Higher-order Nonlinear Terms Research Articles

Patterns of propagation of high-order nonlinear dispersion wave modelled by the generalised KP equation

A wide range of two-dimensional nonlinear wave is described by Kadomtsev–Petviashvili (KP) equation. We obtained the classification of travelling wave patterns to the generalised KP equation with high-order nonlinear dispersive and dissipative terms. Among these patterns, some new phenomena can be acquired for the first time. Representations of wave propagation patterns were achieved by taking specific values of parameters. This means that all these patterns can be realised under appropriate physical conditions.

Analysis of a novel finite element method for a modified Cahn–Hilliard–Hele–Shaw system

In this paper, a novel finite element method for solving a modified Cahn–Hilliard–Hele–Shaw system is proposed. The time discretization is based on the convex splitting of the energy functional in the modified Cahn–Hilliard equation, i.e., the high-order nonlinear term and the linear term in the chemical potential are treated explicitly and implicitly, respectively. Designing in this way leads to solving a linear system at each time step, which is much efficient compared to solving a nonlinear system resulting from most existing schemes. The proposed scheme is proved to be unconditionally energy stable and optimally convergent for the phase variable. Numerical results are presented to support our theoretical analysis.

Nonlinear analysis of a piezoelectric energy harvester in body undulatory caudal fin aquatic unmanned vehicles

Body caudal fin (BCF) locomotion aquatic unmanned vehicles (AUVs) are making critical transitions to full autonomous systems but have challenge in endurance. Therefore, engineering solutions need to be found to maximize the mission capability of these systems. The body actuation of these AUVs offers a platform for a piezoelectric energy harvester to scavenge electrical energy from the mechanical motion. The motion of the animals which these AUVs are inspiring from can be represented in a spatio-temporal function with different envelope constraints to represent an Anguilliform, Subcarangiform, Carangiform, and Thunniform motion. A better approximation of the strain is accounted for by considering the higher-order nonlinear terms due to the large deformation of these systems. Gibbs function is utilized to better approximate the nonlinear constitutive relations of the piezoelectric material. Parametric studies are carried out to investigate how the nonlinear strain and nonlinear piezoelectric material properties affects the performance of the BCF energy harvesters under each motion for various length and placement conditions. The results show that considering the higher-order strain is needed due to the underestimation of the harvested power when linear assumptions are employed for these types of undulatory motion. It is shown that assuming that load resistance, Ropt≈1/Cpω, over-approximates the resistance for the nonlinear piezoelectric material model. This analysis shows the importance of including the nonlinearities due to the piezoelectric material and large deformations in order to accurately estimate the levels of the harvested power and its optimal configuration.

Soliton and breather solutions for a fifth-order modified KdV equation with a nonzero background

We investigate an integrable fifth-order modified KdV equation, which contains the fifth-order dispersion and relevant higher-order nonlinear terms, on the line with a nonzero constant background via the Riemann–Hilbert method. As an application, the explicit soliton and breather solutions are obtained. Some figures are also presented to depict the soliton and breather behaviors of the fifth-order modified KdV equation.

Analysis of nonlinear vibration energy harvesters using a complex dynamic frequency method

To understand the complicated dynamic behavior of a Nonlinear Piezoelectric Energy Harvester (NPEH), this paper develops an improved Complex Dynamic Frequency (CDF) method based on complex normal form. CDF introduces a dynamic frequency factor and establishes a set of algebraic equations in handling the effect of higher-order nonlinear terms in a wide frequency band to obtain periodic responses of NPEH. Numerical and experimental studies verify that the proposed CDF gives consistent and accurate predictions of the systems with both weak and strong nonlinearity. Furthermore, through an implicit relationship between magnet arrangement and output performance, one may effectively control the sweep frequency with softening and hardening characteristics. That is a major breakthrough toward the further nonlinear design for broad bandwidth harvesters. As the application, the experimental results reveal the high response profiles can be in a wide frequency range from 10.8 Hz to 24.5 Hz for the NPEH developed that allows an output power of 9 times higher than the conventional linear structure.

On local-in-space blow-up scenarios for a weakly dissipative rotation-Camassa–Holm equation

ABSTRACT In this paper, we mainly devote to investigate a weakly dissipative shallow water wave model. By overcoming the difficulties caused by the complicated higher-order nonlinear terms and dissipative term, we present blow-up scenarios with dissipative parameter in the non-periodic case under different initial conditions. Meanwhile, we obtain the blow-up result with in the periodic case.

Optical propagation patterns in nonlocal parabolic law medium

The model of optical propagation in nonlocal parabolic law medium is described by nonlinear Schrödinger equation with high-order nonlinear terms. Exact optical propagation patterns are constructed by proposing a general complex trial equation method. These results show rich optical propagation patterns of the model.

Assimilation of All-Sky SEVIRI Infrared Brightness Temperatures in a Regional-Scale Ensemble Data Assimilation System

AbstractEnsemble data assimilation experiments were performed to assess the ability of satellite all-sky infrared brightness temperatures and different bias correction (BC) predictors to improve the accuracy of short-range forecasts used as the model background during each assimilation cycle. Satellite observations sensitive to clouds and water vapor in the upper troposphere were assimilated at hourly intervals during a 3-day period. Linear and nonlinear conditional biases were removed from the infrared observations using a Taylor series polynomial expansion of the observation-minus-background departures and BC predictors sensitive to clouds and water vapor or to variations in the satellite zenith angle. Assimilating the all-sky infrared brightness temperatures without BC degraded the forecast accuracy based on comparisons to radiosonde observations. Removal of the linear and nonlinear conditional biases from the satellite observations substantially improved the results, with predictors sensitive to the location of the cloud top having the largest impact, especially when higher-order nonlinear BC terms were used. Overall, experiments employing the observed cloud-top height or observed brightness temperature as the bias predictor had the smallest water vapor, cloud, and wind speed errors, while also having less degradation to temperatures than occurred when using other predictors. The forecast errors were smaller during these experiments because the cloud-height-sensitive BC predictors were able to more effectively remove the large conditional biases for lower brightness temperatures associated with a deficiency in upper-level clouds in the model background.

Higher-order realizable algebraic Reynolds stress modeling based on the square root tensor

In this study, realizable algebraic Reynolds stress modeling based on the square root tensor [Phys. Rev. E \textbf{92}, 053010 (2015)] is further developed for extending its applicability to more complex flows. In conventional methods, it was difficult to construct an algebraic Reynolds stress model satisfying the realizability conditions when the model involves higher-order nonlinear terms on the mean velocity gradient. Such higher-order nonlinear terms are required to predict turbulent flows with three-dimensional mean velocity. The present modeling based on the square root tensor enables us to make the model always satisfy the realizability conditions, even when it involves higher-order nonlinearity. To construct a realizable algebraic Reynolds stress model applicable to turbulent flows with three-dimensional mean velocity, a quartic-nonlinear eddy-viscosity model is proposed. The performance of the model is numerically verified in a turbulent channel flow, a homogeneous turbulent shear flow, and an axially rotating turbulent pipe flow. The present model gives a good result in each turbulent flow. Note that the mean swirl flow in an axially rotating turbulent pipe flow is reproduced because the present model involves cubic nonlinearity. Such a higher-order realizable algebraic Reynolds stress model, involving quartic nonlinearity on the mean velocity, is expected to be useful in numerically stable predictions of turbulent flows with three-dimensional mean velocity.

Solitons for the modified (2 + 1)-dimensional Konopelchenko\u2013Dubrovsky equations

In the paper, we consider the modified (2 + 1)-dimensional Konopelchenko–Dubrovsky equations which possess high order nonlinear terms. Under the aid of Maple, we derive the exact traveling wave solutions of the mKDs by the auxiliary equation approach. Under some special conditions, Jacobi elliptic function solutions, degenerated triangular function solutions, and solitons for the mKD equations are constructed.

Competing effects of Kerr nonlinearity and K-photon absorptions on continuous-wave laser inscriptions

Material processing with laser offers a reliable tool in a large variety of micromachining technology, ranging from on-disc inscriptions to cell ablations through DNA combing and imprinting for fabrications of micro and nanofluidic devices. In these applications lasers are designed to operate in specific regimes characterized by their powers and wavelengths, so understanding characteristic properties of the distinct possible laser operation regimes turns out to be a relevant step toward optimization of their uses as well as improvement of the technology. In this work we examine the stability of a model of femtosecond laser intended for laser inscriptions in nonlinear transparent media, taking into consideration the laser-induced material damage and multiphoton ionization. The model approximates the laser dynamics by a complex Ginzburg–Landau equation with a Kerr plus a high-order nonlinear term accounting for K-photon ionization, coupled to a Drude equation for time evolution of the electron plasma density. Following the Benjamin–Feir instability theory it is shown that depending on characteristic parameters of the model, multiphoton ionization processes can stabilize continuous-wave or pulse regimes. These parameters include the group-delay dispersion, the linear and nonlinear gains, the phase shift per roundtrip and the plasma ionization rate.

Adaptive time-stepping algorithms for molecular beam epitaxy: Based on energy or roughness

Due to the small parameter, high order derivatives and nonlinear term, it takes a long time to reach the steady state in the simulation of the molecular beam epitaxy (MBE) model. In this work, based on the numerical scheme in Qiao et al. (2011) and the good properties it holds, we introduce two types of adaptive time-stepping methods, in which the physical quantities like energy or roughness are involved to produce time steps. Similarly to the spatial adaptive method, we take equidistribution of the physical quantities in time direction to control the simulation error. The numerical experiment shows that the computational time is significantly saved by these methods.

Asymptotic Stability of Monotone Decreasing Kink Profile Solitary Wave Solutions for Generalized KdV-Burgers Equation

In this paper, we focus on studying the asymptotic stability of the monotone decreasing kink profile solitary wave solutions for the generalized KdV-Burgers equation. We obtain the estimate of the first-order and second-order derivatives for monotone decreasing kink profile solitary wave solutions, and overcome the difficulties caused by high-order nonlinear terms in the generalized KdV-Burgers equation in the estimate by using L2 energy estimating method and Young inequality. We prove that the monotone decreasing kink profile solitary wave solutions are asymptotically stable in H1. Moreover, we obtain the decay rate of the perturbation ψ in the sense of L2 and L∞ norm, respectively, which are (1 + t) −1/2 and (1 + t) −1/4 by using Gargliado-Nirenberg inequality.

Discrete-Time Implementation of Homogeneous Differentiators

The discrete-time version of Levant's arbitrary order robust exact differentiator, which is a forward Euler discretized version of the continuous-time algorithm enhanced by linear higher order terms, is extended by taking into account also nonlinear higher order terms. The resulting differentiator preserves the asymptotic accuracies with respect to sampling and noise known from the continuous-time algorithm. It is demonstrated in a simulation example and by differentiating a measured signal that the nonlinear higher order terms allow reducing the high-frequency switching amplitude whenever the (n+1)th derivative of the signal to be differentiated vanishes, leading to an improvement in the precision.

Sloshing dynamics of a liquid contained in a rectangular basin subjected to a uniform electric field

The 3-D motion of a weak-viscous dielectric liquid inside a partially-filled excited rectangular container in the presence of a uniform electric field is investigated using potential viscous theory. A Mathieu-like equation prescribe the evolution of modes is reached. The interaction between two surface modes is considered. According to linear approach, the normal field and dielectric constant destabilize the flow whereas surface tension stabilizes it. However, the wave number plays a dual role according to the waves kind. The discussion for fixed points as well as the phase space and time evolution of modes predicts a case of global instability due to the resonance condition βˆ1=0. Also, it reveals an unstable role of the forcing indicator whereas detuning parameter tends to enhance the stability. The approximate and numerical solutions of Mathieu-like equation compare well in regard with a qualitative behaviour. The greater the viscosity is, the less important disturbances become. The conclusion with respect to field as well as relative permittivity agrees with that of the linear approach. The paper concludes that considering more higher-order nonlinear terms can increase the accuracy of the results of the analytical method. In addition, a device can be designed to achieve optimal conditions for model stability by controlling available physical parameters.

Two improved displacement shallow water equations and their solitary wave solutions

Two improved displacement shallow water equation (DSWE) are constructed by applying the Hamilton variational principle under the shallow water approximation. The first one is the extended DSWE (EDSWE), which contains a higher order nonlinear term omitted in the original DSWE; and the second one is the fully nonlinear DSWE (FNDSWE) which contains all the nonlinear terms. By using the exp-function method, the EDSWE is analyzed and different types of waves, including the regular solitary wave, the loop solitary wave, the solitary wave with a sharp peak, and the periodic wave, are obtained. The exact solitary wave solution of the FNDSWE is also obtained. It is proved that under the shallow water approximation, the particle trajectory of the solitary wave is a parabolic curve.

Deep postbuckling and nonlinear bending behaviors of nanobeams with nonlocal and strain gradient effects

In this paper, multi-scale modeling for nanobeams with large deflection is conducted in the framework of the nonlocal strain gradient theory and the Euler-Bernoulli beam theory with exact bending curvature. The proposed size-dependent nonlinear beam model incorporates structure-foundation interaction along with two small scale parameters which describe the stiffness-softening and stiffness-hardening size effects of nanomaterials, respectively. By applying Hamilton’s principle, the motion equation and the associated boundary condition are derived. A two-step perturbation method is introduced to handle the deep postbuckling and nonlinear bending problems of nanobeams analytically. Afterwards, the influence of geometrical, material, and elastic foundation parameters on the nonlinear mechanical behaviors of nanobeams is discussed. Numerical results show that the stability and precision of the perturbation solutions can be guaranteed, and the two types of size effects become increasingly important as the slenderness ratio increases. Moreover, the in-plane conditions and the high-order nonlinear terms appearing in the bending curvature expression play an important role in the nonlinear behaviors of nanobeams as the maximum deflection increases.

FEMTOSECOND LASER PULSE PROPAGATION IN NOVEL FIBER: TELLURITE GLASS CODOPED ER3+/TM3+

In this work we investigated propagation of ultrashort laser pulses in Tellurite glass codoped Er3+/Tm3+. We derived a general propagation equation of pulses which includes the linear and nonlinear effects to all orders. We studied in the specific case of Kerr media and obtained an ultrashort pulse propagation equation called a generalized nonlinear Schrödinger equation. The impact of the third order dispersion, the higher-order nonlinear terms self-steepening, and stimulated Raman scattering are explicitly analyzed.

Prediction method for energy consumption per ton of fused magnesium furnaces using data driven and mechanism model

The electric energy consumed in every ton of acceptable product, namely energy consumption per ton (ECT), is an important overall index for the production process of a fused magnesium furnace. The furnace is the equipment for producing the fused magnesia. The ECT value depends on the current in the smelting process. The optimal operation for a fused magnesium furnace is supposed to have the ECT as low as possible, where the key is to predict ECT accurately. By introducing an unknown high-order nonlinear term, this paper builds a dynamic ECT model for different production batches based on the static ECT model for one batch. The average current is taken as the input of the dynamic ECT model, which is composed of the unknown high-order nonlinear term and a nonlinear model with unknown parameters. The order of the nonlinear term is determined by the distance correlation and the nonlinear term is estimated by the stochastic configuration network, while the parameters of the nonlinear model is identified by the least square method. The estimation of the nonlinear term alternates with the parameter identification. This paper proposes a prediction method for ECT, which is composed of the order identification of the nonlinear term, the alternating identification of the model and the ECT prediction model. The simulation experiments are conducted by the on-site data, and the results verify the effectiveness of the proposed prediction method.

Stochastic P-bifurcation in a nonlinear viscoelastic beam model with fractional constitutive relation under colored noise excitation

In this paper, we study the stochastic P-bifurcation problem for an axially moving bistable viscoelastic beam with fractional derivatives of high-order nonlinear terms under colored noise excitatio...