studies shed light on the existence, stability and dynamics of such standing and traveling states in 1+1 dimensions, and pave the way for exploration of corresponding configurations in higher dimensions.

Unveiling multi-wave patterns: dynamic characterization and sensitivity analysis of the Yu-Toda-Sasa-Fukuyama model in lattice and liquid

In this study, an examination of the Yu-Toda-Sasa-Fukuyama equation is undertaken, a model that characterizes elastic waves in a lattice or interfacial waves in a two layer liquid. Our emphasis lies in conducting a comprehensive analysis of this equation through various viewpoints, including the examination of soliton dynamics, exploration of bifurcation patterns, investigation of chaotic phenomena, and a thorough evaluation of the model’s sensitivity. Utilizing a simplified version of Hirota’s approach, multi-soliton pattens, including 1-wave, 2-wave, and 3-wave solitons, are successfully derived. The identified solutions are depicted visually via 3D, 2D, and contour plots using Mathematica software. The dynamic behavior of the discussed equation is explored through the theory of bifurcation and chaos, with phase diagrams of bifurcation observed at the fixed points of a planar system. Introducing a perturbed force to the dynamical system, periodic, quasi-periodic and chaotic patterns are identified using the RK4 method. The chaotic nature of perturbed system is discussed through Lyapunov exponent analysis. Sensitivity and multistability analysis are conducted, considering various initial conditions. The results acquired emphasize the efficacy of the methodologies used in evaluating solitons and phase plots across a broader spectrum of nonlinear models.

Robust inverse scattering analysis of discrete high-order nonlinear Schrödinger equation

Abstract In this study, we present the rigorous theory of the robust inverse scattering method for the discrete high-order nonlinear Schrödinger (HNLS) equation with a nonzero boundary condition (NZBC). Using the direct scattering problem, we deduce the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and scattering matrix. We also formulate the inverse scattering problem using the matrix Riemann–Hilbert problem (RHP). Furthermore, utilizing the loop group theory, we construct the multi-fold Darboux transformation (DT) within the framework of the robust inverse scattering transform. Additionally, we develop the corresponding Bäcklund transformation (BT) to obtain the multi-fold lattice soliton solutions. To derive the high-order rational solutions, we further construct the high-order DT. Finally, we theoretically and graphically analyze these solutions, which exhibit lattice breather waves, W-shape lattice solitons, high-order lattice rogue waves (RW), and their interactions.

Exchange regime of the spin waves in two-dimensional honeycomb-like ferromagnetic lattices

Exchange regime of the spin waves in two-dimensional honeycomb-like ferromagnetic lattices

Solitary waves in electro-mechanical lattices

We introduce and study both analytically and numerically a class of microelectromechanical chains aiming to turn them into transmission lines of solitons. Mathematically, their analysis reduces to the study of a spatially one-dimensional nonlinear Klein–Gordon equation with a model dependent onsite nonlinearity induced by the electrical forces. Since the basic solitons appear to be unstable for most of the force regimes, we introduce a stabilizing algorithm and demonstrate that it enables a stable and persisting propagation of solitons. Among other fascinating nonlinear formations induced by the presented models, we mention the “meson”: a stable square shaped pulse with sharp fronts that expands with a sonic speed, and “flatons”: flat-top solitons of arbitrary width.

Shock waves in semi-infinite photonic lattices

We investigate the existence, stability and propagation dynamics of shock waves existing in defocusing saturable media modulated by a semi-infinite photonic lattice. Such surface waves consist of modulationally stable pedestals in bulk media and decaying oscillatory tails in semi-infinite photonic lattices. Due to Bragg-type reflection, they are strongly localized at the edge of the optical lattice. The kink steepness, pedestal height and localization degree can be controlled by propagation constant, saturable degree and lattice depth. Two types of kinks, i.e., “out-of-phase” and two branches of “in-phase” shock waves are revealed. Out-of-phase shock waves are stable in a substantial part of their existence domain. While the lower-branch in-phase waves with low energy flow are stable in almost their whole existence domain, the higher-branch in-phase waves are completely unstable. We thus show the first example of stable in-phase shock waves in nonlinear optics. Our findings provide new insight into the dynamics of semi-localized nonlinear surface shock waves or kink solitons.

A modal decomposition approach to topological wave propagation

The design of valley-Hall topological insulators is typically informed by analyzing the primitive unit cell, using the Berry curvature of the Bloch-eigenmodes to predict energy localization on the domain boundary, or the dispersion diagram of the semi-infinite lattice to predict the group velocity of topological waves. However, practical systems are finite in size and the propagating topological wave is composed of a finite bandwidth as guaranteed by the Fourier uncertainty principle. Hence, the propagation predicted by the ideally infinite lattice for a given frequency deviate from practical topological systems driven by finite bandwidth wave packets. In this work, we demonstrate that the propagating topological waves in valley Hall lattices can be interpreted using the linear degenerate modal basis. We show that a small subset of closely spaced modes comprise the topological waves, which enables the construction of analytical reduced-order models that accurately capture the topological wave. This result is related to the sparse density of the modal spectrum inside the topological band gap, which allows for the energy to be carried nearly completely by a narrow band of spectrally isolated eigenmodes that are spatially localized along the domain boundary. This result is then employed to refine group velocity predictions of propagating topological waves with respect to input signal bandwidth and frequency by matching the modal spectrum to the supercell dispersion diagram describing the semi-infinite system. Moreover, we utilize the damped modal spectrum to characterize variations in wave group velocity in damped topological lattices and predict edge-to-bulk transitions. Hence, this work establishes a framework for accurately characterizing undamped and damped topological wave propagation in quantum valley Hall systems using the machinery of classical dynamics.

Analysis of transmission and reflection characteristics of linear plane waves in pantographic lattices

Analysis of transmission and reflection characteristics of linear plane waves in pantographic lattices

Impact of nonstoichiometry on the mechanical properties and thermal conductivity of gadolinium zirconate ceramics

Low elastic properties, high hardness, and low thermal conductivity are desirable for a rare-earth zirconate thermal barrier coating material. Realizing nonstoichiometry is a significant way to improve the elastic properties, hardness and thermal conductivity of rare-earth zirconate to further meet the requirements of thermal barrier coating materials, nonstoichiometric Gd2-xZr2+xO7+0.5x (x = 0, ±0.125, and ±0.25) was studied by first-principles calculations first and then verified by the solid-state reaction methods. Calculations revealed that excess Gd3+ and Zr4+ decrease the elastic modulus, sound velocity, and Debye temperature of Gd2Zr2O7, and increase its ductility. The elastic properties of the material exhibit significant anisotropy, which further increases with increasing Gd3+ and Zr4+ contents, resulting from lattice distortion caused by excess Gd3+ and Zr4+ entering the lattice. The hardness of Gd2Zr2O7 increases with a moderate excess of Zr4+, while it decreases with an excess of Zr4+ and Gd3+. In addition, with the disordered occupation of lattice atoms, the anharmonicity of lattice waves is increased, and phonon scattering is enhanced as the concentrations of Gd3+ and Zr4+ increase, resulting in the reduction of the phonon thermal conductivity. Moreover, Zr4+ is more conducive to the decrease in the thermal conductivity. The experimental consequences match well with the calculation consequences. This work is expected to provide support for improving mechanical properties and thermal conductivity of rare-earth zirconate materials.

Spin waves in two-dimensional placoid-like ferromagnetic lattices

The topology of placoid scales is explored as a model of a two-dimensional ferromagnetic lattice. We start from Heisenberg’s Hamiltonian, adopting in low temperature regime and performing the study of free modes of the magnetic excitations. The Holstein–Primakoff approximation was employed, restricting our attention to the bilinear terms in the quantized second operators. The dispersion relation was obtained and the properties of the magnetic modes were studied in terms of the physical fit parameter J2, which reveals the energy dynamics for different regions of the Brillouin Zone. A comparison was made with the energy maps from the square and triangular lattices in order to verify the equivalence of the effects from the placoid geometry. We concluded that this topology imprints a significant energy growing in the regime of low wave vectors with more effective rates than those observed in the triangular lattice, a fact that can be explained by the presence of 4 sites in the unit cell.

Instabilities of interacting matter waves in optical lattices with Floquet driving

Instabilities of interacting matter waves in optical lattices with Floquet driving

Effects of nonlinearity and a new nonlinear resonance in two-path phonon transmittance in lattices with two-dimensional arrays of atomic defects.

The paper is devoted to analytical and numerical studies of the effects of nonlinearity on the two-path phonon interference in the transmission through two-dimensional arrays of atomic defects embedded in a lattice. The emergence of transmission antiresonance (transmission node) in the two-path system is demonstrated for the few-particle nanostructures, which allow us to model both linear and nonlinear phonon transmission antiresonances. The universality of destructive-interference origin of transmission antiresonances of waves of different nature, such as phonons, photons, and electrons, in two-path nanostructures and metamaterials is emphasized. Generation of the higher harmonics as a result of the interaction of lattice waves with nonlinear two-path atomic defects is considered, and the full system of nonlinear algebraic equationsis obtained to describe the transmission through nonlinear two-path atomic defects with an account for the generation of second and third harmonics. Expressions for the coefficients of lattice energy transmission through and reflection from the embedded nonlinear atomic systems are derived. It is shown that the quartic interatomic nonlinearity shifts the antiresonance frequency in the direction determined by the sign of the nonlinear coefficient and enhances in general the transmission of high-frequency phonons due to third harmonic generation and propagation. The effects of the quartic nonlinearity on phonon transmission are described for the two-path atomic defects with a different topology. Transmission through the nonlinear two-path atomic defects is also modeled with the simulation of the phonon wave packet, for which the proper amplitude normalization is proposed and implemented. It is shown that the cubic interatomic nonlinearity red shifts in general the antiresonance frequency for longitudinal phonons independently of the sign of the nonlinear coefficient, and the equilibrium interatomic distances (bond lengths) in the atomic defects are changed by the incident phonon due to cubic interatomic nonlinearity. For longitudinal phonons incident on a system with the cubic nonlinearity, the new narrow transmission resonance on the background of a broad antiresonance is predicted to emerge, which we relate to the opening of the additional transmission channel for the phonon second harmonic through the nonlinear defect atoms. Conditions of the existence of the new nonlinear transmission resonance are determined and demonstrated for different two-path nonlinear atomic defects. A two-dimensional array of embedded three-path defects with an additional weak transmission channel, in which a linear analog of the nonlinear narrow transmission resonance on the background of a broad antiresonance is realized, is proposed and modeled. The presented results provide better understanding and detailed description of the interplay between the interference and nonlinearity in phonon propagation through and scattering in two-dimensional arrays of two-path anharmonic atomic defects with a different topology.

Modular development of deep potential for complex solid solutions

The multicomponent oxide solid solution is a versatile platform to tune the delicate balance between competing spin, charge, orbital, and lattice degrees of freedom for materials design and discovery. The development of compositionally complex oxides with superior functional properties has been largely empirical and serendipitous, in part due to the exceedingly complex chemistry and structure of solid solutions that span a range of length scales. The usage of classical molecular dynamics (MD), a powerful statistical method, in computer-aided materials design has not yet reached the same level of sophistication as that in computer-aided drug design because of the limited availability and accuracy of classical force fields for solids. Here, we introduce the strategy of ``modular development of deep potential'' (ModDP) that enables a systematic development and improvement of deep-neural-network-based model potential, termed as deep potential, for complex solid solutions with minimum human intervention. The converged training database associated with an end-member material is treated as an independent module and is reused to train the deep potential of solid solutions via a concurrent learning procedure. We apply ModDP to obtain classical force fields of two technologically important solid solutions, ${\mathrm{Pb}}_{x}{\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{TiO}}_{3}$ and ${\mathrm{Hf}}_{x}{\mathrm{Zr}}_{1\ensuremath{-}x}{\mathrm{O}}_{2}$. For both materials' systems, a single model potential is capable of predicting various properties of solid solutions including temperature-driven and composition-driven phase transitions over a wide range of compositions. In particular, the deep potential of ${\mathrm{Pb}}_{x}{\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{TiO}}_{3}$ reproduces a few known topological textures such as polar vortex lattice and electric dipole waves in ${\mathrm{PbTiO}}_{3}/{\mathrm{SrTiO}}_{3}$ superlattices, paving the way for MD investigations on the dynamics of topological structures in response to external stimuli. MD simulations of ${\mathrm{Hf}}_{x}{\mathrm{Zr}}_{1\ensuremath{-}x}{\mathrm{O}}_{2}$ reveal a substantial impact of composition variation on both the phase transition temperature and the nature of the high-temperature nonpolar phase.

Diffraction and transmission antiresonances of lattice waves in sparse two-dimensional arrays of defect atoms

The paper theoretically studies the influence of diffraction on the transmission of displacement waves in discrete lattices through crystal planes with sparse two-dimensional arrays of periodically embedded point (atomic) defects by analytical lattice dynamics calculations and molecular dynamics simulations. Transmission through the sparse two-dimensional arrays of atomic defects results in destructive interference of the waves propagating through two paths and the appearance of antiresonances (transmission nodes) in transmission spectrum. It is shown that the diffracted waves can substantially reduce or enhance the phonon transmission and can produce an additional narrow transmission antiresonance, depending on the properties of atomic defects and interatomic interactions. Specific features of the diffraction in discrete lattices as compared with those in a continua are discussed. The influence of positional disorder in defect atoms distribution on lattice waves diffraction and transmission antiresonances is considered. It is demonstrated by the lattice dynamics calculations and molecular dynamics simulations that the diffraction and transmission antiresonances of lattice waves are not suppressed by moderate disorder in the defect atoms distribution. The presented results can be used in the design of novel devices for high-frequency acoustic switching, filtering and interferometry.

Formation of a Diffraction Field During Dynamic Scattering of Slow Neutrons in a Deformed Crystal

The problem of diffraction of slow neutrons with a de Broglie wavelength λ ~10-10 m in a crystal with a deformed lattice is considered. Based on the Hamilton equations describing the dynamic diffraction of slow neutrons, the features of the formation of a dynamic diffraction field in a lattice in various regions of the deformation field characteristics, including weak fields, slowly changing fields and rapidly changing fields are discussed. Using a particular example of a deformation field with a displacement field quadratic in coordinates, analytical solutions are obtained for the wave functions of transmitted and reflected lattice waves.

Rogue waves in quantum lattices with correlated disorder

We investigate the outbreak of anomalous quantum wave-function amplitudes in a one-dimensional tight-binding lattice featuring correlated diagonal disorder. Such rogue-wave-like behavior is fostered by a competition between localization and mobility. The effective correlation length of the disorder is ultimately responsible for bringing the local disorder strength to a minimum, fueling the occurrence of extreme events of much higher amplitudes, especially when compared to the case of uncorrelated disorder. Our findings are valid for a class of discrete one-dimensional systems and reveal profound aspects of the role of randomness in rogue-wave generation.