Nonlinear Drift Research Articles

An insight into the solitonic features of the nonlinear generalized higher-order Schrödinger equation using the solver method

For many nonlinear applications described by the dynamics of nonlinear Schrödinger equation with higher-order terms (HONLSE) such as nonlinear optics, space plasma physics molecular biology, astrophysics, quantum mechanics, superfluid, fluid mechanics, and fiber optics communications, a unique closed-form solution have been obtained using energy equation. In addition, some new solitary solutions HONLSE have been obtained via the unified solver method. The resultant solutions behave as breathers, super solitons, envelope breathers, blow up, localized super waves, periodical super shock, train super solitons, and shock structures. The modulations of Kerr nonlinear, chromatic dispersive, and wave packet drift parameters on the wave characteristics of the obtained solutions have been investigated. It was reported that the model parameters affect the amplitude, steepness, and width of the resultant structures. The provided solution can be used as a box solver for a variety of natural science systems described by distinct nonlinear equations.

Emergence of condensation patterns in kinetic equations for opinion dynamics

In this work, we define a class of models to understand the impact of population size on opinion formation dynamics, a phenomenon usually related to group conformity. To this end, we introduce a new kinetic model in which the interaction frequency is weighted by the kinetic density. In the quasi-invariant regime, this model reduces to a Kaniadakis–Quarati-type equation with nonlinear drift, originally introduced for the dynamics of bosons in a spatially homogeneous setting. From the obtained PDE for the evolution of the opinion density, we determine the regime of parameters for which a critical mass exists and triggers blow-up of the solution. Therefore, the model is capable of describing strong conformity phenomena in cases where the total density of individuals holding a given opinion exceeds a fixed critical size. In the final part, several numerical experiments demonstrate the features of the introduced class of models and the related consensus effects.

Closed-Form Formula for the Conditional Moment-Generating Function Under a Regime-Switching, Nonlinear Drift CEV Process, with Applications to Option Pricing

An analytical derivation of the conditional moment-generating function (MGF) for a regime-switching nonlinear drift constant elasticity of variance process is established. The proposed model incorporates both regime-switching mechanisms and nonlinear drift components to better capture market phenomena such as volatility smiles and leverage effects. Regime-switching models can match the tendency of financial markets to often change their behavior abruptly and the phenomenon that the new behavior of financial variables often persists for several periods after such a change. Closed-form formulas for the MGF under various conditions, which are then applied for option pricing, are also derived. The efficacy and accuracy of the results are validated through a discrete Markov chain simulation. The results obtained from the proposed formulas completely match with those from MC simulations, while requiring significantly less computational time.

Effect of adiabatic trapping of electrons on the nonlinear evolution of ion temperature gradient driven drift mode in a dispersive plasma

We have theoretically investigated the effect of adiabatic trapping of electrons on ion temperature gradient (ITG) driven nonlinear drift mode in a warm and dispersive electron-ion plasma. For this purpose, we have incorporated the gradients in the background plasma density, ion temperature and ambient magnetic field and derived two different nonlinear partial differential equations (NLPDEs). One of them contains only fractional nonlinearity while the other one incorporates the effect of both quadratic and fractional nonlinearities. We have obtained the exact solutions of these NLPDEs by using the functional variable method. We have used the graphical analysis to carry out the parametric study of the obtained solutions for the Tokamak plasma parameters. We have shown that the amplitude and the width of these nonlinear structures depend on the plasma parameters like T e , T i and η i . This work may be helpful to understand the effect of electron trapping on the low frequency drift type modes in laboratory and space plasmas.

Two-Dimensional Time-Fractional Nonlinear Drift Reaction–Diffusion Equation Arising in Electrical Field

Two-Dimensional Time-Fractional Nonlinear Drift Reaction–Diffusion Equation Arising in Electrical Field

Comparing Weights According to Weighing Cycles with Nonlinear Drift of the Comparator

Cycle weighing is used by metrologists around the world to eliminate drift in comparator readings when comparing the mass of reference weights. Various types of cycles and their descriptions are given in the International Recommendation on legal metrology OIML R111-1-2004 adopted in the Russian Federation as a national standard. They consider cycles under the assumption of linear drift of comparator readings. The article discusses weighing cycles that eliminate nonlinear drift of the comparator. Drift models are proposed in the form of exponential and multinomial laws, which are characterized by a fast dependence of readings at the beginning of the cycle and a slower one at the end of the cycle. The derivation of formulas for the difference between the mass of the verified and the original standard according to four comparator readings at equal time intervals is given. The formulas consist of two components: the first coincides with the formula for linear drift, and the second plays the role of a correction for the drift deviation from linearity. The formulas are also correct for linear drift, since in this case the second components become zero. Estimates of the differences in the mass of the compared weights, taking into account the nonlinearity of the comparator drift, make it possible to estimate the measurement uncertainty for linear and nonlinear drift and become relevant with a constant increase in weighing accuracy, especially when comparing high-precision mass standards.

Large deviation principle for stochastic FitzHugh–Nagumo lattice systems

This paper is devoted to the large deviation principle of stochastic FitzHugh–Nagumo lattice system, where the nonlinear drift and diffusion terms are locally Lipschitz continuous. Based on the well-posedness and the convergence of the solutions of the controlled system associated with the considered system, by the weak convergence method, we establish the large deviation principle in C([0,T],l2×l2) for such infinite dimensional system.

Invariant measures of stochastic delay complex Ginzburg-Landau equations

This paper is concerned with the existence of invariant measures for stochastic delay complex Ginzburg-Landau equations defined on the entire integer set. When the nonlinear drift and diffusion terms are globally Lipschitz continous, the existence of invariant measures of the equation is proved by estiblishing the tightness of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem.

Coupled nonlinear drift and IAWs in streaming O–H plasma of upper ionosphere

Nonlinear structures formed by the coupled drift wave (DW) and ion acoustic waves (IAWs) are studied in a magnetized inhomogeneous collisionless bi-ion plasma with ions shear flow along the ambient magnetic field B=B0ẑ. The electrons are assumed to follow double spectral index (r, q) distribution in which r shows the flat top nature, while q is responsible for the shape of the distribution at the tail. A nonlinear differential equation is derived, and its solutions in the form of double layers (DLs) and solitons are obtained in different limits. It is pointed out that the presence of (0.4%) protons in the oxygen plasma of ionosphere should not be ignored because acoustic speeds corresponding to oxygen and hydrogen ions have small ratio of about four and drift wave frequency may lie in the same range. It is found that only the rarefactive solitons can be formed by the nonlinear DW and IAWs in the inhomogeneous oxygen hydrogen (O–H) plasma. However, the theoretical model predicts that both compressive and rarefactive DLs may be formed. The linear instabilities of low-frequency electrostatic waves due to field-aligned shear flow of ions have also been investigated.

Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data

We study the periodic homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary condition posed in an unbounded perforated domain. The nonlinear problem is associated with the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. We are interested in deriving rigorously the upscaled model equations and the corresponding effective coefficients for the case when the microscopic dynamics are linked to a particular choice of characteristic length and time scales that lead to an exploding nonlinear drift. The main mathematical difficulty lies in proving the two-scale compactness and strong convergence results needed for the passage to the homogenization limit. To cope with the situation, we use the concept of two-scale compactness with drift, which is similar to the more classical two-scale compactness result but it is defined now in moving coordinates. We provide as well a strong convergence result for the corrector function, starting this way the search for the order of the convergence rate of the homogenization process for our target nonlinear drift problem.

Continuous–discrete derivative-free extended Kalman filter based on Euler–Maruyama and Itô–Taylor discretizations: Conventional and square-root implementations

In this paper, we continue to study the derivative-free extended Kalman filtering (DF-EKF) framework for state estimation of continuous–discrete nonlinear stochastic systems. Having considered the Euler–Maruyama and Itô–Taylor discretization schemes for solving stochastic differential equations, we derive the related filters’ moment equations based on the derivative-free EKF principal. In contrast to the recently derived MATLAB-based continuous–discrete DF-EKF techniques, the novel DF-EKF methods preserve an information about the underlying stochastic process and provide the estimation procedure for a fixed number of iterates at the propagation steps. Additionally, the DF-EKF approach is particularly effective for working with stochastic systems with highly nonlinear and/or nondifferentiable drift and observation functions, but the price to be paid is its degraded numerical stability (to roundoff) compared to the standard EKF framework. To eliminate the mentioned pitfall of the derivative-free EKF methodology, we develop the conventional algorithms together with their stable square-root implementation methods. In contrast to the published DF-EKF results, the new square-root techniques are derived within both the Cholesky and singular value decompositions. A performance of the novel filters is demonstrated on a number of numerical tests including well- and ill-conditioned scenarios.

Model reduction for stochastic systems with nonlinear drift

In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such nonlinearities in high dimensional settings occur, e.g., when stochastic reaction diffusion equations are discretized in space. We provide a brief discussion around existence, uniqueness and stability of solutions. (Almost) stability then is the basis for new concepts of Gramians that we introduce and study in this work. With the help of these Gramians, dominant subspace is identified leading to a balancing related highly accurate reduced order SDE. We provide an algebraic error criterion and an error analysis of the propose model reduction schemes. The paper is concluded by applying our method to spatially discretized reaction diffusion equations.

An extended Hasegawa–Mima equation for nonlinear drift wave turbulence in general magnetic configurations

We propose an extended Hasegawa–Mima equation describing the evolution of nonlinear drift wave turbulence in general magnetic configurations. Such HMGM equation can be derived within the kinetic framework of guiding center motion or from a two-fluid model of an ion-electron plasma by application of a drift wave turbulence ordering that does not involve conditions on spatial derivatives of magnetic field and plasma density. The HMGM equation is therefore appropriate to describe the evolution of drift wave turbulence in strongly inhomogeneous magnetized plasmas, such as magnetospheric and stellarator plasmas, involving complex magnetic field geometries and non-uniform plasma density distributions. We find conservation laws (mass, energy, and generalized enstrophy) of the HMGM equation, study its algebraic (Hamiltonian) structure, and prove a nonlinear stability criterion for steady solutions through the energy-Casimir method. We then apply these results to describe drift waves and infer the existence of stable toroidal zonal flows with radial shear in dipole magnetic fields.

Exact higher-order moments for linear non-homogeneous stochastic differential equation

This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the nth exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô’s formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments.

Development of high-performance long-pulse discharge in KSTAR

High-performance long-pulse plasma operation is essential for producing economically viable fusion energy in tokamak devices. To achieve such discharges in KSTAR, firstly, the rapid increase in the temperature of plasma-facing components was mitigated. The temperature increase of the poloidal limiter, especially, was associated with beam-driven fast ion orbit loss and the discrepancy of the equilibrium reconstructed with heated magnetic probes of signal drift. The fast ions lost to the poloidal limiter were reduced by optimizing the plasma shape and the composition of neutral beam injection (NBI). This nonlinear signal drift was successfully reduced by a new thermal shielding protector on the magnetic probes. Secondly, a lower loop voltage approach was implemented to reduce a poloidal flux consumption rate. A plasma current of 400 kA and a line-averaged electron density of ∼2.0 × 1019 m−3 were chosen by considering the L–H power threshold, fast ion orbit loss, and beam shine-through power loss for low loop voltage in KSTAR. In addition, the application of electron cyclotron heating also helped maintain the plasma with low loop voltage (∼25 mV) by enhancing the NBI-driven current and achieving a high poloidal beta (β P) state. KSTAR has achieved a long pulse (∼90 s) operation with the high performance of β P ⩽ 2.7, thermal energy confinement enhancement factor (H98y2) ∼ 1.1, and fraction of non-inductive current (f NI) ⩽ 0.96. Still, gradual degradation of the plasma performance has been observed over time in the discharges. In one of the long-pulse discharges, β P reduced by ∼18% over the time of ∼8τ R (current relaxation time, τ R ∼ 5 s) and ∼1067τ E,th (thermal energy confinement time, τ E,th ∼ 45 ms). The degradation may be closely associated with weak, yet growing, and persistent toroidal Alfvén eigenmodes and their effect on fast ion confinement.

Gas-phase odor mixture quantification based on relative comparison method using multiple olfactory receptors

The gas-phase odor biosensors exhibit outstanding sensitivity and selectivity towards target ligands. Current odor detections, however, were only operated under one or multiple separated olfactory receptors (ORs) stimulated by a single component odor. Meanwhile, the nonlinear characteristic, drift, and aging problems in biosensors were the main barriers for better quantification performance. Here, we proposed a gas-phase odor biosensor for odor mixture quantification based on relative comparison method using cell expressing ORs, the liquid thickness control and liquid exchange maintained repeatable OR responses. We repeatedly compared the known and its adjacent unknown odor response thus mitigating the biosensor drift interference and sensor nonlinearity characteristic. We first conducted the odor quantification of one odor component in a mixture of five odors. The existence of other odors did not influence the quantification of Or56a towards geosmin component in the mixture. Then, two types of OR were utilized for quantifying odor mixture with or without adding the OR label information to the individual cells. With-label quantification was more precise and efficient than without-label one. Lastly, we simultaneously expressed two ORs on one cell line for wider detection range and quantified the mixture even if OR has cross sensitivity. We believe the odor mixture quantification methods established in this study will extend the application scenarios of odor biosensors especially in rapid on-site odor detection.

Remaining Useful Life Prediction of Lithium-Ion Battery Based on Adaptive Fractional Lévy Stable Motion with Capacity Regeneration and Random Fluctuation Phenomenon

The capacity regeneration phenomenon is often overlooked in terms of prediction of the remaining useful life (RUL) of LIBs for acceptable fitting between real and predicted results. In this study, we suggest a novel method for quantitative estimation of the associated uncertainty with the RUL, which is based on adaptive fractional Lévy stable motion (AfLSM) and integrated with the Mellin–Stieltjes transform and Monte Carlo simulation. The proposed degradation model exhibits flexibility for capturing long-range dependence, has a non-Gaussian distribution, and accurately describes heavy-tailed properties. Additionally, the nonlinear drift coefficients of the model can be adaptively updated on the basis of the degradation trajectory. The performance of the proposed RUL prediction model was verified by using the University of Maryland CALEC dataset. Our forecasting results demonstrate the high accuracy of the method and its superiority over other state-of-the-art methods.

Damped quantum drift Zakharov-Kuznetsov equation for a dissipative inhomogeneous partially degenerate plasma with quantizing magnetic field

Damped quantum drift Zakharov-Kuznetsov (DQDZK) equation is investigated in two dimensions for a spatially inhomogeneous and dissipative plasma with adiabatic electron trapping in the presence of quantizing magnetic field and partial degeneracy. Linear and nonlinear analysis is presented in detail. For the 2D case, it is observed that the angular frequency of linear drift ion acoustic wave (DIAW) exhibits a cut-off point that gets modified by Landau quantization and partial degeneracy of the trapped electrons. It is observed that the inclusion of ion-neutral collisions leads to the formation of DQDZK equation for the system under consideration. The DQDZK is solved analytically using the hyperbolic tangent method and a decaying solitary wave solution is obtained. It is shown that the effects of quantizing magnetic field, partial degeneracy, and obliqueness significantly modify the properties of the nonlinear drift ion acoustic structures. The results presented here can be gainfully employed to understand the linear and nonlinear wave propagation in dense degenerate plasmas present in astrophysical compact objects like white dwarfs and neutron stars.

Drift kinetic theory of neoclassical tearing modes in tokamak plasmas: polarisation current and its effect on magnetic island threshold physics

A nonlinear 4-dimensional drift island theory derived in (Imada et al 2019 Nucl. Fusion 59 046016 and references therein) provides qualitative predictions of the plasma response to a stationary neoclassical tearing mode (NTM) magnetic island in a low beta, large aspect ratio tokamak plasma. (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001) refines a model for the magnetic drift frequency and exploits the limit of rare collisions, reducing this theory to 3-dimensional and thus providing a more accurate treatment of the trapped-passing boundary layer. The drift island theory is further improved in (Dudkovskaia et al 2023 Nucl. Fusion 63 016020) by introducing plasma shaping and finite beta effects. In the present paper, an improved model is adopted to resolve the drift island separatrix boundary layer, allowing one to investigate the polarisation current contribution that exists around the magnetic island separatrix, including in the presence of the background electric field. In particular, different magnetic topologies from both sides of the separatrix generate a radial discontinuity in the distribution function gradient there, when collisions are neglected. Allowing for collisional dissipation in the leading order distribution function around the separatrix resolves this discontinuity, smoothing the density distribution. The overall effect of the polarisation current on the NTM threshold is then combined from the outer contributions that exist outside the layer, as well as the separatrix layer piece, and self-consistently accounts for the electrostatic potential reconstructed from plasma quasi-neutrality. The corresponding NTM threshold is quantified and compared with previous predictions of (Dudkovskaia et al 2021 Plasma Phys. Control. Fusion 63 054001, Dudkovskaia et al 2023 Nucl. Fusion 63 016020).

A New Nonlinear Ion Drift Model of Memristor Element and its Versatile Analog Reconfigurable Realizations

While using polynomial functions to define window functions is an initial approach in studying the memristor element, it is susceptible to generating imaginary results. However, using window functions, including the trigonometric function, is a current field of research on the memristor element. This paper uses the trigonometric Blackman window function to present a new memristor element model and investigates its nonlinear ion drift model properties. The motivation of this study is the usage of the trigonometric Blackman window function, which presents a more detailed definition and leads to more accurate results in windowing operations. The Blackman window function can address the issues of border locking and terminal state. Numerical simulations have verified this proposed structure. Additionally, the analog realizations of the memristor element constructed with the Blackman window function have been achieved on a Field Programmable Analog Array, which offers fast prototyping, serving as an alternative approach for emulating memristors.