Higher-order Nonlinearities Research Articles

Fast Sparse Grid Simulations of Fifth Order WENO Scheme for High Dimensional Hyperbolic PDEs

The weighted essentially non-oscillatory (WENO) schemes, especially the fifth order WENO schemes, are a popular class of high order accurate numerical methods for solving hyperbolic partial differential equations (PDEs). However when the spatial dimensions are high, the number of spatial grid points increases significantly. It leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as a fifth order WENO scheme. How to achieve fast simulations by high order WENO methods for high spatial dimension hyperbolic PDEs is a challenging and important question. In the literature, sparse-grid technique has been developed as a very efficient approximation tool for high dimensional problems. In a recent work [Lu, Chen and Zhang, Pure and Applied Mathematics Quarterly, 14 (2018) 57-86], a third order finite difference WENO method with sparse-grid combination technique was designed to solve multidimensional hyperbolic equations including both linear advection equations and nonlinear Burgers’ equations. Numerical experiments showed that WENO computations on sparse grids achieved comparable third order accuracy in smooth regions of the solutions and nonlinear stability as that for computations on regular single grids. In application problems, higher than third order WENO schemes are often preferred in order to efficiently resolve the complex solution structures. In this paper, we extend the approach to higher order WENO simulations specifically the fifth order WENO scheme. A fifth order WENO interpolation is applied in the prolongation part of the sparse-grid combination technique to deal with discontinuous solutions. Benchmark problems are first solved to show that significant CPU times are saved while both fifth order accuracy and stability of the WENO scheme are preserved for simulations on sparse grids. The fifth order sparse grid WENO method is then applied to kinetic problems modeled by high dimensional Vlasov based PDEs to further demonstrate large savings of computational costs by comparing with simulations on regular single grids. Several open problems are discussed at last.

Parameter modulation of periodic waves and solitons in metamaterials with higher-order dispersive and nonlinear effects

A generalized nonlinear Schrodinger equation with higher-order dispersive and nonlinear effects is presented to govern the pulse propagation in negative index materials. Via the mapping method, Jacobian elliptic function solutions and their corresponding soliton solutions are found. Parameter modulation of higher-order dispersive effect and nonlinearities for periodic waves and solitons is studied. With the addition of the third dispersion parameter, or the quintic nonlinear parameter, the change of real and imaginary parts of periodic waves and solitons is same; namely, their amplitudes increase, and yet their widths decrease. Moreover, phase transitions of solutions with parameter modulation are found.

Robust Forecasting-Aided State Estimation in Power Distribution Systems With Event-Triggered Transmission and Reduced Mixed Measurements

This paper is concerned with the forecasting aided state estimation (FASE) problem for power distribution networks (PDNs) subject to nonlinear measurements and limited communication resources. For the sake of using bandwidth more effectively, a special closed-loop event triggering condition is first designed with the hope to reduce unnecessary data transmission from the measurement side to the remote estimator. And an event-based robust filtering algorithm is proposed to improve the estimation performance of estimator with the nonlinear intermittent observations. In this algorithm, the measurement output function with high-order nonlinearity is reasonably reduced to a quadratic alternative form, in order to avoid the computation burden caused by the use of the second-order extended Kalman filter (SOEKF), the quadratic nonlinear function is further linearized by Taylor series with quadratic norm bound. Furthermore, an upper bound of the filtering error covariance containing some uncertainties (includes quadratic term and non-triggering errors) is derived and subsequently minimized at each time step. The desired filter gain is obtained by recursively solving a set of Riccati-like matrix equations, which is easy to implement via online computation. Finally, the effectiveness of the proposed algorithm is demonstrated using the standard IEEE 13-bus and 123-bus distribution test feeders with both phasor measurement unit (PMU) and conventional measurements.

Assessing additive effects of air pollutants on mortality rate in Massachusetts

BackgroundWe previously found additive effects of long- and short-term exposures to fine particulate matter (PM2.5), ozone (O3), and nitrogen dioxide (NO2) on all-cause mortality rate using a generalized propensity score (GPS) adjustment approach. The study addressed an important question of how many early deaths were caused by each exposure. However, the study was computationally expensive, did not capture possible interactions and high-order nonlinearities, and omitted potential confounders.MethodsWe proposed two new methods and reconducted the analysis using the same cohort of Medicare beneficiaries in Massachusetts during 2000–2012, which consisted of 1.5 million individuals with 3.8 billion person-days of follow-up. The first method, weighted least squares (WLS), leveraged large volume of data by aggregating person-days, which gave equivalent results to the linear probability model (LPM) method in the previous analysis but significantly reduced computational burden. The second method, m-out-of-n random forests (moonRF), implemented scaling random forests that captured all possible interactions and nonlinearities in the GPS model. To minimize confounding bias, we additionally controlled relative humidity and health care utilizations that were not included previously. Further, we performed low-level analysis by restricting to person-days with exposure levels below increasingly stringent thresholds.ResultsWe found consistent results between LPM/WLS and moonRF: all exposures were positively associated with mortality rate, even at low levels. For long-term PM2.5 and O3, the effect estimates became larger at lower levels. Long-term exposure to PM2.5 posed the highest risk: 1 μg/m3 increase in long-term PM2.5 was associated with 1053 (95% confidence interval [CI]: 984, 1122; based on LPM/WLS methods) or 1058 (95% CI: 988, 1127; based on moonRF method) early deaths each year among the Medicare population in Massachusetts.ConclusionsThis study provides more rigorous causal evidence between PM2.5, O3, and NO2 exposures and mortality, even at low levels. The largest effect estimate for long-term PM2.5 suggests that reducing PM2.5 could gain the most substantial benefits. The consistency between LPM/WLS and moonRF suggests that there were not many interactions and high-order nonlinearities. In the big data context, the proposed methods will be useful for future scientific work in estimating causality on an additive scale.

Higher order dynamic mode decomposition of wind pressures on square buildings

Wind pressures on buildings with different aspect ratios were investigated via higher-order dynamic mode decomposition (HODMD) in this study. Taken’s embedding theorem was used to augment the spatial dimensionality of the original snapshot matrix by introducing time delay coordinates. HODMD was applied to reveal the spatial-temporal evolution characteristics of fluctuating wind pressures on building surfaces. It was found that HODMD can successfully extract the modes and corresponding frequencies from the random pressure field. As the aspect ratio increases, the main mode is more dominant than other modes. Moreover, comparisons between HODMD and proper orthogonal decomposition (POD) suggests that the first-order mode shapes extracted by HODMD and POD are generally similar, and both can reflect the main characteristics of the fluctuating pressure field. However, the higher-order mode shapes extracted by HODMD and POD are different due to higher-order nonlinearities. The modes of HODMD oscillate at a fixed frequency while those of POD resemble random signals, indicating the physical meaning of HODMD is more evident. Furthermore, HODMD is proven to be superior to POD in reconstructing pressure fields with less root mean square errors, because HODMD reconstructs the pressure field directly while POD primarily aims to rebuild the energy field.

Tunable parametric amplification of a graphene nanomechanical resonator in the nonlinear regime

Parametric amplification is widely used in nanoelectro-mechanical systems to enhance the transduced mechanical signals. Although parametric amplification has been studied in different mechanical resonator systems, the nonlinear dynamics involved receives less attention. Taking advantage of the excellent electrical and mechanical properties of graphene, we demonstrate electrical tunable parametric amplification using a doubly clamped graphene nanomechanical resonator. By applying external microwave pumping with twice the resonant frequency, we investigate parametric amplification in the nonlinear regime. We experimentally show that the extracted coefficient of the nonlinear Duffing force α and the nonlinear damping coefficient η vary as a function of external pumping power, indicating the influence of higher-order nonlinearity beyond the Duffing (∼x 3) and van der Pol (∼) types in our device. Even when the higher-order nonlinearity is involved, parametric amplification still can be achieved in the nonlinear regime. The parametric gain increases and shows a tendency of saturation with increasing external pumping power. Further, the parametric gain can be electrically tuned by the gate voltage with a maximum gain of 10.2 dB achieved at the gate voltage of 19 V. Our results will benefit studies on nonlinear dynamics, especially nonlinear damping in graphene nanomechanical resonators that has been debated in the community over past decade.

An Exact Solution to the Quadratic Damping Strong Nonlinearity Duffing Oscillator

The nonlinear equations of motion such as the Duffing oscillator equation and its family are seldom addressed in intermediate instruction in classical dynamics; this one is problematic because it cannot be solved in terms of elementary functions before. Thus, in this work, the stability analysis of quadratic damping higher-order nonlinearity Duffing oscillator is investigated. Hereinafter, some new analytical solutions to the undamped higher-order nonlinearity Duffing oscillator in the form of Weierstrass elliptic function are obtained. Posteriorly, a novel exact analytical solution to the quadratic damping higher-order nonlinearity Duffing equation under a certain condition (not arbitrary initial conditions) and in the form of Weierstrass elliptic function is derived in detail for the first time. Furthermore, the obtained solutions are camped to the Runge–Kutta fourth-order (RK4) numerical solution.

Interplay of multiphoton absorption, Raman scattering, and third-order dispersion in soliton fiber lasers

We theoretically investigate the generation of dissipative solitons (DSs) and interactions between them in a fiber laser with higher-order dispersion and nonlinearity, multiphoton absorption, and gain dispersion or spectral filtering. A random component of the group-velocity dispersion (GVD) is taken into account too. The DSs are stabilized by the dynamical balance of the dispersion terms by the cubic–quintic nonlinearity, along with the balancing of the losses by a linear gain. Novel findings are presented for effects of the third-order GVD and intra-pulse stimulated Raman scattering on the formation and interactions of DSs in the system. A possibility of all-optical control of interactions between DSs by means of phase and temporal shifts between them is elaborated. The stability of the DSs against relatively large random-noise perturbations is explored too.

A new partial differential equation for image inpainting

A considerable interest in the inpainting problem have attracted many researchers in applied mathematics community. In fact in the last decade, nonlinear high order partial dierential equations have payed a central role in high quality inpainting developments. In this paper, we propose a technique for inpainting that combines an anisotropic diusion process with an edge-corner enhancing shock ltering. This technique makes use of a partial differential equation that is based on a nonlinear structure tensor which increases the accuracy and robustness of the coupled diusion and shock ltering. A methodology of partition and adjustment is used to estimate the contrast parameters that control the strength of the diffusivity functions. We focus on restoring large missing regions in grey scale images containing complex geometries parts. Our model is extended to a three dimensional case, where numerical experimentations were carried out on lling brain multiple sclerosis lesions in medical images. The efficiency and the competitiveness of the proposed algorithm is numerically compared to other approaches on both synthetic and real images.

Analysis of the CCFD method for MC-based image denoising problems

Image denoising using mean curvature leads to the problem of solving a nonlinear fourth-order integro-differential equation. The nonlinear fourth-order term comes from the mean curvature regularization functional. In this paper, we treat this high-order nonlinearity by reducing the nonlinear fourth-order integro-differential equation to a system of first-order equations. Then a cell-centered finite difference scheme is applied to this system. With a lexicographical ordering of the unknowns, the discretization of the mean curvature functional leads to a block pentadiagonal matrix. Our contributions are fourfold: (i) we give a new method for treating the high-order nonlinearity term; (ii) we express the discretization of this term in terms of simple matrices; (iii) we give an analysis for this new method and establish that the error is of first order; and (iv) we verify this theoretical result by illustrating the convergence rates in numerical experiments.

Revisiting Soliton Dynamics in Fiber Optics Under Strict Photon-Number Conservation

We revisit the complex interplay between the Raman-induced frequency shift (RIFS) and the effect of self-steepening (SS) in the propagation of solitons, and in the framework of an equation that ensures strict conservation of the number of photons. The generalized nonlinear Schrödinger equation (GNLSE) is shown to severely fail in preserving the number of photons for sub-100-fs solitons, leading to a large overestimation of the frequency shift. Furthermore, when considering the case of a frequency-dependent nonlinear coefficient, the GNLSE also fails to provide a good estimation of the time shift experienced by the soliton. We tackle these shortcomings of the GNLSE by resorting to the recently introduced photon-conserving GNLSE (pcGNLSE) and study the interplay between the RIFS and self-steepening. As a result, we make apparent the impact of higher-order nonlinearities on short-soliton propagation and propose an original and direct method for the estimation of the second-order nonlinear coefficient.

Third-order Riemann pulses in optical fibers.

We introduce the concept of third-order Riemann pulses in nonlinear optical fibers. These pulses are generated when properly tailored input pulses propagate through optical fibers in the presence of higher-order dispersion and Kerr nonlinearity. The local propagation speed of these optical wave packets is governed by their local amplitude, according to a rule that remains unchanged during propagation. Analytical and numerical results exhibit a good agreement, showing controllable pulse steepening and subsequent shock wave formation. Specifically, we found that the pulse steepening dynamic is predominantly determined by the action of higher-order dispersion, while the contribution of group velocity dispersion is merely associated with a shift of the shock formation time relative to the comoving frame of the pulse evolution. Unlike standard Riemann waves, which exclusively exist within the strong self-defocusing regime of the nonlinear Schrödinger equation, such third-order Riemann pulses can be generated under both anomalous and normal dispersion conditions. In addition, we show that the third-order Riemann pulse dynamics can be judiciously controlled by a phase chirping parameter directly included in the initial chirp profile of the pulse.

Peregrine Solitons of the Higher-Order, Inhomogeneous, Coupled, Discrete, and Nonlocal Nonlinear Schrödinger Equations

This study reviews the Peregrine solitons appearing under the framework of a class of nonlinear Schrödinger equations describing the diverse nonlinear systems. The historical perspectives include the various analytical techniques developed for constructing the Peregrine soliton solutions, followed by the derivation of the general breather solution of the fundamental nonlinear Schrödinger equation through Darboux transformation. Subsequently, we collect all forms of nonlinear Schrödinger equations, involving systematically the effects of higher-order nonlinearity, inhomogeneity, external potentials, coupling, discontinuity, nonlocality, higher dimensionality, and nonlinear saturation in which Peregrine soliton solutions have been reported.

Efficient generation of extreme terahertz harmonics in three-dimensional Dirac semimetals

Frequency multiplication of terahertz signals on a solid state platform is highly sought-after for the next generation of high-speed electronics and the creation of frequency combs. Solutions to efficiently generate extreme harmonics (up to the $31^{\rm{st}}$ harmonic and beyond) of a terahertz signal with modest input intensities, however, remain elusive. Using fully nonperturbative simulations and complementary analytical theory, we show that 3D Dirac semimetals (DSMs) have enormous potential as compact sources of extreme terahertz harmonics, achieving energy conversion efficiencies beyond $10^{-5}$ at the $31^{\rm{st}}$ harmonic with input intensities on the order of $10$ MW/cm$^2$, over $10^5$ times lower than in conventional THz high harmonic generation systems. Our theory also reveals a fundamental feature in the nonlinear optics of 3D DSMs: a distinctive regime where higher-order optical nonlinearity vanishes, arising as a direct result of the extra dimensionality in 3D DSMs compared to 2D DSMs. Our findings should pave the way to the development of efficient platforms for high-frequency terahertz light sources and optoelectronics based on 3D DSMs.

Implementation of Dual-Nonlinearity Mechanism for Bandwidth Extension of MEMS Multi-Modal Energy Harvester

This paper presents a planar architecture implementing dual-nonlinearity mechanism for broadening bandwidth of a two-degree-of-freedom MEMS energy harvester. The proposed device consists of an electromagnetic-based inner resonator for power generation, and an outer resonator for accessory frequency tuning. In comparison with conventional 2DOF configuration, different high-order nonlinearity is subjected to the first two resonances by nonlinear springs incorporated into the inner and outer resonator, respectively, achieving double hardening responses with increased bandwidth at elevated excitation. Moreover, independent tuning of the resonant frequency and working bandwidth of both resonances could be achieved through concordant control of inner and outer nonlinear resonator, and with further adjust of mass and frequency ratio, the low-efficiency power generation range separating adjacent resonances could be effectively reduced. It's numerically shown that with proper nonlinearity at both oscillation stages, the approximate convergence of two nonlinear resonances with broadband and narrow low-efficiency range could be achieved following the mass ratio of μ <; 0.07 and linear frequency ratio of 0.7 <; α <; 1 between inner and outer stage. At acceleration of 1 g, test result further confirms that under dual-nonlinearity with nonlinearity ratio of 0.5, a continuous energy harvesting spectrum covering two nonlinear responses is realized with mass ratio of 0.02 and frequency ratio of 1, showing 3-dB bandwidth of 65 Hz. Power output of 127.7 μW and normalized power density of 1064 μWcm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> g <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> are achieved with the optimum load resistance of 37.5 Ω. This mechanism could be further applied to multi-DOF VEH for ultra-wide bandwidth of energy harvesting. [2020-0231]

Backstepping Fuzzy Sliding Mode Control for the Antiskid Braking System of Unmanned Aerial Vehicles

This paper proposes a backstepping fuzzy sliding mode control method for the antiskid braking system (ABS) of unmanned aerial vehicles (UAVs). First, the longitudinal dynamic model of the UAV braking system is established and combined with the model of the electromechanical actuator (EMA), based on reasonable simplification. Subsequently, to overcome the higher-order nonlinearity of the braking system and ensure the lateral stability of the UAV during the braking process, an ABS controller is designed using the barrier Lyapunov function to ensure that the slip ratio can track the reference value without exceeding the preset range. Then, a power fast terminal sliding mode control algorithm is adopted to realize high-performance braking pressure control, which is required in the ABS controller, and a fuzzy corrector is established to improve the dynamic adaptation of the EMA controller in different braking pressure ranges. The experimental results show that the proposed braking pressure control strategy can improve the servo performance of the EMA, and the hardware in loop (HIL) experimental results indicate that the proposed slip ratio control strategy demonstrates a satisfactory performance in terms of stability under various runway conditions.

Pockels soliton microcomb

Kerr soliton microcombs have recently emerged as a prominent topic in integrated photonics and have enabled new horizons for optical frequency metrology. Kerr soliton microcombs, as the name suggests, are based on high-order cubic optical nonlinearity. It is desirable to exploit quadratic photonic materials, namely Pockels materials, for soliton generation and on-chip implementation of 1f–2f comb self-referencing. Such quadratically driven solitons have been proposed theoretically, but have not yet been observed in a nanophotonic platform, despite recent progress in quadratic comb generation in free-space and crystalline resonators. Here, we report photonic-chip-based Pockels microcomb solitons driven by three-wave mixing in an aluminium nitride microring resonator. In contrast to typical Kerr solitons, the Pockels soliton features unity soliton generation fidelity, two-by-two evolution of multi-soliton states, favourable tuning dynamics and high pump-to-soliton conversion efficiency. Photonic-chip-based microcomb solitons driven by Pockels nonlinearity—the quadratic χ(2) effect—instead of the Kerr soliton are demonstrated in an aluminium nitride microring resonator with a conversion efficiency of 17%.

On the approximate solutions to a damped harmonic oscillator with higher-order nonlinearities and its application to plasma physics: semi-analytical solution and moving boundary method

In this paper, a strongly nonlinear oscillator equation with higher-order nonlinear (cubic) restoring force, namely damped Duffing equation with higher-order nonlinearities, has been solved and analyzed analytically and numerically using some effectiveness and high-accuracy methods. Firstly, the physical oscillator system consisting of a spring–mass moves under cubic resorting force with constant spring is considered for getting damped Duffing equation. After that a semi-analytical solution is obtained in terms of the Jacobian elliptic function cn. Also, the numerical approximate solution using Runge–Kutta-4th (RK4) procedure is carried out and compared to the semi-analytical solution. It is found that the agreement between both solutions is very good especially if the coefficient of the cubic nonlinear term becomes larger than the coefficient of the linear one. However, we made some improvements to the semi-analytical solution using the numerical moving boundary method to get higher accuracy solution. Furthermore, the error analysis is estimated for both the semi-analytical solution, improved semi-analytical solution, and RK4 solution. The method that is used for getting the semi-analytical solution is elementary, i.e., allows to obtain only integrable case without using any Lie group theory. In another meaning, our approach gives exact/analytical solution for the integrable case such as undamped Duffing equation. Moreover, the t-intercepts and amplitude points are estimated precisely. Furthermore, the plasma oscillations are discussed in the framework of damped Duffing equation. Firstly, we reduced the fluid basic equations of pair-ion plasmas with Maxwellian electrons using reductive perturbation technique to a modified korteweg de-Vries Burgers (mKdVB) equation, and by using traveling wave transformation the mKdVB equation is converted to the damped Duffing equation. After that we compared between all mentioned solutions using plasma parameters.

Few-cycle optical pulses in negative index materials with dispersive permittivity and permeability

Propagation of few-cycle optical pulses in nonlinear optical Kerr (cubic) and non-Kerr (quintic) type metamaterials, exhibiting frequency-dependent dielectric susceptibility and magnetic permeability, is considered. Considering the theory of electromagnetic waves from Maxwell’s equations, a new nonlinear evolution equation describing the combined influences of higher-order nonlinearities and higher-order linear and nonlinear dispersions, appropriate for electromagnetic ultrashort pulse propagation in negative index materials, is derived beyond the slowly varying envelope approximation. A fully numerical simulation of the newly derived model equation, based on the lossy Drude model, shows the propagation of soliton-like stable few-cycle optical pulses under some parameter values. The change in types of self-steepening parameters induces structural changes of the initial input pulse, characterized by a soliton molecule made of either asymmetric or symmetric optical pulses. Also, the mutual balancing between Kerr and non-Kerr nonlinearities and higher-order dispersions is found to support the formation of soliton-molecules in both the normal and anomalous group velocity dispersion regimes.

Theoretical study of high order and saturable Kerr media nonlinearities in Z-scan

Based on Fresnel–Kirchhoff diffraction theory, the effect of high order nonlinearities and saturable Kerr media in Z-scan experiments are theoretically investigated. Analytical expressions were obtained for the Z-scan curve, which can be strongly affected by the pump intensity. Moreover, empirical relationships were found to describe the intensity dependence of the peak-valley distance and the transmittance difference, allowing an accurate determination of parameters such as high-order nonlinearities, the saturation intensity and the confocal parameter.