Nonlinear Dispersion Relation Research Articles

Simultaneous energy harvesting and vibration control in a nonlinear metastructure: A spectro-spatial analysis

Considerable attention has recently been given to the study of simultaneous energy harvesting and vibration attenuation in metastructures. However, only linear metastructures were investigated although nonlinear metastructures and nonlinear electromechanical devices offer superior interesting wave propagation phenomena (e.g., birth of solitary waves, tunable bandgap, acoustic nonreciprocity) and broadband energy harvesting. In this paper, we investigate the wave propagation in a weakly nonlinear metastructure with electromechanical resonators. Explicit expressions are derived for the nonlinear dispersion relations using the method of multiple scales. These expressions are validated via direct numerical integration. We carried out parametric studies to investigate the role of different parameters of the electromechanical resonators on the linear and nonlinear band structure. To obtain further detailed information on the nonlinear wave propagation, we employ spectro-spatial analysis on the numerical simulations. This spectro-spatial analysis can reveal the output voltage distortion due to different types of nonlinearities. The results indicate that nonlinear chain can enhance energy harvesting through the birth of solitary wave and without degrading the boundary of the bandgap. The results also suggest that such a system is suitable for designing electromechanical diodes and rectifiers.

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids.

On the example of the Poynting–Thomson–Zener rheological model for solids, which exhibits both dissipation and wave propagation, with nonlinear dispersion relation, we introduce and investigate a finite difference numerical scheme. Our goal is to demonstrate its properties and to ease the computations in later applications for continuum thermodynamical problems. The key element is the positioning of the discretized quantities with shifts by half space and time steps with respect to each other. The arrangement is chosen according to the spacetime properties of the quantities and of the equations governing them. Numerical stability, dissipative error, and dispersive error are analyzed in detail. With the best settings found, the scheme is capable of making precise and fast predictions. Finally, the proposed scheme is compared to a commercial finite element software, COMSOL, which demonstrates essential differences even on the simplest—elastic—level of modeling.

Quasi-stationary solutions in gravity theories with modified dispersion relations and Finsler–Lagrange–Hamilton Geometry

Modified gravity theories, MGTs, with modified (nonlinear) dispersion relations, MDRs, encode via indicator functionals possible modifications and effects of quantum gravity; in string/brane, noncommutative and/or nonassociative gravity theories etc. MDRs can be with global and/or local Lorentz invariance violations, LIVs, determined by quantum fluctuations, random, kinetic, statistical and/or thermodynamical processes etc. Such MGTs with MDRs and corresponding models of locally anisotropic spacetime and curved phase spaces can be geometrized in an axiomatic form for theories constructed on (co) tangent bundles with base spacetime Lorentz manifolds. In certain canonical nonholonomic variables, the geometric/physical objects are defined equivalently as in generalized Einstein-Finsler and/or Lagrange-Hamilton spaces. In such Finsler like MGTs, the coefficients of metrics and connections depend both on local Lorentz spacetime coordinates and, additionally, on (co) fiber velocity and/or momentum type variables. The main goal of this work is to elaborate on a nonholonomic diadic "shell by shell" formulation of MGTs with MDRs, with a conventional (2+2)+(2+2) splitting of total phase space dimensions, when the (dual) modified Einstein-Hamilton equations can be decoupled in general forms. We show how this geometric formalism allows us to construct various classes of exact and parametric solutions determined by generating and integration functions and effective sources depending, in principle, on all phase space coordinates. This work provides a self-consistent geometric and analytic method for constructing in our further partner papers different types of black hole solutions for theories with MDRs and LIVs and elaborating various applications in modern cosmology and astrophysics.

Negative Group Velocity Plasmons Propagating in Waveguides Composed of Graphene Metamaterials

Fast light which demonstrates negative group velocity, is achieved by the anomalous dispersion or photon tunneling. However, many applications based on the fast light are limited due to the disadvantages of inferior tunability or nonlinear dispersion relationship of the fast light-carrying medium. In this paper, we propose the graphene plasmonic crystal waveguides whose guiding and claddings are composed of the graphene plasmonic metamaterials, where the backward propagating plasmonic modes corresponding to negative group velocity are observed. The dispersion relation and the group velocity of three types of graphene plasmonic crystal waveguides are investigated by varying the materials and the geometrical parameters of the graphene plasmonic crystal waveguides. Numerical experiments are designed to verify the authenticity of a fast plasmon in the graphene plasmonic crystal waveguides. Our proposed graphene plasmonic crystal waveguides might find significant applications in the fields of nanophotonics, on-chip electromagnetic field manipulation in deep nanoscale, and the technique of high density plasmonic integrated plasmonic circuit in the future.

Nonlinear dynamics of coupled transverse-rotational waves in granular chains.

The nonlinear dynamics of coupled waves in one-dimensional granular chains with and without a substrate is theoretically studied accounting for quadratic nonlinearity. The multiple time scale method is used to derive the nonlinear dispersion relations for infinite granular chains and to obtain the wave solutions for semi-infinite systems. It is shown that the sum frequency and difference frequency components of the coupled transverse-rotational waves are generated due to their nonlinear interactions with the longitudinal wave. Nonlinear resonances are not present in the chain with no substrate where these frequency components have low amplitudes and exhibit beating oscillations. In the chain positioned on a substrate two types of nonlinear resonances are predicted. At resonance, the fundamental frequency wave amplitudes decrease and the generated frequency component amplitudes increase along the chain, accompanied by the oscillations due to the wave number asynchronism. The results confirm the possibility of a highly efficient energy transfer between the waves of different frequencies, which could find applications in the design of acoustic devices for energy transfer and energy rectification.

Explicit dispersion relation for strongly nonlinear flexural waves using the homotopy analysis method

Using the homotopy analysis method, we present an explicit frequency-versus-wavenumber nonlinear dispersion relation for a flexural elastic beam. In our analysis, we employ the Euler–Bernoulli kinematic hypothesis and consider both a conventional transverse motion model and an inextensional planar motion model. As an example, we consider geometric nonlinearity in the form of Green–Lagrange strain. The underlying constitutive relation is formulated by linearly relating the second Piola–Kirchhoff stress to the Green–Lagrange strain, although the method is directly applicable to material nonlinearities as well. The derived analytical solution, for each model, is obtained to Mth-order accuracy and is verified by comparing with a numerical result brought about by laboriously finding the roots of the corresponding travelling-wave implicit relation governing the nonlinear elastodynamics. This is the first derivation of an explicit dispersion relation for an elastic beam undergoing strongly nonlinear finite flexural deformation. The derived relation characterizes the nature of a traveling cosine-like nonlinear wave throughout its stable pre-breaking state.

Nonlinear Evolution and Breakup of the Cavitating Liquid Jet Surrounded by the Rotary Compressible Air.

A nonlinear dispersion relationship has been established to study the surface evolution and breakup of the cavitating liquid jet with cavitation bubbles surrounded by the rotary air, and the built dispersion relationship and its solution are validated by comparing with the results in the reference. The effects of air rotation, fluid compressibility, and bubble volume fraction on jet morphology are investigated mainly. Air rotation changes the dominant mode of perturbation wave on the jet surface, and more uneven corrugated flows are formed at the interface with the increase of gas rotational strength. The fluid compressibility has little impact on jet morphology in the circumferential direction, while it has some impacts on the axial morphology, especially on the arrangement of droplets. Cavitation bubbles will affect the jet morphology, while the effect is smaller than gas rotation. Both swirling gas and fluid compressibility promote the jet breakup, while the influence of compressibility on jet breakup is obviously greater than gas rotation. In addition, the bubble volume fraction will promote the breakup of the cavitating liquid jet; however, this kind of promoting impact decreases with the increase of the cavitation bubble volume fraction.

Interaction of Rossbi waves with a jet flow: basic equations and verification for Antarctic circumpolar current

The article focuses on the interaction of Rossby waves in the ocean with zonal jet flows. A new approach is proposed to show that nonlinearity in the long-wave approximation exactly compensates the Doppler shift. A new dispersion relation for the Rossby waves interacting with the jets is deduced from the nonlinear theory. The conclusion is verified using satellite altimetry data of the Antarctic Circumpolar Current (ACC). For the ACC area, we compare empirical velocities obtained from the altimetry data with theoretical phase velocities of Rossby waves calculated from nonlinear dispersion relation using the equivalent beta effect. The comparison shows that the new dispersion relation based on the nonlinear approach is capable of describing both the westward and the eastward propagation of mesoscale eddies in the field of sea level anomalies that can be identified as manifestation of Rossby waves in the ocean.

Effect of long-range 1/r interaction on thermal and quantum depletion of a dipolar quasi-two-dimensional Bose gas

The depletion of a quasi-two-dimensional (quasi-2D) dipolar Bose condensed gas in the presence of both contact and long-range 1/r interactions is investigated in the framework of Hartree–Fock–Bogoliubov (HFB) approximation. When the characteristic wavelength of a mode is much larger than the trap size, the dipole-dipole (DD) interaction can be treated as a contact interaction and in the low momentum limit the long-range nature of the 1/r interaction has the dominant contribution and leads to the nonlinear (nonphononic) dispersion relation. We will show that quantum depletion is temperature independent and is determined by the contact, DD and longrange 1/r coupling constants (ɛdd = gd/g and C). The the quantum depletion is affected by long-range 1/r interaction and at large momentum limits the momentum dependence of quantum depletion unaffected by 1/r interaction.

Nonlinear frequency shift of acoustic waves in semiconductor plasmas

The evolution of nonlinear instability of an electron acoustic wave from the coupling of pair modes of slightly different frequencies in semiconductor plasmas is studied. The linear dispersive properties of pair modes, viz., pump wave and sideband wave, are analyzed on employing the quantum hydrodynamic model. The linear fields of pair modes couple through a second order convective term to unfold the nonlinear beat frequency of the electron acoustic mode. The nonlinear dispersion relation leads to the derivation of the growth rate of three-wave parametric instability. The quantum effects of semiconductor electrons are taken through the exchange-correlation potential, the Bohm potential, and statistical Fermi pressure. The growth rate is also analyzed graphically. This study seeks its applications in semiconductor devices.

Internally resonant wave energy exchange in weakly nonlinear lattices and metamaterials.

This paper presents a multiple-scales analysis approach capable of capturing internally resonant wave interactions in weakly nonlinear lattices and metamaterials. Example systems considered include a diatomic chain and a locally resonant metamaterial-type lattice. At a number of regions in the band structure, both the frequency and wave number of one nonlinear plane wave may relate to another in a near-commensurate manner (such as in a 2:1 or 3:1 ratio) resulting in an internal resonance mechanism. As shown herein, nonlinear interactions in the lattice couple these waves and enable energy exchange. Near such internal resonances, previously derived higher-order dispersion corrections for single plane wave propagation may break down, leading to singularities in the predicted nonlinear dispersion relationships. Using the presented multiple-scales approach and the two example systems, this paper examines internal resonance occurring (i) within the same branch and (ii) between different branches of the band structure, resolving the aforementioned singularity issue while capturing energy exchange. The multiple-scales evolution equations, together with a local stability analysis, uncover multiple stable fixed points associated with periodic energy exchange between internally resonant propagating modes. Response results generated using direct numerical simulation verify the perturbation-based predictions for amplitude-dependent dispersion corrections and slow-scale energy exchange; importantly, these comparisons verify the exchange frequency predicted by the multiple-scales approach.

Nonlinear dispersion relation in anharmonic periodic mass-spring and mass-in-mass systems

The study of wave propagation in chains of anharmonic periodic systems is of fundamental importance to understand the response of dynamical absorbers of vibrations and acoustic metamaterials working in the nonlinear regime. Here, we derive an analytical nonlinear dispersion relation for periodic chains of anharmonic mass-spring and mass-in-mass systems resulting from considering the hypothesis of factorization for the spatial and temporal parts of the solution and a periodic distribution function as ansatz of a general solution of the temporal part of the nonlinear equations of motion. A comparison with numerical simulations shows the range of validity of this expression. This work provides a tool to design and study nonlinear dynamics for some classes of seismic metamaterials such as composite foundations, perturbative metamaterials, and metasurfaces.

Interaction between Rossby Waves and a Jet Flow: Basic Equations and Verification for the Antarctic Circumpolar Current

Abstract—The article focuses on the interaction of Rossby waves in the ocean with zonal jet flows. A new approach is proposed to show that nonlinearity in the long-wave approximation exactly compensates the Doppler shift. A new dispersion relation for the Rossby waves interacting with the jets is deduced from the nonlinear theory. The conclusion is verified using satellite altimetry data of the Antarctic Circumpolar Current (ACC). For the ACC area, we compare empirical velocities obtained from the altimetry data with theoretical phase velocities of Rossby waves calculated from nonlinear dispersion relation using the equivalent beta effect. The comparison shows that the new dispersion relation based on the nonlinear approach is capable of describing both the westward and the eastward propagation of mesoscale eddies in the field of sea level anomalies that can be identified as manifestation of Rossby waves in the ocean.

Inflationary perturbation spectrum in extended effective field theory of inflation

The effective field theory (EFT) of inflation provides a natural framework to study the new physical effects on primordial perturbations. Recently a healthy extension of the EFT of inflation with high-order operators has been proposed, which avoids ghosts and meanwhile leads to a nonlinear dispersion relation of the scalar perturbations. This paper is devoted to studying the effects of these high-order operators by using the uniform asymptotic approximation method. In particular, we first construct the approximate analytical solution to the mode function of the scalar perturbations. Because of the presence of the high-order operators, the perturbation modes usually experience a period of non-adiabatic evolution before they cross the Hubble radius, which could lead to the production of excited states and modifications of the primordial perturbation spectrum. However, we show that the modified power spectrum is still nearly scale-invariant and the presence of the high-order operators can only affect the overall amplitude of the spectrum. In particular, after showing explicitly the impact of these new effects on particle production rate and perturbation spectrum, we explore their origin in detail.

Inflationary Cosmology with Quantum Gravitational Effects and Swampland Conjectures**Supported by National Natural Science Foundation of China under Grant No. 11675143 and the Fundamental Research for the Provincial Universities of Zhejiang in China under Grant No. RF-A2019015

Recently proposed two swampland criteria that arising from string theory landscape leads to the important challenge of the realization of single-field inflationary models. Especially one of swampland criteria which implies a large tensor-to-scalar ratio is strongly in tension with recent observational results. In this paper, we explore the possibility the swampland conjectures could be compatible with single-field inflationary scenarios if the effects due to the quantum theory of gravity are considered. We show that the quantum gravitational effects due to the nonlinear dispersion relation provides significant modifications on the amplitude of both the scalar and tensor perturbation spectra. Such modifications could be either raise or reduce the perturbation spectra depending on the values of the parameters in the nonlinear terms of the dispersion relations. Therefore, these effects can reduce the tensor-to-scalar ratio to a smaller value, which helps to relax the tension between the swampland conjecture and observational data.

An improved Lagrangian model for the time evolution of nonlinear surface waves

Accurate real-time simulations and forecasting of phase-revolved ocean surface waves require nonlinear effects, both geometrical and kinematic, to be accurately represented. For this purpose, wave models based on a Lagrangian steepness expansion have proved particularly efficient, as compared to those based on Eulerian expansions, as they feature higher-order nonlinearities at a reduced numerical cost. However, while they can accurately model the instantaneous nonlinear wave shape, Lagrangian models developed to date cannot accurately predict the time evolution of even simple periodic waves. Here, we propose a novel and simple method to perform a Lagrangian expansion of surface waves to second order in wave steepness, based on the dynamical system relating particle locations and the Eulerian velocity field. We show that a simple redefinition of reference particles allows us to correct the time evolution of surface waves, through a modified nonlinear dispersion relationship. The resulting expressions of free surface particle locations can then be made numerically efficient by only retaining the most significant contributions to second-order terms, i.e. Stokes drift and mean vertical level. This results in a hybrid model, referred to as the ‘improved choppy wave model’ (ICWM) (with respect to Nouguier et al.’s J. Geophys. Res., vol. 114, 2009, p. C09012), whose performance is numerically assessed for long-crested waves, both periodic and irregular. To do so, ICWM results are compared to those of models based on a high-order spectral method and classical second-order Lagrangian expansions. For irregular waves, two generic types of narrow- and broad-banded wave spectra are considered, for which ICWM is shown to significantly improve wave forecast accuracy as compared to other Lagrangian models; hence, ICWM is well suited to providing accurate and efficient short-term ocean wave forecast (e.g. over a few peak periods). This aspect will be the object of future work.

Generalised uncertainty relations from superpositions of geometries

Phenomenological approaches to quantum gravity implement a minimum resolvable length-scale but do not link it to an underlying formalism describing geometric superpositions. Here, we introduce an intuitive approach in which points in the classical spatial background are delocalised, or ‘smeared’, giving rise to an entangled superposition of geometries. The model uses additional degrees of freedom to parameterise the superposed classical backgrounds. Our formalism contains both minimum length and minimum momentum resolutions and we naturally identify the former with the Planck length. In addition, we argue that the minimum momentum is determined by the de Sitter scale, and may be identified with the effects of dark energy in the form of a cosmological constant. Within the new formalism, we obtain both the generalised uncertainty principle (GUP) and extended uncertainty principle (EUP), which may be combined to give an uncertainty relation that is symmetric in position and momentum. Crucially, our approach does not imply a significant modification of the position-momentum commutator, which remains proportional to the identity matrix. It therefore yields generalised uncertainty relations without violating the equivalence principle, in contradistinction to existing models based on nonlinear dispersion relations. Implications for cosmology and the black hole uncertainty principle correspondence are briefly discussed, and prospects for future work on the smeared-space model are outlined.

Active Dual-Band Power Divider and Active Quad-Plexer Based on Traveling Wave Amplification and D-CRLH Transmission Line

A travelling wave structure with two novel functionalities is presented in this brief. The structure benefits from the nonlinear dispersion relation of dual-composite right/left-handed (D-CRLH) transmission line (TL). It consists of two distributed amplifiers (DAs) that have one common gate line and two drain lines. The conventional microstrip TL is used for gate line while the D-CRLH unit cells (UCs) are used to make the drain TLs. When the operating frequencies of D-CRLH TLs of both drain lines are the same, the structure acts as a directional dual-band active power divider (PD). The RF input signal depending on the input signal frequency, is divided into two forward or reverse output ports after amplification. If the D-CRLH TLs have different operating frequencies with no common band, the structure can act as an active quad-plexer or directional quad band amplifier. In this situation, four amplified signals with different frequency bands can be extracted from four different physical output ports. Analysis and design procedure of the structure is provided and a fabricated prototype is measured which shows a good agreement between the simulated results and the measurement results.

Discretized dynamics of exchange spin wave bulk and edge modes in honeycomb nanoribbons with armchair edge boundaries

We develop a field theory to study the dynamics of long wavelength exchange spin wave excitations on honeycomb nanoribbons characterized by armchair edge boundaries and the Néel antiferromagnetic ordering state. Appropriate boundary conditions are established by requiring that the bulk and edge spins precess with the same frequency for any given spin wave eigenmode in these systems. A set of characteristic boundary equations, common for bulk and edge spin wave modes, are hence derived. The equations of motion for the spin dynamics are then solved to determine the propagating and evanescent exchange spin wave modes. We obtain the expected discretized bulk spin waves spectrum due to the finite width of the nanoribbon. For an isotropic magnetic nanoribbon, the Dirac cone is reduced to a single linear dispersion curve due to this discretization. The number and wavelengths of allowed bulk modes for isotropic and anisotropic nanoribbons are determined from the derived characteristic boundary equations. As witnessed by our numerical results for different examples it is shown that the characteristics of these modes depend on the width of the nanoribbon and its antiferromagnetic anisotropy. Further, anisotropic nanoribbons, even those with the slightest anisotropy, present evanescent modes with non-linear dispersion relations. The spatial variation of the amplitudes of the evanescent exchange spin waves across the finite widths of the nanoribbons, is found to be strongly dependent on the system magnetic anisotropy and its width. The developed theoretical approach is general and can be applied for nanoribbons with all types of boundary edges.

Evanescent modes and step-like acoustic black holes

We consider a model of an acoustic black hole formed by a quasi-one dimensional Bose–Einstein condensate with a step-like horizon. This system is analyzed by solving the corresponding Bogoliubov–de Gennes equation with an appropriate matching condition at the jump. When the step is between a subsonic and supersonic flow, a sonic horizon develops and in addition to the scattering coefficients we compute the distribution of the accompanying analogue Hawking radiation. Additionally, in response to the abrupt variation in flow and non-linear Bogoliubov dispersion relation, evanescent solutions of the Bogoliubov–de Gennes equation also appear and decay out from the horizon. We bound this decay length and show that these modes produce a modulation of observables outside the event horizon by their interference with outgoing Hawking flux. We go further and find specific superpositions of ingoing eigenmodes which exhibit coherent cancellation of the Hawking flux outside the horizon but nevertheless have evanescent support outside the black hole. We conclude by speculating that when quasiparticle interactions are included, evanescent modes may yield a leakage of information across the event horizon via interactions between the real outgoing Hawking flux and the virtual evanescent modes, and that we may expect this as a generic feature of models which break Lorentz invariance at the UV (Planck) scale.