Evolution Equations Research Articles

Constraints for twist-two alien operators in QCD

Abstract Parton evolution equations in QCD are controlled by the anomalous dimensions of gauge-invariant twist-two spin-N quark and gluon operators. Under renormalization, these mix with gauge-variant operators of the same quantum numbers, referred to as alien operators. Our work addresses the systematic study of these alien operators at arbitrary spin N, using generalized BRST symmetry relations to derive their couplings and Feynman rules at all values of N. We observe how the all-N structure of the generalized (anti-)BRST constraints relates the couplings of alien operators with n + 1 gluons to those with n gluons. Realizing a bootstrap, we present all one-loop results necessary for performing the operator renormalization up to four loops in QCD.

On a class of evolution equations in Lipschitz subdomains of Riemannian manifolds

We study the existence, uniqueness, analyticity, and continuous dependence of mild solutions for a class of evolutionary integrodifferential equations with free boundary conditions on L p -spaces in Lipschitz subdomains of Riemannian manifolds. Our strategy is to use the Laplace transform theory to ensure the existence of an analytic resolvent to the Hodge–Laplacian in the space of all ℓ-differential forms with p-th power integrable coefficients.

QCD evolution of entanglement entropy

Abstract Entanglement entropy has emerged as a novel tool for probing nonperturbative quantum chromodynamics (QCD) phenomena, such as color confinement in protons. While recent studies have demonstrated its significant capability in describing hadron production in deep inelastic scatterings, the QCD evolution of entanglement entropy remains unexplored. In this work, we investigate the differential rapidity-dependent entanglement entropy within the proton and its connection to final-state hadrons, aiming to elucidate its QCD evolution. Our analysis reveals a strong agreement between the rapidity dependence of von Neumann entropy, obtained from QCD evolution equations, and the corresponding experimental data on hadron entropy. These findings provide compelling evidence for the emergence of a maximally entangled state, offering new insights into the nonperturbative structure of protons.

An Efficient Analysis of Amplitude and Phase Dynamics in Networked MEMS-Colpitts Oscillators

Abstract This paper explores the interactions of both phase and amplitude in a network of N MEMS-Colpitts oscillators that are resistively coupled. The numerical simulations of the extensive networks of oscillators, required for emerging applications such as clock synchronization and neuromorphic computing, become computationally prohibitive as the number of oscillators increases. This complicates the design and evaluation of such systems, as understanding the effects of changes in coupling and design parameters can require many simulations. This study employs the method of multiple scales (MS) in combination with the harmonic balance method to convert the coupled differential equations governing the system of oscillators into a set of nonlinear evolution equations for the amplitude and phase of the oscillators. The amplitude and phase evolve on a timescale that is slow, commensurate with the damping in the system, compared with the fast timescale of the oscillation frequencies. The approach used in this study to describe the amplitude and phase dynamics offers significant computational efficiency (gains of 10× to 50× are shown) compared to direct numerical integration while maintaining an accurate representation of the response. The results of the presented simulations demonstrate the effect of coupling strength on the dynamics of the network, accounting for both phase and amplitude dynamics.

Diffusion-driven flows in a nonlinear stratified fluid layer

Diffusion-driven flow is a boundary layer flow arising from the interplay of gravity and diffusion in density-stratified fluids when a gravitational field is non-parallel to an impermeable solid boundary. This study investigates diffusion-driven flow within a nonlinearly density-stratified fluid confined between two tilted parallel walls. We introduce an asymptotic expansion inspired by the centre manifold theory, where quantities are expanded in terms of derivatives of the cross-sectional averaged stratified scalar (such as salinity or temperature). This technique provides accurate approximations for velocity, density and pressure fields. Furthermore, we derive an evolution equation describing the cross-sectional averaged stratified scalar. This equation takes the form of the traditional diffusion equation but replaces the constant diffusion coefficient with a positive-definite function dependent on the solution's derivative. Numerical simulations validate the accuracy of our approximations. Our investigation of the effective equation reveals that the density profile depends on a non-dimensional parameter denoted as $\gamma$ representing the flow strength. In the large $\gamma$ limit, the system is approximated by a diffusion process with an augmented diffusion coefficient of $1+\cot ^{2}\theta$ , where $\theta$ signifies the inclination angle of the channel domain. This parameter regime is where diffusion-driven flow exhibits its strongest mixing ability. Conversely, in the small $\gamma$ regime, the density field behaves like pure diffusion with distorted isopycnals. Lastly, we show that the classical thin film equation aligns with the results obtained using the proposed expansion in the small $\gamma$ regime but fails to accurately describe the dynamics of the density field for large $\gamma$ .

Nonlinear Rayleigh wave propagation in a three-layer sandwich structure in dual-phase-lag

Plane, nonlinear Rayleigh wave propagation is investigated in a three-layer sandwich structure of a thermoelastic medium, within the frame of dual-phase-lag theory. The thermal conductivity is taken as a linear function of temperature. This induces nonlinearity in the evolution equations for the heat flux components. A particular solution is found in the form of Poincaré expansion in a small parameter that reflects the fluctuations of temperature about a steady value. This solution is discussed and plots are provided for a special case when both external faces of the structure are traction-free and under prescribed temperature regime. It is noted, in particular, that some quantities of practical interest suffer jumps at the interfaces. Some of the jumps appear only starting from the second order of approximation. The existence of jumps may be favourably used to carry out some measurements of material parameters.

Modulational instability for a cubic-quintic model of coupled Gross-Pitaevskii equations with residual nonlinearities

Abstract We demonstrate the existence of modulational instability (MI) in both trapped miscible and immiscible two component Bose-Einstein condensates. The study is addressed theoretically and numerically in the framework of one-dimensional coupled Gross-Pitaevskii equations incorporating intra- and interspecies cubic-quintic nonlinearities with higher order ones. Using the time-dependent variational approach, we derive the new Euler-Langrange equations for the time evolution of the phase and amplitude of the modulational perturbation as well as the effective potential and the instability criteria of the system. We examine the effects of higher-order nonlinearities on the instability dynamics of the condensates. We show that the modulational properties of the chosen wave numbers are significantly modified. Direct numerical simulations run by the split step Fourier method confirm the analytical predictions.

The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations

In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discretizing the second-order center difference quotient. In addition, we strictly prove that the proposed ADI difference scheme has unique solvability and is unconditionally stable and convergent. The extrapolation method is further applied to improve the numerical solution accuracy. Finally, two numerical examples are given to verify our theoretical results.

Relativistic and Heavy Ion Effects on Single‐ and Multi‐Soliton Structures of Ion‐Acoustic Waves

ABSTRACTThe fundamental characteristics of ion‐acoustic waves (IAWs) due to the relativistic effects are investigated in a collisionless unmagnetized plasma system consisting of relativistic positively charged inertial heavy ions, kappa‐distributed electrons, and Maxwellian light ions. The reductive perturbation method (RPM) is used to derive the nonlinear evolution equation like the Korteweg‐de Vries (KdV) equation. The production of single‐, double‐, and triple‐IAW solitons (IAWSs) is also investigated. It is found that for the effects of concerned parameters only rarefactive (IAWSs) are produced. The results obtained in this study are useful not only to understand the space plasma phenomena such as plasma sheet boundary layer of earth's magnetosphere, laser‐plasma interaction, but also to validate the laboratory experiments.

Review of Some Modified Generalized Korteweg–de Vries–Kuramoto–Sivashinsky Equations (Part II)

In part I of this work to appear in Foudations-MDPI 2024, some existence and uniqueness results for the solutions of some equations were reviewed, such as the Korteweg–de Vries equation (KdV), the Kuramoto–Sivashinsky equation (KS), the generalized Korteweg–de Vries–Kuramoto–Sivashinsky equation (gKdV-KS), and the nonhomogeneous boundary value problem for the KdV-KS equation in quarter plane. The main objective of this paper is to review some results of the existence of global attractors for the evolution equations with nonlinearity of the form N(ux), where ux denotes the derivative of u with respect to x, focusing in particular on the Kuramoto–Sivashinsky equation in one and two dimensions. In order to illustrate the general abstract results, we have chosen to discuss in detail the existence of global attractors for the Kuramoto–Sivashinsky (KS) equation in 1D and 2D. Once a global attractor is obtained, the question arises whether it has special regularity properties. Then we give an integrated version of the homogeneous steady state Kuramoto–Sivashinsky equation in Rn. This work ends with a change from rectangular to polar coordinates in the three-dimensional KS equation to give an energy estimate in this case.

Micromechanics-based variational phase-field modeling of fatigue fracture

In this paper, we extend the micromechanics-based phase-field model to simulate fatigue failure. The coupling of a micromechanics-based framework with the phase-field approach helps to differentiate between failure modes, by distinguishing between open and closed microcracks. This integrated framework links continuum field variables, such as plastic strain and damage variable, to micromechanical mechanisms like frictional sliding and microcrack opening. We first improve the algorithm’s stability during loading-unloading in the tensile regime through a modification of the plasticity evolution equations. Next, we incorporate fatigue damage accumulation and deterioration due to cyclic loading into the micromechanics-based phase-field model. A fatigue degradation function, driven by free energy accumulation, is introduced to degrade the fracture energy upon reaching a specified threshold during cyclic loading. Various cyclic loads are applied to benchmark tests, both with and without imperfections (e.g. holes, inclusions, voids), under plane strain conditions to capture diverse failure modes. The results demonstrate the model’s capability to accurately describe tensile, shear, and mixed-mode fracture under cyclic loading. Furthermore, the model effectively simulates key features of fatigue behavior, including crack nucleation, growth, and coalescence.

Multi-axial fatigue of high-strength concrete: Model-enabled interpretation of punch-through shear test response

This paper aims to fill the gap in multi-axial fatigue characterization of concrete by presenting a comprehensive numerical analysis of data from the punch-through shear test (PTST), which allows control over combined shear and compression loads in the tested ligament. To accurately reproduce the degradation process in this multi-axial configuration, a material model must capture two key effects: (i) dissipative behavior at the inter-aggregate level during subcritical pulsating loading and (ii) fatigue-induced tri-axial stress redistribution in the test ligament. The recently published MS1 model by the authors effectively incorporates these features, thereby enabling a deeper interpretation of the experimental data. The former effect is seamlessly integrated into the model using the Lemaitre-type fatigue hypothesis, facilitating the direct derivation of general evolution equations. The latter effect is efficiently represented through the microplane homogenization framework. The model’s predictions for PTST response under monotonic shear loading with varying levels of radial confinement align well with experimental results. Similarly, the simulated fatigue response accurately reproduces the experimentally observed S-N (Wöhler) curves. Further studies demonstrate the model’s ability to predict the effect of fatigue loading sequence. A thermodynamically-based formulation provides an energetic interpretation of this effect, offering a quantitative breakdown of energy dissipation attributed to individual dissipative mechanisms over the lifetime of the material. The studies show that damage-induced dissipation remains almost constant regardless of different fatigue loading histories.

An enhanced numerical model for considering coupled strain-softening and seepage effects on rock masses surrounding a submarine tunnel

The seepage of groundwater and the strain-softening of rock mass in a submarine tunnel expand the plastic region of rock, thereby affecting its overall stability. It is therefore essential to study the stress and strain fields in the rocks surrounding the submarine tunnel by considering the coupled effect of strain-softening and seepage. However, the evolution equation for the hydro-mechanical parameters in the existing fully coupled solution is a uniform equation that is unable to reproduce the characteristics of rock mass in practice. In this study, an updated numerical procedure for the submarine tunnel is derived by coupling strain-softening and seepage effect based on the experimental results. According to the hydro-mechanical coupling theory, the hydro-mechanical parameters such as elastic modulus, Poisson's ratio, Biot's coefficient and permeability coefficient of rocks are characterized by the fitting equations derived from the experimental data. Then, the updated numerical procedure is deduced with the governing equations, boundary conditions, seepage equations and fitting equations. The updated numerical procedure is verified accurately compared with the previous analytical solution. By utilizing the updated numerical procedure, the characteristics of stress field and the influences of initial pore water pressure, Biot's coefficient, and permeability coefficient on the stress, displacement and water-inflow of the surrounding rocks are discussed. Regardless of the variations in hydro-mechanical parameters, the stress distribution has a similar trend. The initial permeability coefficient exerts the most significant influence on the stress field. With the increases in initial pore water pressure and Biot's coefficient, the plastic region expands, and the water-inflow and displacement increase accordingly. Given the fact that the stability of the tunnel is more sensitive to the seepage force controlled by the hydraulic parameters, it is suggested to dewater the ground above the submarine tunnel to control the initial pore water pressure.

A mean curvature flow arising in adversarial training

We connect adversarial training for binary classification to a geometric evolution equation for the decision boundary. Relying on a perspective that recasts adversarial training as a regularization problem, we introduce a modified training scheme that constitutes a minimizing movements scheme for a nonlocal perimeter functional. We prove that the scheme is monotone and consistent as the adversarial budget vanishes and the perimeter localizes, and as a consequence we rigorously show that the scheme approximates a weighted mean curvature flow. This highlights that the efficacy of adversarial training may be due to locally minimizing the length of the decision boundary. In our analysis, we introduce a variety of tools for working with the subdifferential of a supremal-type nonlocal total variation and its regularity properties.

Analysis of the wave solutions of the nonlinear evolution equations help of advanced exp⁡(-φ(ξ))-expansion scheme

With the assistance of emblematic calculation programming, the current paper explores the specific voyaging wave arrangements from the general (2+1)- layered nonlinear development conditions by utilizing the advanced exp⁡(-φ(ξ))-expansion with time-partial boundaries. As a result, the used technique is effectively utilized and recently created some precise voyaging wave arrangements. The recently created arrangements have been communicated regarding mathematical and exaggerated capabilities. The created arrangements have been inquired into their comparing condition with the guide of the emblematic calculation programming Maple. The elements of nonlinear wave arrangements are analyzed and shown by maple18 in 3D, and 2D plots, and form plots with explicit upsides of the mind boggling boundaries are plotted. The advanced exp⁡(-φ(ξ))-expansion strategy is a solid treatment for looking through fundamental nonlinear waves that enhance an assortment of water waves in the long-frequency system that emerges in designing fields.

Bilinear optimal control for a fractional diffusive equation

We consider a bilinear optimal control for an evolution equation involving the fractional Laplace operator of order 0<s<1. We first give some existence and uniqueness results for the considered evolution equation. Next, we establish some weak maximum principle results allowing us to obtain more regularity of our state equation. Then, we consider an optimal control problem which consists to bring the state of the system at final time to a desired state. We show that this optimal control problem has a solution and we derive the first- and second-order optimality conditions.

Reduced order modeling of turbulent reacting flows on low-rank matrix manifolds

A new low-rank approximation, referred to as time-dependent principal component analysis (t-PCA), is developed for reduced-order modeling (ROM) of scalar transport in turbulent reactive flows. In t-PCA, the evolution of the composition matrix is constrained to a low-rank matrix manifold, similar to that in standard PCA. Specifically, the t-PCA approximates the composition matrix through the multiplication of two thin, time-dependent matrices that represent spatial and composition subspaces. The evolution equations for these subspaces are derived by projecting the composition transport equation onto the tangent space of the low-rank matrix manifold. While the evolution equations for the spatial subspace in both PCA and t-PCA are similar, there are differences in how the composition subspace is obtained: (i) In t-PCA, the composition subspace is time-dependent, whereas in PCA, it is static. (ii) The t-PCA does not require any prior data, and an evolution equation for the composition subspace is derived. In PCA, the composition subspace is obtained from data. The t-PCA can be regarded as an on-the-fly low-rank approximation that can adapt to changes in the flow instantaneously. It is shown that the low-rank t-PCA approximations achieve residual levels lower than those obtained via PCA. For demonstrations and a comparative assessment of the ROMs, simulations are conducted of a non-premixed CO/H2 flame in a temporally evolving jet. Two cases are considered, based on the mechanisms previously suggested for combustion kinetics of this flame: (i) the GRI-Mech 3.0 model involving 53 species for a two-dimensional flame, (ii) the skeletal syngas model involving 11 species for a three-dimensional turbulent flame. The results are appraised via a posteriori comparisons against data generated via full-rank direct numerical simulation (DNS) of the same flame, and also with the PCA-reduced data from the DNS. It is shown that t-PCA yields excellent predictions of various features of the thermo-chemistry field.

Controllability of some nonlocal‐impulsive Volterra evolution systems via measures of noncompactness

This study investigates the controllability of a Volterra evolution equation with impulsive terms and nonlocal initial conditions. With the aid of the resolvent operator generated by the linear part of the equation, mild solutions can be defined. Notably, the resolvent operator lacks compactness and equicontinuity. Additionally, the compactness of the impulsive and nonlocal functions is not required. Sufficient conditions for controllability are obtained through measures of noncompactness in Banach spaces. Functional differential equations and hyperbolic partial differential equations can be solved with these results. An example is given to illustrate the validity of the presented results.

Recursive dynamic state estimation for power systems with an incomplete nonlinear DAE model

AbstractPower systems are highly complex, large‐scale engineering systems subject to many uncertainties, which makes accurate mathematical modeling challenging. This article introduces a novel centralized dynamic state estimator designed specifically for power systems where some component models are missing. Including the available dynamic evolution equations, algebraic network equations, and phasor measurements, the least squares criterion is applied to estimate all dynamic and algebraic states recursively. The approach generalizes the iterated extended Kalman filter and does not require static network observability, relying on the network topology and parameters. Furthermore, a topological criterion is established for placing phasor measurement units (PMUs), termed topological estimability, which guarantees the uniqueness of the solution. A numerical study evaluates the performance under short circuits in the network and load changes and shows superior tracking performance compared to robust procedures from the literature with computational times in accordance with the typical PMU sampling rates.

Electromagnetism and Maxwellian Evolution Equations in terms of Darboux Frame in Minkowski Space with Abnormalities

Abstract Electromagnetic wave propagation is often thought of as the transport of polarised light and this behaviour is well defined by Maxwell's equations when propagating in an optical fiber. In this paper, we examine the $\textbf{q}-direction$ and $\textbf{n}-direction$ Berry's phase equation along a Darboux-framed optical fibre in Minkowski space. Afterwards, $\textbf{q}-direction$ and $\textbf{n}-direction$ for the electromagnetic curves of Rytov parallel transport laws are determined to shown in the application section, the connections of the electromagnetic curve with the anholonomic coordinates for Maxwell's equation by Maxwellian evolution that cause are visualised and illustrated with the MAPLE program.