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We study motility-induced phase separation in symmetric and asymmetric active binary mixtures. We start with the coarse-grained run-and-tumble bacterial model that provides evolution equations for the density fields ρ i ( r ⃗ , t). Next, we study the phase separation dynamics by solving the evolution equations using the Euler discretization technique. We characterize the morphology of domains by calculating the equal-time correlation function C(r, t) and the structure factor S(k, t), both of which show dynamical scaling. The form of the scaling functions depends on the mixture composition and the relative activity of the species, Δ. For k → ∞, S(k, t) follows Porod's law: S(k, t) ∼ k−(d+1) and the average domain size L(t) shows a diffusive growth as L(t) ∼ t1/3 for all mixtures. [ABSTRACT FROM AUTHOR]

We present dynamic density functional theory (DDFT) incorporating general inhomogeneous, incompressible, time-dependent background flows and inertia, describing externally driven passive colloidal systems out of equilibrium. We start by considering the underlying nonequilibrium Langevin dynamics, including the effect of the local velocity of the surrounding liquid bath, to obtain the nonlinear, nonlocal partial differential equations governing the evolution of the (coarse-grained) density and velocity fields describing the dynamics of colloids. In addition, we show both with heuristic arguments, and by numerical solution, that our equations and solutions agree with existing DDFTs in the overdamped (high friction) limit. We provide numerical solutions that model the flow of hard spheres, in both unbounded and confined domains, and compare with previously derived DDFTs with and without the background flow. [ABSTRACT FROM AUTHOR]

In this study, Cu–10W and 2 wt%Y 2 O 3 /Cu–10W composites were successfully prepared via mechanical ball milling and spark plasma sintering. Hot compression experiments on the two materials were carried out in the temperature range of 300°C-600 °C and in the strain rate range of 0.01 s−1–10 s−1. The hot deformation activation energy of the two materials was calculated and a constitutive equation was established. The optimum hot deformation process parameters were obtained according to a hot processing map. The effects of Y 2 O 3 on the thermal deformation behavior of the Cu–10W composites and their microstructures were investigated, and the results showed that Y 2 O 3 could enhance the stability of Cu–10W at high temperatures. The activation energies of thermal deformation of Cu–10W and 2 wt%Y 2 O 3 /Cu–10W were 159.56 kJ/mol and 129.85 kJ/mol, respectively. Moreover, Y 2 O 3 improved the interfacial bonding strength of Cu and W in the Cu–10W material, effectively preventing the generation of microcracks and pores during the high-temperature compression process; Y 2 O 3 provides more nucleation sites in the recrystallization process, which is favorable for the excitation of recrystallization. A low strain rate at high temperatures favors the occurrence of recrystallization, and the 2 wt%Y 2 O 3 /Cu–10W composites tend to exhibit discontinuous dynamic recrystallization (DDRX) under high-Z-value conditions and continuous dynamic recrystallization (CDRX) under low-Z-value conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study the wave equation on infinite graphs. On one hand, in contrast to the wave equation on manifolds, we construct an example for the non-uniqueness for the Cauchy problem of the wave equation on graphs. On the other hand, we obtain a sharp uniqueness class for the solutions of the wave equation. The result follows from the time analyticity of the solutions to the wave equation in the uniqueness class. In the last part, we extend the result to a wide class of linear evolution equations. [ABSTRACT FROM AUTHOR]

In this article, the potential Kadomtsev-Petviashvili (pKP) type coupled system with variable coefficients is studied, which have many applications in wave phenomena and soliton interactions in a two-dimensional space with time. In this framework, Hirota bilinear form is applied to acquire diverse types of interaction lump solutions from the foresaid equation. Abundant lump, lump-periodic, lump-soliton and multi soliton solutions to the pKP system are presented by the Hirota bilinear form and a mixture of exponentials and trigonometric functions. Lump, lump-periodic, lump-soliton and multi soliton solutions are studied with the usage of symbolic computation. In addition, the symbolic computation and the applied methods for governing model are investigated. The movement role of the waves is investigated, and the theoretical analysis of the acquired solutions is discussed using the bilinear technique of all produced solutions with 2D and 3D plots with respective parameters. The computational difficulties and outcomes highlight the clarity, effectiveness, and simplicity of the approaches, suggesting that these schemes can be applied to a variety of dynamic and static nonlinear equations governing evolutionary phenomena in computational physics as well as to other real-world situations and a wide range of academic fields. We used software Maple 2024 "(http://www.maplesoft.com/)". [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents a systematic approach to investigating the existence of solutions for fractional integro-differential equation systems incorporating delay and relaxation impulsive terms. By employing suitable definitions of fractional derivatives, we establish physically interpretable boundary conditions. To account for abrupt state changes, impulsive conditions are integrated into the model. The system is transformed into an equivalent integral equation, facilitating the application of Banach and Schaefer fixed-point theorems to prove the existence and uniqueness of solutions. The practical applicability of our findings is demonstrated through an illustrative example. [ABSTRACT FROM AUTHOR]

We show that braces whose adjoint groups are powerful and powerful left nilpotent pre-Lie rings are in one-to-one correspondence and that they are right nilpotent under some cardinality assumptions. [ABSTRACT FROM AUTHOR]

A new vector-product Lie algebra is constructed for a reductive homogeneous space, which can lead to the presentation of two corresponding loop algebras. As a result, two integrable hierarchies of evolution equations are derived from a new form of zero-curvature equation. These hierarchies can be reduced to the heat equation, a special diffusion equation, a general linear Schrödinger equation, and a nonlinear Schrödinger-type equation. Notably, one of them exhibits a pseudo-Hamiltonian structure, which is derived from a new vector-product identity proposed in this paper. [ABSTRACT FROM AUTHOR]

We develop a cluster‐based model order reduction (called C‐pRBMOR) approach for efficient homogenization of bones, compatible with a large variety of generalized standard material (GSM) models. To this end, the pRBMOR approach based on a mixed incremental potential formulation is extended to a clustered version for a significantly improved computational efficiency. The microscopic modeling of bones falls into a mixed incremental class of the GSM framework, originating from two potentials. An offline phase of the C‐pRBMOR approach includes both a clustering analysis spatially decomposing the micro‐domain within an RVE and a space–time decomposition of the microscopic plastic strain fields. A comparative study on two different clustering approaches and two algorithms for mode identification is additionally conducted. For an online analysis, a cluster‐enhanced version of evolution equations for the reduced variables is derived from an effective incremental variational formulation, rendering a very small set of nonlinear equations to be numerically solved. Several numerical examples show the effectiveness of the C‐pRBMOR approach. A striking acceleration rate beyond 104 against conventional FE computations and that beyond 103 against the original pRBMOR approach are observed. [ABSTRACT FROM AUTHOR]

We study the cubic Szegö equation which is an integrable nonlinear nondispersive and nonlocal evolution equation. In particular, we present a direct approach for obtaining the multiphase and multisoliton solutions as well as a special class of periodic solutions. Our method is substantially different from the existing one which relies mainly on the spectral analysis of the Hankel operator. We show that the cubic Szegö equation can be bilinearized through appropriate dependent variable transformations and then the solutions satisfy a set of bilinear equations. The proof is carried out within the framework of an elementary theory of determinants. Furthermore, we demonstrate that the eigenfunctions associated with the multiphase solutions satisfy the Lax pair for the cubic Szegö equation, providing an alternative proof of the solutions. Last, the eigenvalue problem for a periodic solution is solved exactly to obtain the analytical expressions of the eigenvalues. [ABSTRACT FROM AUTHOR]

In this article, we derive the convergence rate for the Wong–Zakai equation of some approximation of stochastic evolution equations with multiplicative noise. To be more precise, the diffusion coefficient in front of the noise is the multiplication operator, and, is therefore not bounded, a situation not treated in the literature. Since our motivation comes from problems in numerical ling, we consider a finite, high-dimensional problem approximating a stochastic evolution equation on a random time grid. By imposing suitable stability conditions on the drift term and the time grid, we achieve a convergence rate in the mean square of order $ \min \{1-\delta,2-2\gamma \} $ min { 1 − δ , 2 − 2 γ } , for some $ 0 \lt \delta \lt 1 $ 0 < δ < 1 and $ 0 \lt \gamma \lt 1/2 $ 0 < γ < 1 / 2. [ABSTRACT FROM AUTHOR]

We develop a rheological model to approximate the nonlinear rheology of wormlike micelles using two constitutive models to represent a structural transition at high shear rates. The model is intended to describe the behavior of semidilute wormlike micellar solutions over a wide range of shear rates whose parameters can be determined mainly from small-amplitude equilibrium measurements. Length evolution equations are incorporated into reactive Rolie-Poly entangled-polymer rheology and dilute reactive-rod rheology, with a kinetic exchange between the two models. Although the micelle length is remarkably reduced during flow, surprisingly, we propose that they are not shortened by stress-enhanced breakage, which remains thermally driven. Instead, we hypothesize that stretching energy introduces a linear potential that decreases the rate of recombination and reduces the mean micelle length. This stress-hindered recombination approach accurately describes transient stress-growth upon start-up shear flow, and it predicts a transition of shear viscosity and alignment response observed at high shear rates. The proposed mechanism applies only when self-recombination occurs frequently. The effect of varying the relative rate of self-recombination on the rheology of wormlike micelles at high shear rates is yet to be explored. [ABSTRACT FROM AUTHOR]

This study aims to establish a strain instanton equation and damage factor evolution law for gypsum specimens by considering damping. First, damping energy is calculated based on the single-degree-of-freedom vibration model, and the instantaneous strain equation is obtained based on the stress balance equation. Second, the dissipation energy is divided into damping and damage energies, and a damage-factor correction algorithm is obtained. Third, cyclic loading and unloading tests were performed at different loading rates and stress amplitudes to verify the accuracy of the strain equation. Finally, the specimens' magnitude curves and crack characteristics were monitored using moment–tensor acoustic emission simulations. The factors influencing the damping energy and strain equations, energy and damage evolution laws of the specimens, and damage patterns of the specimens at different loading rates were analysed. The results show that the instantaneous strain equation and the modified damage factor considering the damping effect can effectively reflect the deformation law and damage state of the specimens. In contrast, the damage to the specimens in the lower limit of the variable stress experiment was lower than that in the lower limit of the constant stress experiment. As the loading rate increases, the damage energy density of the specimen decreases, and the damage factor within a single cycle gradually decreases. As the loading rate increases, the number of crack events in the model increases significantly, size becomes more uniform, and sequentially exhibits dense and sparse distribution patterns, percentage of shear cracks decreases significantly, number of mixed cracks increases significantly, brittle behaviour of the specimen becomes obvious, and a complete damage state is attained known as the 'crushed' state. This study provides a theoretical reference for damage assessments of viscoelastic–plastic materials subjected to perturbing loads. [ABSTRACT FROM AUTHOR]

This study investigates the long-time vorticity dynamics of the multi-cellular configurations of the two-dimensional (2D) Taylor–Green vortex (TGV). The pseudo-spectral method is used to solve the incompressible Navier–Stokes equation to analyze the evolution of TGV arrays. The focus is on understanding vortex interactions leading to vortex filamentation and stripping (forward cascade) during primary instability; merger and reconnection (inverse cascade) among the TGV vortical cells subsequently. Here, consideration of multiple cells avoids imposing symmetries at the smallest periodic length scale, and thereby affecting disturbance growth. The initial condition is taken from the analytic solution of the TGV, and Fourier spectral method is employed to track the interactions of the initial doubly-periodic vortices. The full sequence of evolution from one equilibrium state to another for the TGV is not addressed before, as reported here to fill this gap for multiple TGV cells in both directions. By studying various vortical interactions in the ensemble, here we report the enstrophy and energy spectra for different number of TGV cells. This is crucial in understanding the very long-time evolution process, at post-critical Reynolds numbers for the 2D TGV problem in the same physical domain, (0 ≤ (x , y) ≤ 4 π) having (4 × 4) and (6 × 6) cells. Reported results show the evolution of these vortical cells from original configurations to finally a (1 × 1) vortical cells — the universal state not demonstrated before. • The present work carries out the long-time direct simulation of the 2D TGV problem in its multi-cellular configurations which has previously never been reported (as long as up to t=6000). Presented results are for cases starting with (4 × 4)- and (6 × 6)-vortical cells at t =0. • The Fourier pseudo-spectral method along with the four stage, fourth order Runge–Kutta (RK4) time integration method has previously not been used to perform direct simulation of the 2D TGV problem studied here over a very long time. • Such high accuracy method helps not only capture the primary instability followed by the inverse cascading of energy, but also this leads finally to the unique configuration with (1 × 1)-cells. • The final state is seen to be the universal configuration, irrespective of the initial number of cells. This is noted to be the case for (2 × 2)-cells, (4 × 4)-cells, (6 × 6)-cells and (8 × 8)-cells. The last case has not been shown here to keep the size of the manuscript within limit. [ABSTRACT FROM AUTHOR]

The similarity transformation (ST) method is applied to reduce the Lax pair for some nonlinear partial differential equations (NLPDEs) into a system of ordinary differential equations (ODEs) to obtain its similarity solutions. Then the ODE system is considered to find the analytical solutions of the PDE by plotting the acquired similarity solutions. The method is applied to the three different equations named as; Modified Boussinesq (MBQ) equation, Kadomtsev-Petviashvili (KP) equation, and (2+1) - Korteweg-de-Vries breaking type (KDV-BS) equation. The Lie transformation method is utilized to convert the modified Boussinesq equation's Lax Pair into a system of ordinary differential equations and obtain the analytical solutions of this equation. Likewise, this method is used for the KP and (2+1)-dimensional KdV Lax Pairs. The Lie vectors are optimized through the commutation operation. The reduction of the Lax pair instead of the original equation reveals a new solution. The applied method is effective in spreading the solution of NLPDEs. [ABSTRACT FROM AUTHOR]

We consider the problem of sampling from a distribution governed by a potential function. This work proposes an explicit score-based MCMC method that is deterministic, resulting in a deterministic evolution for particles rather than a stochastic differential equation evolution. The score term is given in closed form by a regularized Wasserstein proximal, using a kernel convolution that is approximated by sampling. We demonstrate fast convergence on various problems and suggest improved dimensional dependence of mixing time bounds for the case of Gaussian distributions compared to the unadjusted Langevin algorithm (ULA) and the Metropolis-adjusted Langevin algorithm (MALA). We additionally derive closed-form expressions for the distributions at each iterate for quadratic potential functions, characterizing the variance reduction. Empirical results demonstrate that the particles behave in an organized manner, lying on level set contours of the potential. Moreover, the posterior mean estimator of the proposed method is shown to be closer to the maximum a-posteriori estimator compared to ULA and MALA in the context of Bayesian logistic regression. Additional examples demonstrate competitive performance for Bayesian neural network training. [ABSTRACT FROM AUTHOR]

In this paper, we study the contribution of color correlations to Double Parton Scattering (DPS). We show that there is a specific class of Feynman diagrams related to so-called 1 → 2 processes when the contribution of these color correlations is not Sudakov suppressed with the transverse scales. The effective absence of Sudakov suppression gives hope that although they are small relative to color singlet correlations, they eventually can be observed. [ABSTRACT FROM AUTHOR]

The presence of an abundant population of low-frequency photons at high redshifts (such as a radio background) can source leading order effects on the evolution of the matter and spin temperatures through rapid free–free absorptions. This effect, known as soft photon heating, can have a dramatic impact on the differential brightness temperature, |$\Delta T_{\rm b}$|⁠ , a central observable in 21-cm cosmology. Here, we introduce a semi-analytic framework to describe the dynamics of soft photon heating, providing a simplified set of evolution equations and a useful numerical scheme which can be used to study this generic effect. We also perform quasi-instantaneous and continuous soft photon injections to elucidate the different regimes in which soft photon heating is expected to impart a significant contribution to the global 21-cm signal and its fluctuations. We find that soft photon backgrounds produced after recombination with spectral index |$\gamma \gt 3.0$| undergo significant free–free absorption, and therefore this heating effect cannot be neglected. The effect becomes stronger with steeper spectral index, and in some cases the injection of a synchrotron-like spectrum (⁠|$\gamma = 3.6$|⁠) can suppress the amplitude of |$\Delta T_{\rm b}$| relative to the standard model prediction (where an additional radio background is absent), making the global 21-cm signal even more difficult to detect in these scenarios. [ABSTRACT FROM AUTHOR]

ABSTRACT In practical sensor networks, sensor nodes may operate with different sampling periods and initial sampling time instants, and their observations may also be nonuniform. Unfortunately, the research on distributed state estimation problems over such asynchronous sensor networks is very limited. Thus, this article focuses on the distributed consensus filtering problem over sensor networks with asynchronous measurements. First, the asynchronous measurement from each sensor is synchronized by the continuous‐time state evolution equation to a unified filtering fusion time instant within a given filtering period. After measurement synchronization, the statistical characteristics of measurement noise change. The cross‐correlations between the converted measurement noises are analyzed, as well as one‐step correlations between the converted measurement noises and the process noise. Second, an optimal asynchronous distributed consensus filter is designed based on synchronization measurements under the criterion of minimum mean‐square error with the above correlations between various types of noises taken into account. Meanwhile, a suboptimal distributed consensus filtering algorithm is further proposed to reduce computational complexity. Finally, based on the Lyapunov function method, the stability of the estimation error is theoretically demonstrated with an appropriate selection of consensus filtering gain and validated through simulations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate global properties of a class of evolution differential operators defined on a product of tori and spheres. We present a comprehensive characterization of global solvability and hypoellipticity, providing necessary and sufficient conditions that involve Diophantine conditions and the connectedness of sublevel sets associated with the coefficients of the operator. Furthermore, we recover well‐known results from existing literature and introduce novel contributions. [ABSTRACT FROM AUTHOR]

The continuum level hydrodynamic equations of a simple fluid were derived by Irving and Kirkwood directly from discrete particle dynamics using statistical mechanics almost 75 years ago. Their elegant derivation demonstrated the fundamental molecular basis of macroscopic fluid flow and culminated in molecular expressions for the stress tensor and heat current density that have since been employed in countless molecular simulations to date. In this article, an alternative derivation is presented, which leads to more general expressions for the fundamental transport relationships and which arrives at them in a more straightforward chain of consistency that ensues directly from the Principle of Least Action. The main point of departure from the Irving–Kirkwood derivation is the application of a transformation mapping of the total momentum of each individual particle onto the sum of its peculiar momentum and its momentum relative to the local velocity field. This mapping provides a phase-space distribution function applicable in the space of particle positions and peculiar momentum, from which a noncanonical Poisson bracket can be derived in terms of the same set of microscopic variables. For a given dynamic variable, expressed in terms of particle positions and peculiar momenta, the expectation value of the noncanonical Poisson bracket of the dynamic variable is shown to correspond to the evolution equation of the expectation value of the dynamic variable. This allows for a direct derivation of all macroscopic density evolution equations (mass, momentum, and energy density fields) using a systematic procedure free of assumptions concerning the macroscopic state of the system. Furthermore, an explicit expression of the time evolution of the entropy density at the hydrodynamic level is derived following the same procedure. Finally, in the limit of short-range interparticle interactions, a molecular-based expression for the local stress tensor as properly defined from continuum mechanics is developed at the hydrodynamic level that elucidates the continuum mechanics connection of the general stress expression of Irving and Kirkwood. [ABSTRACT FROM AUTHOR]

This article introduces two kinds of processing techniques to solve Jacobian elliptic equations and obtain rich periodic wave solutions. Then, the equation was used as an auxiliary equation to solve the (3+1)-dimensional modified Korteweg de Vries–Zakharov–Kuznetsov (mKDV-ZK) equation. Combined with the mapping method, a large number of new types of exact periodic wave solutions were obtained, many of which were rarely found in previous research. Numerical simulations have demonstrated the evolution of various periodic waves in (3+1)-dimensional mKDV-ZK. The solutions and wave phenomena obtained in this article will help expand our understanding of the equation. [ABSTRACT FROM AUTHOR]

Timed event graphs (TEGs) are mathematical models for the dynamics of discrete systems. A TEG is a directed graph consisting of two kinds of nodes, called places and transitions, and directed arcs between places and transitions. Each place can contain tokens that move around by firings of transitions, which is determined by the holding times assigned to places. In the weighted timed event graphs (WTEGs), the weights are assigned to the arcs and these weights represent the number of the tokens consumed or produced by the firing of the adjacent transitions. Unlike usual timed event graphs, weights on arcs make it difficult to describe the evolution of systems in simple mathematical formulae. The main purpose of this paper is to derive max-plus algebraic evolution equations for unitary WTEGs. We focus on the numbers indicating how many times each transition fires to reach the original token configuration again. Such numbers are known as the minimal T-semiflow and are computed from arc weights. We expand the dimension of the firing time vector by dividing the original variables into several variables, but it does not depend on the number of tokens. We also show that max-plus coefficient matrices in the evolution equation provide important information about the dynamics of a unitary WTEG. For example, the nilpotency of the coefficient matrix in the highest order term means the liveness of the WTEG. In this case, the average cycle time of the WTEG is computed as the eigenvalue of a certain max-plus matrix. [ABSTRACT FROM AUTHOR]

Technical surfaces are generally not perfectly smooth, but usually exhibit roughness. Additionally, as micro production techniques continue to evolve, geometrical surface structures at the micro scale can be manufactured, which presents a challenge for wetting models that are commonly formulated for smooth surfaces. A phase field model for surface wetting, proposed by Diewald et al., serves as a basis for this investigation. On smooth surfaces, the width of the gas‐liquid interface can be widened for scale bridging purposes, as this allows for accurate computations on coarse grids. However, it was observed that this interface scaling impacts the results when the model is applied to rough surfaces, and therefore a free choice of the interface width is not always sensible. In this work, the ability of the model to deal with sinusoidally shaped surfaces is investigated. An Allen–Cahn evolution equation is used to determine static equilibrium configurations for droplets on such surfaces. [ABSTRACT FROM AUTHOR]

Stellar-mass binary black holes may have circumbinary discs if formed through common-envelope evolution or within gaseous environments. Discs can drive binaries into wider and more eccentric orbits, while gravitational waves harden and circularize them. We combine cutting-edge evolution prescriptions for disc-driven binaries with well-known equations for gravitational-wave-driven evolution, and study the evolution of stellar-mass binary black holes. We find that binaries are driven by their disc to an equilibrium eccentricity, |$0.2\lesssim e_\mathrm{eq}~\lesssim 0.5$|⁠ , that dominates their evolution. Once they transition to the GW-dominated regime their eccentricity decreases rapidly; we find that stellar-mass binary black holes with long-lived discs will likely be observed in LISA with detectable eccentricities |${\sim} 10^{-2}$| at 0.01 Hz, with the precise value closely correlating with the binary's initial mass ratio. This may lead stellar-mass binary black holes with CBDs observed in LISA to be confused with dynamically-formed binary black holes. [ABSTRACT FROM AUTHOR]

The columnar-flow approximation allows the development of computationally efficient numerical models tailored to the study of the rapidly rotating dynamics of Earth's fluid outer core. In this paper, we extend a novel columnar-flow formulation, called Plesio-Geostrophy (PG) by including thermal effects and viscous boundary conditions. The effect of both no-slip and stress-free boundaries, the latter being a novelty for columnar-flow models, are included. We obtain a set of fully 2-D evolution equations for fluid flows and temperature where no assumption is made regarding the geometry of the latter, except in the derivation of an approximate thermal diffusion operator. To test the new PG implementation, we calculated the critical parameters for onset of thermal convection in a spherical domain. We found that the PG model prediction is in better agreement with unapproximated, 3-D calculations in rapidly rotating regimes, compared to another state-of-the-art columnar-flow model. [ABSTRACT FROM AUTHOR]

In this work, we study the Chafee–Infante differential equation (CIDE) for various types of nonlinear waves (NW) like breather waves (BW), periodic cross-kink (PCKW), lump waves (LW), rogue waves (RW), periodic wave (PW), lump soliton (LS) with one kink, LS with two kink and periodic-cross lump waves (PCLW). The CIDE is a nonlinear evolution equation (NLEE) introduced by Nathaniel Chafee and Ettore Infante. This equation has been widely investigated in the context of pattern generation and soliton solutions, and it plays an essential role in understanding the complex nonlinear dynamics across a wide range of scientific fields. BW is a localized, non-dispersive solution to NW equations, maintaining shape and amplitude during travel. An LS is a type of solitary wave solution in NLEEs. In mathematics, RW refers to rare and unusually large solutions of NW equations. These waves have garnered significant attention due to their unpredictable and extreme nature. A PW is a type of wave that repeats its shape and behavior at regular intervals in both time and space. These waves are characterized by their periodicity, which means that they have a consistent and predictable pattern of oscillation. We'll offer a graphical explanation of our newly discovered answers toward the conclusion. [ABSTRACT FROM AUTHOR]

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The focus of this article lies in the existence of global mild solutions for Hilfer fractional stochastic evolution equations (HFSEEs) on the positive semi-axis (0 , + ∞) with a derivative order greater than 3 2 and less than 2. A significant challenge, which also presents novelty here, is to extend the generalized Ascoli–Arzela (A–A) theorem established in Zhou and He (2022) to the stochastic case under the associated sine family { S (t) } t ≥ 0 not necessary compact. Then, by relying on our newly established generalized A–A theorem, standard stochastic analysis theory and Schauder's fixed point principle, we derive the existence of global mild solutions for the addressed system. Finally, we demonstrate the feasibility of our obtained results through an illustrative example. [ABSTRACT FROM AUTHOR]

Concrete structures such as bridge decks and road pavements are subjected to repetitive loading and are susceptible to fatigue failure. A simplified stress–strain analysis method that can simulate concrete behavior with a sound physical basis, acceptable prediction precision, and reasonable computation cost is urgently needed to address the critical issue of high-cycle fatigue in structural engineering. An equivalent fatigue constitutive model at discrete loading cycles incorporated into the concrete damaged plasticity model (CDPM) in Abaqus is proposed based on fatigue damage evolution. A damage variable is constructed from maximum fatigue strains, and fatigue damage evolution is described by a general equation whose parameters' physical meaning and value range are identified. With the descending branch of the monotonic stress–strain curve as the envelope of fatigue residual strength and fatigue damage evolution equation as shape function, fatigue residual strength, residual stiffness, and residual strain are calculated. The equivalent fatigue constitutive model is validated through comparison with experimental data, where satisfactory simulation results were obtained for axial compression and flexural tension fatigue. The model's novelty lies in integrating the fatigue damage evolution equation with CDPM, explicitly explaining performance degradation caused by fatigue damage. The proposed model could accommodate various forms of concrete constitution and fatigue stress states and has a broad application prospect for fatigue analysis of concrete structures. [ABSTRACT FROM AUTHOR]

The primary aim of this article is to explore the approximate controllability of second-order impulsive hemivariational inequalities with initial conditions in Hilbert space. The mild solution was initially derived using the properties of the cosine and sine family of operators, Clarke's subdifferential, and the fact that the related linear equation has an evolution operator. The results of the approximate controllability of the considered systems are then taken into account using the fixed-point theorem method. An application is provided to support our theoretical findings. [ABSTRACT FROM AUTHOR]

We study semilinear third‐order (in time) evolution equations with fractional Laplacian (−Δ)σ$$ {\left(-\Delta \right)}^{\sigma } $$ and power nonlinearity |u|p$$ {\left|u\right|}^p $$, which was proposed by Bezerra–Carvalho–Santos (J. Evol. Equ. 2022) recently. In this manuscript, we obtain a new critical exponent p=pcrit(n,σ):=1+6σmax{3n−4σ,0}$$ p={p}_{\mathrm{crit}}\left(n,\sigma \right):= 1+\frac{6\sigma }{\max \left\{3n-4\sigma, 0\right\}} $$ for n⩽103σ$$ n\leqslant \frac{10}{3}\sigma $$. Precisely, the global (in time) existence of small data Sobolev solutions is proved for the supercritical case p>pcrit(n,σ)$$ p>{p}_{\mathrm{crit}}\left(n,\sigma \right) $$, and energy solutions blow up in finite time even for small data if 1

This work deals with a cosmological model consisting of cold dark matter and dark energy without any interaction. The study has been done both at the background level and at the perturbative level for the homogeneous and isotropic FLRW spacetime. By suitable transformation of the variables, the cosmic evolution equations are converted into an autonomous system. A discrete dynamical system analysis has been employed for cosmic inferences. [ABSTRACT FROM AUTHOR]

We explore a cosmological model inspired by the modified Kaniadakis entropy and disclose the influences of the modified Friedmann equations on the evolution of the Universe. We find that in modified Kaniadakis cosmology with only pressure-less matter, one can reproduce the accelerated Universe without invoking any kind of dark energy. We delve into the evolution of the Universe during the radiation-dominated era as well. We also investigate the behavior of the scale factor and the deceleration parameter for a multiple-component Universe consisting of pressure-less matter and a cosmological constant/dark energy. Interestingly enough, the predicted age of the Universe in the modified Kaniadakis cosmology becomes larger compared to the standard cosmology which may alleviate the age problem. Furthermore, the findings reveal that in the modified Kaniadakis cosmology, the transition from a decelerated phase to an accelerated Universe occurs at higher redshifts compared to the standard cosmology. These results shed light on the potential implications of incorporating Kaniadakis entropy into cosmological models and provide valuable insights into the behavior of the Universe in different cosmological scenarios. Moreover, they emphasize the crucial role of modifications to the geometry component and the significance of such modifications in understanding the dynamics of the Universe. [ABSTRACT FROM AUTHOR]

Problems without initial conditions for evolution equations and variational inequalities appear in the modeling of different non-stationary processes within many fields of science, such as ecology, economics, physics, cybernetics, etc., if these processes started a long time ago and initial conditions do not affect them in the actual time moment. Thus, we can assume that the initial time is minus infinity. In the case of linear and weakly nonlinear evolution equations and variational inequalities, standard initial conditions should be replaced with the behavior of the solution as the time variable goes to minus infinity. However, for some strongly nonlinear evolution equations and variational inequalities, this problem has a unique solution in the class of functions without behavior restriction as the time variable goes to minus infinity. In this study, the correctness of the problem without initial conditions for such types of variational inequalities from a new class, or more precisely, for sub-differential inclusions with functionals, is investigated. Moreover, estimates of solutions are obtained. The results are new and mostly theoretical. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global (in time) existence of small data solutions to the Cauchy problem for the following structurally damped σ‐evolution model with nonlinear memory term: utt+(−Δ)σu+μ(−Δ)σ2ut=∫0t(t−τ)−γ|ut(τ,·)|pdτ,with σ>0. In particular, for γ∈((n−σ)/n,1), we find the sharp critical exponent, under the assumption of small data in L1. Dropping the L1 smallness assumption of initial data, we show how the critical exponent is consequently modified for the problem. In particular, we obtain a new interplay between the fractional order of integration 1−γ in the nonlinear memory term and the assumption that initial data are small in Lm, for some m>1. [ABSTRACT FROM AUTHOR]

We consider a fractional‐in‐time evolution equation arising in the theory of ion‐sound waves in plasma, where the spatial variable varies on the half‐line. We first provide sufficient conditions for which there exist no global‐in‐time weak solutions and obtain an upper bound of the lifespan. Next, we find a class of initial data for which local‐in‐time weak solutions do not exist, that is, an instantaneous blow‐up of weak solutions occurs. The proofs of our results are based on some integral inequalities and the test function method. [ABSTRACT FROM AUTHOR]

In this work, a new method to determine and to describe the evolution of physical systems (in classical or quantum regimes) is proposed. The main idea is based in the observation that the complete determination of dynamics of the system is connected to the nonlinear realization (NLR) of the symmetry group associated with the symmetry of the Hamiltonian H (t) of the system. The Maurer–Cartan forms, when they are parametrically associated to the projection on the hypersurface induced by the group in question via pullback, of the Hamiltonian components (based on the generators of the symmetry group by which it is defined) give rise to a fundamental system of equations by which the precise evolution can be determined. Such a system of equations under certain conditions that we will specify in this paper coincides with the system obtained through the Wei–Norman method, thus showing explicitly that our proposal is more general due the universal character of equations from the NLRs approach. As example, the important problem of the Caldirola–Kanai inverted Hamiltonian is completely solved, showing at the same time a new interpretation of what NLRs refer to. [ABSTRACT FROM AUTHOR]

We investigate the viscoelastic relaxation to equilibrium of an isolated disordered planar epithelium described using the cell vertex model. In its standard form, the model is formulated as coupled evolution equations for the locations of vertices of confluent polygonal cells. Exploiting the model's gradient-flow structure, we use singular-value decomposition to project modes of deformation of vertices onto modes of deformation of cells. Expressing the dynamics in terms of operators of a discrete calculus, we show how eigenmodes of discrete Laplacian operators (specified by constitutive assumptions related to dissipation and mechanical energy) provide a spatial basis for evolving fields, and demonstrate how the operators can incorporate approximations of conventional spatial derivatives. We relate the spectrum of relaxation times to the eigenvalues of the Laplacians, modified by corrections that account for the fact that the cell network (and therefore the Laplacians) evolve during relaxation to an equilibrium prestressed state, providing the monolayer with geometric stiffness. While dilational modes of the Laplacians capture rapid relaxation in some circumstances, showing diffusive dynamics, geometric stiffness is typically a dominant source of monolayer rigidity, as we illustrate for monolayers exposed to unsteady stretching deformations. [ABSTRACT FROM AUTHOR]

We introduce models for viscoelastic materials, both solids and fluids, based on logarithmic stresses to capture the elastic contribution to the material response. The matrix logarithm allows to link the measures of strain, that naturally belong to a multiplicative group of linear transformations, to stresses, that are additive elements of a linear space of tensors. As regards the viscous stresses, we simply assume a Newtonian constitutive law, but the presence of elasticity and plastic relaxation makes the materials non‐Newtonian. Our aim is to discuss the existence of weak solutions for the corresponding systems of partial differential equations in the nonlinear large‐deformation regime. The main difficulties arise in the analysis of the transport equations necessary to describe the evolution of tensorial measures of strain. For the solid model, we only need to consider the equation for the left Cauchy–Green tensor, while for the fluid model, we add an evolution equation for the elastically‐relaxed strain. Due to the tensorial nature of the fields, available techniques cannot be applied to the analysis of such transport equations. To cope with this, we introduce the notion of charted weak solution, based on non‐standard a priori estimates, that lead to a global‐in‐time existence of solutions for the viscoelastic models in the natural functional setting associated with the energy inequality. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKEs) for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as Schrödinger equations or multidimensional KdV‐type equations. However, our situation here is different since the water waves equations are quasilinear and solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss. This is the first of two papers in which we design a new strategy to address this issue. We investigate solutions of the gravity water waves system in two dimensions. In the irrotational case, this system can be reduced to an evolution equation on the one‐dimensional interface, which is a large torus TR${\mathbb {T}}_R$ of size R≥1$R\ge 1$. Our first main result is a deterministic energy inequality, which provides control of (possibly large) Sobolev norms of solutions for long times, under the condition that a certain L∞$L^\infty$‐type norm is small. This energy inequality is of “quintic” type: if the L∞$L^\infty$ norm is O(ε)$O(\varepsilon)$, then the increment of the high‐order energies is controlled for times of the order ε−3$\varepsilon ^{-3}$, consistent with the approximate quartic integrability of the system. In the second paper in this sequence, we will show how to use this energy estimate and a propagation of randomness argument to prove a probabilistic regularity result up to times of the order ε−3R3/2−${\varepsilon }^{-3} R^{3/2-}$, in a suitable scaling regime relating ε→0${\varepsilon }\rightarrow 0$ and R→∞$R \rightarrow \infty$. For our second main result, we combine the quintic energy inequality with a bootstrap argument using a suitable Z$Z$‐norm of Strichartz‐type to prove that deterministic solutions with Sobolev data of size ε≪1${\varepsilon }\ll 1$ are regular for times of the order ε−3min(ε−3,R3/4)${\varepsilon }^{-3} \min \big ({\varepsilon }^{-3}, R^{3/4} \big)$. In particular, on the real line, solutions exist for times of order O(ε−6)$O(\varepsilon ^{-6})$. This improves substantially on all the earlier extended lifespan results for 2D gravity water waves with small Sobolev data. [ABSTRACT FROM AUTHOR]

We develop an abstract theory of flows of geometric 퐻-structures, i.e., flows of tensor fields defining 퐻-reductions of the frame bundle, for a closed and connected subgroup H ⊂ SO ⁢ ( n ) H\subset\mathrm{SO}(n) , on any connected and oriented 푛-manifold with sufficient topology to admit such structures. The first part of the article sets up a unifying theoretical framework for deformations of 퐻-structures, by way of the natural infinitesimal action of GL ⁢ ( n , R ) \mathrm{GL}(n,\mathbb{R}) on tensors combined with various bundle decompositions induced by 퐻-structures. We compute evolution equations for the intrinsic torsion under general flows of 퐻-structures and, as applications, we obtain general Bianchi-type identities for 퐻-structures, and, for closed manifolds, a general first variation formula for the L 2 L^{2} -Dirichlet energy functional ℰ on the space of 퐻-structures. We then specialise the theory to the negative gradient flow of ℰ over isometric 퐻-structures, i.e., their harmonic flow. The core result is an almost-monotonocity formula along the flow for a scale-invariant localised energy, similar to the classical formulas by Chen–Struwe [M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Differential Geom. 28 (1988), 3, 485–502; Y. M. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), 1, 83–103] for the harmonic map heat flow. This yields an 휀-regularity theorem and an energy gap result for harmonic structures, as well as long-time existence for the flow under small initial energy, relative to the L ∞ L^{\infty} -norm of initial torsion, in the spirit of Chen–Ding [Y. M. Chen and W. Y. Ding, Blow-up and global existence for heat flows of harmonic maps, Invent. Math. 99 (1990), 3, 567–578]. Moreover, below a certain energy level, the absence of a torsion-free isometric 퐻-structure in the initial homotopy class imposes the formation of finite-time singularities. These seemingly contrasting statements are illustrated by examples on flat 푛-tori, so long as the set [ S n , SO ⁢ ( n ) / H ] [\mathbb{S}^{n},\mathrm{SO}(n)/H] of homotopy classes of maps S n → SO ⁢ ( n ) / H \mathbb{S}^{n}\to\mathrm{SO}(n)/H contains more than one element and the universal cover of SO ⁢ ( n ) / H \mathrm{SO}(n)/H is a sphere, e.g. when n = 7 n=7 and H = G 2 H=\mathrm{G}_{2} , or n = 8 n=8 and H = Spin ⁢ ( 7 ) H=\mathrm{Spin}(7) . [ABSTRACT FROM AUTHOR]

For temperatures below and beyond the Curie temperature, the stochastic Landau-Lifshitz-Bloch equation describes the evolution of spins in ferromagnetic materials. In this work, we consider the stochastic Landau-Lifshitz-Bloch equation driven by a real valued Wiener process and show Wong-Zakai type approximations for the same. We consider non-zero contribution from the helicity term in the energy. First, using a Doss-Sussmann type transform, we convert the stochastic partial differential equation into a deterministic equation with random coefficients. We then show that the solution of the transformed equation depends continuously on the driving Wiener process. We then use this result, along with the properties of the said transform to show that the solution of the originally considered equation depends continuously on the driving Wiener process. [ABSTRACT FROM AUTHOR]

A novel evolving surface finite element method, based on a novel equivalent formulation of the continuous problem, is proposed for computing the evolution of a closed hypersurface moving under a prescribed velocity field in two- and three-dimensional spaces. The method improves the mesh quality of the approximate surface by minimizing the rate of deformation using an artificial tangential motion. The transport evolution equations of the normal vector and the extrinsic Weingarten matrix are derived and coupled with the surface evolution equations to ensure stability and convergence of the numerical approximations. Optimal-order convergence of the semidiscrete evolving surface finite element method is proved for finite elements of degree k ≥ 2. Numerical examples are provided to illustrate the convergence of the proposed method and its effectiveness in improving mesh quality on the approximate evolving surface. [ABSTRACT FROM AUTHOR]

In this paper, we provide several techniques to explicitly calculate fractional powers of 2×2$2\times 2$ operator matrices Λ=Λ11Λ12Λ21Λ22,$$\begin{equation*}\hspace*{10pc} \Lambda = \def\eqcellsep{&}\begin{bmatrix} \Lambda _{11} & \Lambda _{12} \\ \Lambda _{21} & \Lambda _{22} \end{bmatrix}, \end{equation*}$$focusing on creating a theory that can be applied to distinct situations. To illustrate the abstract results developed, we consider its application in systems of coupled reaction–diffusion equations and in (strongly damped) wave equations. We also discuss how these techniques can be applied to higher order matrices and we specifically calculate the fractional powers of a 4×4$4\times 4$ operator matrix associated to a weakly coupled system of wave equation. In addition, we deal with the applicability of this analysis with respect to solvability, stabilization, regularity, smooth dynamics, and connection with evolutionary classic equation and its fractional counterpart. [ABSTRACT FROM AUTHOR]

A nonseparable wave-like integro-differential equation for the time evolution of the Wigner distribution function in phase space is educed from the corresponding separable kinetic equation. By employing the quantum hydrodynamical description, a non-local evolution wave equation is also derived by synthesizing the Hamilton-Jacobi equation with that of continuity, which predicts the generation of nonlocal and quadrupole quantum phenomena in the propagation of the spatial probability density. [ABSTRACT FROM AUTHOR]

In this article, we explore Freidlin-Wentzell's large deviation principle for a stochastic partial differential equation with the nonlinear diffusion-convection operator in divergence form satisfying p-type growth, involving coercivity assumptions, perturbed by small multiplicative Brownian noise. We use the weak convergence method to prove the Laplace principle, which is equivalent to the large deviation principle in our framework. [ABSTRACT FROM AUTHOR]

The dynamics that govern disease spread are hard to model because infections are functions of both the underlying pathogen as well as human or animal behavior. This challenge is increased when modeling how diseases spread between different spatial locations. Many proposed spatial epidemiological models require trade-offs to fit, either by abstracting away theoretical spread dynamics, fitting a deterministic model, or by requiring large computational resources for many simulations. We propose an approach that approximates the complex spatial spread dynamics with a Gaussian process. We first propose a flexible spatial extension to the well-known SIR stochastic process, and then we derive a moment-closure approximation to this stochastic process. This moment-closure approximation yields ordinary differential equations for the evolution of the means and covariances of the susceptibles and infectious through time. Because these ODEs are a bottleneck to fitting our model by MCMC, we approximate them using a low-rank emulator. This approximation serves as the basis for our hierarchical model for noisy, underreported counts of new infections by spatial location and time. We demonstrate using our model to conduct inference on simulated infections from the underlying, true spatial SIR jump process. We then apply our method to model counts of new Zika infections in Brazil from late 2015 through early 2016. [ABSTRACT FROM AUTHOR]