Nonlinear Dispersion Research Articles

Partial degeneration of finite gap solutions to the Korteweg–de Vries equation: soliton gas and scattering on elliptic backgrounds

We obtain Fredholm type formulas for partial degenerations of theta functions on (irreducible) nodal curves of arbitrary genus, with emphasis on nodal curves of genus one. An application is the study of ‘many-soliton’ solutions on an elliptic (cnoidal) background standing wave for the Korteweg–de Vries (KdV) equation starting from a formula that is reminiscent of the classical Kay–Moses formula for N-solitons. In particular, we represent such a solution as a sum of the following two terms: a ‘shifted’ elliptic (cnoidal) background wave and a Kay–Moses type determinant containing Jacobi theta functions for the solitonic content, which can be viewed as a collection of solitary disturbances on the cnoidal background. The expressions for the travelling (group) speed of these solitary disturbances, as well as for the interaction kernel describing the scattering of pairs of such solitary disturbances, are obtained explicitly in terms of Jacobi theta functions. We also show that genus N + 1 finite gap solutions with random initial phases converge in probability to the deterministic cnoidal wave solution as N bands degenerate to a nodal curve of genus one. Finally, we derive the nonlinear dispersion relations and the equation of states for the KdV soliton gas on the residual elliptic background.

Nonlinearity enhanced wave bandgaps in metamaterial honeycombs embedding spider web-like resonators

Wave propagation in metamaterial honeycombs endowed with periodically distributed nonlinear resonators is addressed. The linear and nonlinear dispersion properties of the metamaterial are investigated. The nonlinear wave propagation equations obtained via a projection method and the Floquet–Bloch theorem are attacked by the method of multiple scales to obtain in closed form the nonlinear manifolds parametrized by the amplitudes, the frequency, and the wave numbers. The effects of the nonlinearity on the frequency bandgaps are thoroughly investigated and the optimization problem of the resonators nonlinearity towards increased bandgap size is tackled to provide a significant practical framework for the design of nonlinear metamaterials.

Influence of coulomb damping on wave propagation behaviors of nonlinear nonconservative phononic chains/lattices

Fascinating nonlinearity-induced behavior of phononic crystals (PCs) has recently become a hot research topic in the community. However, due to the limitations in the analytical modelling of damping in dynamic systems, the study of damped PCs has not received proper attention. In this paper, the influence of Coulomb damping on the wave propagation behavior of cubically nonlinear monoatomic phononic chains is investigated. To do so, the nonlinear dispersion relation is obtained analytically using the well-established multiple scales method and the band structure of the damped nonlinear chains is compared to the ones corresponding to the linear and nonlinear undamped chains. Due to the coupling between the amplitude and the frequency, stemmed from the nonlinear nature of the chain, Coulomb damping can lead to lower dispersion frequencies in the chain. The formulation and results are then expanded to 2D nonlinear lattices. The present manuscript is the first attempt to capture the effect of Coulomb damping on the wave propagation behavior of nonlinear lattices and the results put us one step closer to developing a comprehensive analytical model for the behavior of damped PCs which can in turn lead to invaluable design concepts for nonlinear nonconservative wave-manipulation devices.

Finite-amplitude nonlinear waves in inertial amplification metamaterials: Theoretical and numerical analyses

The motivation of this study is to understand the unique characteristics of finite-amplitude elastic wave propagation in an inertial amplification (IA) system coupled with a Duffing resonator and leverage the unique opportunities offered by the IA mechanism and nonlinearities to achieve the modulation of elastic waves. Based on the assumption of the perturbation method, the effective range of the wave amplitude was investigated, which is usually ignored. Meanwhile, the obvious phenomenon of bandgap transition (between the single resonance bandgap and hybrid bandgap) is observed with varying IA levels. In particular, we demonstrated a new phenomenon of negative wave propagation and an obvious standing-wave mode in this system. Here, an analytical investigation of the amplitude-dependent and IA-dependent nonlinear dispersion is conducted based on the perturbation approach. Numerical simulations using the time-domain finite element method and Gaussian pulse excitation were performed to validate the nonlinear dispersion and negative motion of the wave packet. The research results can be used to tune wave propagation in metamaterials and provide ideas for the design of metamaterials, which can be used in practical circumstances for different energy transfer needs.

Radiative pumping of exciton-polaritons in 2D hybrid perovskites

In addition to their attractive technological applications in photovoltaics and light emitters, the perovskite family of semiconductors has recently emerged as an excellent excitonic material for fundamental studies. Specifically, the 2D hybrid organic-inorganic perovskite (HOIP) offers the added advantage of room temperature investigations owing to their large exciton binding energy. In this work, we strongly couple excitons in 2D HOIP crystals to planar microcavity photons sustaining exciton-polaritons under ambient conditions resulting in a Rabi splitting of 290 meV. Dark excitons directly pump the polariton branch along its dispersion in resonance with the Stokes shifted emission state (radiative pumping), creating a high density of polaritons at higher in-plane momentum (k||). We further probe the nonlinear polariton dispersion dynamics at varying input laser fluence, which indicates efficient polariton-polariton scattering and decay to k|| = 0 from higher k||. The observation of Stokes shift-assisted energy exchange of dark states with lower polaritons coupled with evidence of efficient polariton-polariton scattering makes 2D HOIPs an attractive platform to study exciton-polariton many-body physics and Bose-Einstein like condensation (BEC) at room temperature.

Modulational instability in PT -symmetric Bragg grating structures with saturable nonlinearity

We investigate the nontrivial characteristics of modulational instability (MI) in a system of Bragg gratings with saturable nonlinearity. We also introduce an equal amount of gain and loss into the existing system, which gives rise to an additional degree of freedom, due to the concept of $\mathcal{PT}$ symmetry. We obtain the nonlinear dispersion relation of the saturable model and discover that such dispersion relations for both the conventional and $\mathcal{PT}$-symmetric cases contradict the conventional Kerr and saturable systems by not displaying the typical signature of loop formation in either the upper branch or lower branch of the curve drawn against the wave number and detuning parameter. We employ a standard linear stability analysis to study the MI dynamics of the continuous waves perturbed by an infinitesimal perturbation. The main objective of this paper is twofold: We investigate the dynamics of the MI gain spectrum at the top and bottom of the photonic band gap followed by a comprehensive analysis carried out in the anomalous and normal dispersion regimes. As a result, this perturbed system driven by the saturable nonlinearity and gain or loss yields a variety of instability spectra, which include the conventional sidebands, monotonically increasing gain, the emergence of a single spectrum in either of the Stokes wave-number regions, and so on. In particular, we observe a remarkably peculiar spectrum, which is caused predominantly by the system parameter, though the perturbation wave number boosts the former. We also address the impact of all the physical parameters considered in the proposed model, which include the coupling coefficient, dispersion parameter, and saturable nonlinearity on the phenomenon of MI for different $\mathcal{PT}$-symmetric regimes ranging from the unbroken one to the broken one in greater detail.

Particle creation and entanglement in a dispersive model with step velocity profile

We investigate particle creation and entanglement structure in a dispersive model with subliminal dispersion relation. Assuming the step function spatial velocity profile of the background flow, mode functions for a massless scalar field is exactly obtained by the matching method. Power spectrums of created particles are calculated for the subsonic and the transsonic flow cases. For the transsonic case, the sonic horizon exists and created particles show the Planckian distribution for low frequency region but the thermal property disappears for high frequency region near the cutoff frequency introduced by the non-linear dispersion. For the subsonic case, although the sonic horizon does not exist, the effective group velocity horizon appears due to the non-linear dispersion for high frequency region and approximate thermal property of the power spectrum arises. Relation between particle creation and entanglement between each mode is also discussed.

Widely Tunable Femtosecond Sources with Continuously Tailorable Bandwidth Enabled by Self‐Phase Modulation

AbstractGenerating ultrafast pulses with better spectrotemporal control is crucial for optimizing and characterizing nonlinear light–matter responses, yet it is limited by the gain bandwidth of laser media or the phase‐matching geometry of nonlinear processes. This work proposes a simple approach to independently manage a femtosecond source's spectral location and bandwidth. Self‐phase‐modulation‐enabled spectral broadening is first analyzed, which is potentially energy‐scalable using hollow‐core capillaries or multipass cells. It is demonstrated that the outmost lobes in the broadened spectrum show different dependencies on the initial pulse energy and duration. A simple yet effective toy model is introduced that successfully predicts broadband spectral tuning, and the impact of other nonlinear effects, dispersion, and input pulse asymmetry on the experimental scenario is also discussed. Thus a fiber‐based versatile source is demonstrated, which is compressible down to its transform‐limit duration, as short as 12.2 fs centered at 920 nm. In addition, bandwidth‐dependent third‐harmonic generation spectroscopy is performed from a dielectric metasurface with an optimized nonlinear response, and the dependency of laser bandwidth and pulse duration is investigated on the signal‐to‐background ratio of two‐photon images. It is believed that this demonstration will advance the investigation of bandwidth‐dependent nonlinear spectroscopy and microscopy.

Effects of metallic electrodes on the thermoelectric properties of zigzag graphene nanoribbons with periodic vacancies

We theoretically analyze the thermoelectric properties of graphene quantum dot arrays (GQDAs) with line- or surface-contacted metal electrodes. Such GQDAs are realized as zigzag graphene nanoribbons (ZGNRs) with periodic vacancies. Gaps and minibands are formed in these GQDAs, which can have metallic and semiconducting phases. The electronic states of the first conduction (valence) miniband with nonlinear dispersion may have long coherent lengths along the zigzag edge direction. With line-contacted metal electrodes, the GQDAs have the characteristics of serially coupled quantum dots (SCQDs) if the armchair edge atoms of the ZGNRs are coupled to the electrodes. By contrast, the GQDAs have the characteristics of parallel quantum dots if the zigzag edge atoms are coupled to the electrodes. The maximum thermoelectric power factors of SCQDs with line-contacted electrodes of Cu, Au, Pt, Pd, or Ti at room temperature were similar or greater than 0.186 nW K−1; their figures of merit were greater than three. GQDAs with line-contacted metal electrodes have much better thermoelectric performance than surface contacted metal electrodes.

Propagation of chirped gray solitons in weakly nonlocal media with parabolic law nonlinearity and spatio-temporal dispersion

In our research, we have examined the existence and stability of chirped periodic and solitary waves in a weakly nonlocal nonlinear medium, which exhibits various types of effects, including inter-modal dispersion, nonlinear dispersion, detuning, spatio-temporal, and parabolic law nonlinearity. By studying the nonlinear Schrödinger equation that describes the field dynamics in this system, a class of nonlinearly chirped periodic waves is derived in the presence of all physical processes. In addition, we have obtained solitary waves of the gray type in the long-wave limit of these nonlinear waveforms. We have found that the frequency chirp associated with these optical waves depends on their intensity and its magnitude can be controlled by manipulating the nonlinear dispersion parameter. Furthermore, we have numerically studied the stability of the gray soliton solution under finite initial perturbations. Our results indicate that the nonlinear waves we have identified represent new types of extremely robust chirped localized structures in weakly nonlocal nonlinear parabolic law media.

Nonlinear Waves and Dispersive Equations

Nonlinear dispersive equations are models for nonlinear waves in a wide range of physical contexts. Mathematically they display an interplay between linear dispersion and nonlinear interactions, which can result in a wide range of outcomes from finite time blow-up to solitons and scattering. They are linked to many areas of mathematics and physics, ranging from integrable systems and harmonic analysis to fluid dynamics, geometry and general relativity and probability.

KdV breathers on a cnoidal wave background

Using the Darboux transformation for the Korteweg–de Vries equation, we construct and analyze exact solutions describing the interaction of a solitary wave and a traveling cnoidal wave. Due to their unsteady, wavepacket-like character, these wave patterns are referred to as breathers. Both elevation (bright) and depression (dark) breather solutions are obtained. The nonlinear dispersion relations demonstrate that the bright (dark) breathers propagate faster (slower) than the background cnoidal wave. Two-soliton solutions are obtained in the limit of degeneration of the cnoidal wave. In the small amplitude regime, the dark breathers are accurately approximated by dark soliton solutions of the nonlinear Schrödinger equation. These results provide insight into recent experiments on soliton-dispersive shock wave interactions and soliton gases.

Single Mode Multi-Frequency Factorization Method for the Inverse Source Problem in Acoustic Waveguides

This paper investigates the inverse source problem with a single propagating mode at multiple frequencies in an acoustic waveguide. The goal is to provide both theoretical justification and efficient algorithms for imaging extended sources using the sampling methods. In contrast to the existing far/near-field operator based on the integral over the space variable in the sampling methods, a multi-frequency far-field operator is introduced based on the integral over the frequency variable. This far-field operator is defined in a way to incorporate the possibly nonlinear dispersion relation, a unique feature in waveguides. The factorization method is deployed to establish a rigorous characterization of the range support, which is the support of source in the direction of wave propagation. A related factorization-based sampling method is also discussed. These sampling methods are shown to be capable of imaging the range support of the source. Numerical examples are provided to illustrate the performance of the sampling methods, including an example to image a complete sound-soft block.

Quiescent Optical Solitons for the Concatenation Model with Nonlinear Chromatic Dispersion

This paper recovers quiescent optical solitons that are self-sustaining, localized wave packets that maintain their shape and amplitude over long distances due to a balance between nonlinearity and dispersion. When a soliton is in a state of quiescence, it means that it is stationary in both space and time. Quiescent optical solitons are typically observed in optical fibers, where nonlinearity and dispersion can lead to the formation of solitons. The concatenation model is considered to understand the behavior of optical pulses propagating through nonlinear media. Here, we consider the familiar nonlinear Schrödinger equation, the Lakshmanan–Porsezian–Daniel equation, and the Sasa–Satsuma equation. The current paper also addresses the model with nonlinear chromatic dispersion, a phenomenon that occurs in optical fibers and other dispersive media, where the chromatic dispersion of the material is modified by nonlinear effects. In the presence of nonlinearities, such as self-phase modulation and cross-phase modulation, the chromatic dispersion coefficient becomes a function of the optical intensity, resulting in nonlinear chromatic dispersion. A full spectrum of stationary optical solitons, along with straddled stationary solitons, are obtained. There are four integration schemes that made this retrieval possible. The numerical simulations are also included for these solitons. The parameter constraints also indicate the existence criteria for these quiescent solitons.

Modelling and Analysis of the Dispersal of a Polymeric Pollutant Injected into a Channel Flow of a Newtonian Liquid

The transient dynamics of nonlinear dispersion of a polymeric pollutant ejected by an external source into a laminar flow of a Newtonian liquid flowing through a rectangular channel is investigated. The Boussinesq approximation is assumed for the density variation with pollutant concentration. The governing equations of mass and momentum conservation are coupled to the pollutant concentration equation as well as to the viscoelastic constitutive model for the polymer stresses. The Oldroyd-B viscoelastic constitutive model is employed to model the deformation and characteristics of the polymer stresses. The coupled system of nonlinear partial differential equations is solved numerically using robust and efficient semi-implicit finite difference methods (FDM). Solutions are presented in graphical form for various parameter values. The model can be a useful tool in understanding the dynamics of domestic and industrial pollution situations that may arise from improper discharge of long-chain hydrocarbon products into, say, water drainage systems. The novelty of this investigation is in the modelling of the long-chain hydrocarbon-product pollutants via appropriate viscoelastic (polymeric) constitutive equations. In general, it is observed that parameters which increase (decrease) the flow velocity correspondingly increase (respectively decrease) the wall shear stress. Similarly, it is observed that parameters which increase (decrease) the polymer concentration correspondingly increase (respectively decrease) the mass transfer rates. The wall shear stress and mass transfer are measurable quantities. In this respect, our work offers such measurements as predictive tools to detect the scale of contamination.

New soliton solutions of the mZK equation and the Gerdjikov-Ivanov equation by employing the double [formula omitted]-expansion method

In the electrical transmission lines, the processing of cable signals distribution, computer networks, high-speed computer databases and discrete networks can be investigated by the modified Zakharov-Kuznetsov (mZK) equation as a data link propagation control model in the study of nonlinear Schrödinger type equations as well as in the analysis of the generalized stationary Gardner equation. The proposed Gerdjikov–Ivanov model can be used in the field of nonlinear optics, weakly nonlinear dispersion water waves, quantum field theory etc. In this work, we developed complete traveling wave solutions with specific t-type, kink type, bell-type, singular solutions, and periodic singular solutions to the proposed mZK equation and the Gerdjikov-Ivanov equation with the aid of the double G′/G,1/G- expansion method. These settled solutions are very reliable, durable, and authentic which can measure the fluid velocity and fluid density in the electrically conductive fluid and be able to analysis of the flow of current and voltage of long-distance electrical transmission lines too. These traveling wave solutions are available in a closed format and make them easy to use. The proposed method is consistent with the abstraction of traveling wave solutions.

Analytical method for modal analysis of submerged electric motor stator

Modal analysis of the stator is important for reducing motor noise and diagnosing motor faults. An accurate analytical model is beneficial for the establishment of the relationship between the motor stator parameters and the natural frequencies, which will further aid in optimizing machine designs. Up to now, the existing modal analysis models are mainly aimed at onshore motors. When the motor is submerged in dense fluid, such as those used in the rim-driven thrusters, the vibration behaviors are quite different due to the high characteristic impedance of the fluid. In a bid to analyze the wet natural frequencies of motor stator, this paper proposes analytical models with one, three and five degrees of freedom for different application scenarios based on Donnell, Love and Soedel’s theories. The coupling between the structure and fluid is realized by the acoustic pressure load derived from the acoustic wave function. The resulting non-linear dispersion equations are then solved by Newton’s iterative method. The proposed models are applied to analyze the natural frequencies of motor stators of various sizes and with a wide range of thickness-to-radius and length-to-radius ratios. The results show that the analytical models are insensitive to the stator radius but sensitive to the thickness-to-radius and length-to-radius ratios. For thick and short motor stators, it is necessary to consider the shear deformation and rotatory inertia in the analytical model to compute the natural frequency accurately, especially when the fluid loading is taken into account.

Notes on wave-breaking phenomena for a Fornberg-Whitham-type equation

We investigate the wave-breaking mechanism of solutions to a Fornberg-Whitham-type equation. By means of a weaker conserved L2-norm, we show some wave-breaking criteria for the equation, which improve some blow-up results on the classical Fornberg-Whitham equation. Moreover, our conclusions suggest that the wave breaking for the equation may occur even with small slope of the initial value. To analyze the interaction between nonlinear and nonlocal dispersion terms, we provide two different approaches. The first one is based on a subtle analysis on evolution of the solution u and its gradient ux. The other is to make full use of the known blow-up results on Riccati-type inequalities with t-dependent functions.

Homotopy analysis of wave transformation over permeable seabeds and porous structures

Based on the Homotopy Analysis Method (HAM), a semi-analytical solution method is proposed to solve the modified mild slope equation (MMSE) for waves propagation over permeable seabeds and porous structures. The present method overcomes the mathematical difficulties in treating the nonlinear dispersion equations introduced by the permeability effect, which limits the previous semi-analytical solutions to impermeable seabed and structures only. The proposed method is applied to both 1-D and 2-D axisymmetric wave transformation problems. The method is validated by systematically comparing the semi-analytical solutions of wave amplitudes and reflection coefficients against available experimental data and numerical results, demonstrating the method's capability for predicting wave transformation over both impermeable and permeable seabeds. It is ready to show that, for impermeable conditions, the present semi-analytical solutions degenerate into the earlier solutions without considering the porous effect. The semi-analytical solution method is further extended to examine the porous effects on wave transformation and reflection in various topographical conditions.

Revisiting resources, cleaner energy and sustainable economic performance: the role of cleaner electricity from BRICS economies

Since the last three decades, the natural resources and energy sector has been regarded as a substantial determinant of economic growth and prosperity. However, the literature is still contradictory and scant for BRICS economies and needs more empirical evidence for appropriate policies. This study aims to analyze the influence of natural resources and cleaner energy on economic performance in the presence of consumption expenditure. Using second-generation diagnostic, unit root, and cointegration tests, the results verify the long-run equilibrium association between the variables during 1990-2021. Due to non-linear data dispersion, the study uses the method of moment quantile regression. It concludes that both natural resources and cleaner energy have an asymmetric influence on the economic growth of these nations. The region’s significant factors of economic growth are electricity production from renewable sources and oil rents. Renewable electricity output and mineral rents adversely affect economic growth and performance. Besides, consumption expenditure is also negatively and significantly affecting economic growth. Based on the empirical outcomes, this study also suggested policies that could benefit the BRICS economic growth and performance.