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We develop an accurate weak-interaction theory for the pulses of the generalized Kuramoto–Sivashinsky (gKS) equation. This equation is the simplest prototype that retains the fundamental mechanisms of wave evolution in nonlinear media, namely, dominant nonlinearity, instability/energy supply, stability/energy dissipation and dispersion. The dynamics of the usual (dispersionless) KS equation is chaotic in both space and time. However, sufficiently strong dispersion regularizes the dynamics and the solutions evolve into arrays of interacting pulses, that can form bound states if dispersion does not exceed a certain threshold value. To obtain a theoretical insight into the interaction of the pulses, we represent a solution of the gKS equation as a superposition of pulses and an overlap function, and we carefully derive a coupled system of ordinary differential equations describing the evolution of the locations of the pulses by projecting the dynamics onto translational modes. This approach allows one to analyse bound states of the pulses and to derive a criterion on the existence of a countable infinite or finite number of bound states, depending on the strength of the dispersive term in the equation, which is in agreement with Shilnikov's criterion on the existence of subsidiary homoclinic orbits. We consider in detail two- and three-pulse bound states. We compare the interaction theory with computations of the full equation when the initial condition is a superposition of two or three pulses, and in all cases we find very good agreement. [ABSTRACT FROM PUBLISHER]

We examine binary interaction of two-dimensional solitary pulses in falling liquid films. We make use of a two-field system of equations for the local flow rate and interface position that includes (second-order) viscous dispersion effects. By applying a coherent-structure theory, we obtain a dynamical system for the separation length between the pulses. The system is re-formulated in terms of a potential function which not only allows us to predict bound-state formation but also to interpret physically this formation. Numerical results of the fully non-linear second-order model are in very good agreement with the theoretically predicted distances. In addition, we numerically show that there are different types of dynamics associated with each bound state, including an overdamped, underdamped and undamped oscillating state depending on both the initial separation length and the Reynolds number. [ABSTRACT FROM PUBLISHER]

We investigate the equilibrium of a fluid in contact with a solid boundary through a density-functional theory. Depending on the conditions, the fluid can be in one phase, gas or liquid, or two phases, while the wall induces an external field acting on the fluid particles. We first examine the case of a liquid film in contact with the wall. We construct bifurcation diagrams for the film thickness as a function of the chemical potential. At a specific value of the chemical potential, two equally stable films, a thin one and a thick one, can coexist. As saturation is approached, the thickness of the thick film tends to infinity. This allows the construction of a liquid–gas interface that forms a well-defined contact angle with the wall. [ABSTRACT FROM PUBLISHER]

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The propagation of a premixed laminar flame supported by an exothermic chemical reaction under adiabatic conditions but subject to inhibition through a parallel endothermic chemical process is considered. The temporal stability to longitudinal perturbations of any resulting flames is investigated. The heat loss through the endothermic reaction, represented by the dimensionless parameter α, has a strong quenching effect on wave propagation. The wave speed-cooling parameter (α, c) curves are determined for a range of values of the other parameters. These curves can be monotone decreasing or S-shaped, depending on the values of the parameters β representing the rate at which inhibitor is consumed relative to the consumption of fuel, μ, the ratio of the activation energies of the reactants and the Lewis numbers. This gives the possibility of having either one, two or three different flame velocities for the same value of the cooling parameter α. F o r Lewis numbers close to unity, when there are three solutions, two of them are stable and one is unstable, with two saddle-node bifurcation points on the (α, c) curve. For larger values of the Lewis numbers there is a Hopf bifurcation point on the curve, dividing it into a stable and an unstable branch. The saddle-node and Hopf bifurcation curves are also determined. The two curves have a common, Takens--Bogdanov bifurcation point. [ABSTRACT FROM AUTHOR]

The propagation of a premixed laminar flame supported by an exothermic chemical reaction under adiabatic conditions but subject to inhibition through a parallel endothermic chemical process is considered. The heat loss through the endothermic reaction, represented by the dimensionless parameter α; has a strong quenching effect on wave propagation. The temperature profile can have a front or a pulse structure depending on the relative values of the parameters α and β; the latter represents the rate at which inhibitor is consumed relative to the consumption of fuel. The wave speed–cooling parameter (α) curves are determined for various values of the other parameters. These curves can be monotone decreasing or S‐shaped, depending on the value of β and the ratio of the activation energies of the reactants. This gives the possibility of having either one, two or three different flame velocities for the same value of the cooling parameter α. [ABSTRACT FROM PUBLISHER]

Consider wave propagation through regions of no excitability (gaps) in an otherwise excitable medium. Propagation in the gaps takes place via simple diffusion. We extend the geometric method for a one-gap system developed by Lewis and Keener to the case of two and three gaps, and we obtain conditions for successful wave propagation and failure. We show that, like the one-gap system, steady-state multiplicity for the case of two gaps arises via a limit point bifurcation. We also demonstrate that in some cases the presence of a large number of gaps promotes wavefront propagation. [ABSTRACT FROM AUTHOR]

The mass transport into a fluid bounded by stationary rigid walls in the limit of large Péclet number, Pe, is examined analytically. Two model systems are considered in detail: a stationary cavity and a model involving two concentric rotating cylinders. A macroscopic gradient is imposed between the top and bottom surfaces. It is demonstrated that mass transport into the fluid is enhanced owing to a recirculation zone which is connected to the solid boundary through a boundary layer of thickness O( Pe−1/3) in which cross-stream molecular diffusion is balanced by convection. The associated enhancement is large and scales as Pe1/3. Our asymptotic analysis is found to be in good agreement with numerical solutions of the full transport equation. [ABSTRACT FROM AUTHOR]

The swift progression and expansion of machine learning (ML) have not gone unnoticed within the realm of statistical mechanics. In particular, ML techniques have attracted attention by the classical density-functional theory (DFT) community, as they enable automatic discovery of free-energy functionals to determine the equilibrium-density profile of a many-particle system. Within classical DFT, the external potential accounts for the interaction of the many-particle system with an external field, thus, affecting the density distribution. In this context, we introduce a statistical-learning framework to infer the external potential exerted on a classical many-particle system. We combine a Bayesian inference approach with the classical DFT apparatus to reconstruct the external potential, yielding a probabilistic description of the external-potential functional form with inherent uncertainty quantification. Our framework is exemplified with a grand-canonical one-dimensional classical particle ensemble with excluded volume interactions in a confined geometry. The required training dataset is generated using a Monte Carlo (MC) simulation where the external potential is applied to the grand-canonical ensemble. The resulting particle coordinates from the MC simulation are fed into the learning framework to uncover the external potential. This eventually allows us to characterize the equilibrium density profile of the system by using the tools of DFT. Our approach benchmarks the inferred density against the exact one calculated through the DFT formulation with the true external potential. The proposed Bayesian procedure accurately infers the external potential and the density profile. We also highlight the external-potential uncertainty quantification conditioned on the amount of available simulated data. The seemingly simple case study introduced in this work might serve as a prototype for studying a wide variety of applications, including adsorption, wetting, and capillarity, to name a few. [ABSTRACT FROM AUTHOR]

The number of new daily SARS-CoV-2 infections experienced an abrupt increase during the last quarter of 2020 in almost every European country. The phenomenological explanation offered was a new mutation of the virus, first identified in the UK. We use publicly available data in combination with a time-delayed controlled SIR model, which captures the effects of preventive measures on the spreading of the virus. We are able to reproduce the waves of infection occurred in the UK with a unique transmission rate, suggesting that the new SARS-CoV-2 variant is as transmissible as previous strains. Our findings indicate that the sudden surge in cases was, in fact, related to the relaxation of preventive measures and social awareness. We also simulate the combined effects of restrictions and vaccination campaigns in 2021, demonstrating that lockdown policies are not fully effective to flatten the curve. For effective mitigation, it is critical that the public keeps on high alert until vaccination reaches a critical threshold. [ABSTRACT FROM AUTHOR]

We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. 19, 1 (1978)]. In particular, we study stationary states of the mean field limit for a system of weakly interacting diffusions moving in a multiwell potential energy landscape, coupled via a Curie-Weiss type (quadratic) interaction potential. The location and depth of the local minima of the potential are either deterministic or random. We characterize the structure and nature of bifurcations and phase transitions for this system, by means of extensive numerical simulations and of analytical calculations for an explicitly solvable model. Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov) equation, the minimization of the free-energy functional, and a continuation algorithm for the stationary solutions. [ABSTRACT FROM AUTHOR]

Consider a two-dimensional capped capillary pore formed by capping two parallel planar walls with a third wall orthogonal to the two planar walls. This system reduces to a slit pore sufficiently far from the capping wall and to a single planar wall when the side walls are far apart. Not surprisingly, wetting of capped capillaries is related to wetting of slit pores and planar walls. For example, the wetting temperature of the capped capillary provides the boundary between first-order and continuous transitions to condensation. We present a numerical investigation of adsorption in capped capillaries of mesoscopic widths based on density functional theory. The fluid-fluid and fluid-substrate interactions are given by the pairwise Lennard-Jones potential. We also perform a parametric study of wetting in capped capillaries by a liquid phase by varying the applied chemical potential, temperature, and pore width. This allows us to construct surface phase diagrams and investigate the complicated interplay of wetting mechanisms specific to each system, in particular, the dependence of capillary wetting temperature on the pore width. [ABSTRACT FROM AUTHOR]

We propose high-order well-balanced finite-volume schemes for a broad class of hydrodynamic systems with attractive-repulsive interaction for ces and linear and nonlinear damping. Our schemes are suitable for free energies containing convolutions of an interaction potential with the density, which are essential for applications such as the Keller--Segel model, more general Euler--Poisson systems, or dynamic-density functional theory. Our schemes are also equipped with a nonnegative-density reconstruction which allows for vacuum regions during the simulation. We provide several prototypical examples from relevant applications highlighting the benefit of our algorithms and also elucidate some of our analytical results. [ABSTRACT FROM AUTHOR]

We propose a numerical scheme based on the Chebyshev pseudo-spectral collocation method for solving the integral and integro-differential equations of the density-functional theory and its dynamic extension. We demonstrate the exponential convergence of our scheme, which typically requires much fewer discretization points to achieve the same accuracy compared to conventional methods. This discretization scheme can also incorporate the asymptotic behavior of the density, which can be of interest in the investigation of open systems. Our scheme is complemented with a numerical continuation algorithm and an appropriate time stepping algorithm, thus constituting a complete tool for an efficient and accurate calculation of phase diagrams and dynamic phenomena. To illustrate the numerical methodology, we consider an argon-like fluid adsorbed on a Lennard-Jones planar wall. First, we obtain a set of phase diagrams corresponding to the equilibrium adsorption and compare our results obtained from different approximations to the hard sphere part of the free energy functional. Using principles from the theory of sub-critical dynamic phase field models, we formulate the time-dependent equations which describe the evolution of the adsorbed film. Through dynamic considerations we interpret the phase diagrams in terms of their stability. Simulations of various wetting and drying scenarios allow us to rationalize the dynamic behavior of the system and its relation to the equilibrium properties of wetting and drying. [ABSTRACT FROM AUTHOR]

Thin liquid films are ubiquitous in natural phenomena and technological applications. They have been extensively studied via deterministic hydrodynamic equations, but thermal fluctuations often play a crucial role that needs to be understood. An example of this is dewetting, which involves the rupture of a thin liquid film and the formation of droplets. Such a process is thermally activated and requires fluctuations to be taken into account self-consistently. In this work we present an analytical and numerical study of a stochastic thin-film equation derived from first principles. Following a brief review of the derivation, we scrutinise the behaviour of the equation in the limit of perfectly correlated noise along the wall-normal direction, as opposed to the perfectly uncorrelated limit studied by Grün et al. (J Stat Phys 122(6):1261-1291, 2006). We also present a numerical scheme based on a spectral collocation method, which is then utilised to simulate the stochastic thin-film equation. This scheme seems to be very convenient for numerical studies of the stochastic thin-film equation, since it makes it easier to select the frequency modes of the noise (following the spirit of the long-wave approximation). With our numerical scheme we explore the fluctuating dynamics of the thin film and the behaviour of its free energy in the vicinity of rupture. Finally, we study the effect of the noise intensity on the rupture time, using a large number of sample paths as compared to previous studies. [ABSTRACT FROM AUTHOR]

We study a prototypical example in nonlinear dynamics where transition to self-similarity in a singular limit is fundamentally changed as a parameter is varied. Here, we focus on the complicated dynamics that occur in a generalised unstable thin-film equation that yields finite-time rupture. A parameter, n , is introduced to model more general disjoining pressures. For the standard case of van der Waals intermolecular forces, n = 3 , it was previously established that a countably infinite number of self-similar solutions exist leading to rupture. Each solution can be indexed by a parameter, ϵ = ϵ 1 > ϵ 2 > ⋯ > 0 , and the prediction of the discrete set of solutions requires examination of terms beyond-all-orders in ϵ. However, recent numerical results have demonstrated the surprising complexity that exists for general values of n. In particular, the bifurcation structure of self-similar solutions now exhibits branch merging as n is varied. In this work, we shall present key ideas of how branch merging can be interpreted via exponential asymptotics. • Large class of nonlinear systems involve self-singular singularity formation. • Exponential asymptotics provides a framework for the analysis of such problems. • Framework exemplified with hydrodynamic model prototype of thin-film rupture. • Exponential asymptotics selects discrete solution set and explains branch merging. • Complexity includes topological transitions, emergence, and self-organisation. [ABSTRACT FROM AUTHOR]

Abstract: We study phase field equations in perforated domains for arbitrary free energies. These equations have found numerous applications in a wide spectrum of both science and engineering problems with homogeneous environments. Here, we focus on strongly heterogeneous materials with perforations such as porous media. To the best of our knowledge, we provide the first derivation of upscaled equations for general free energy densities. In view of the versatile applications of phase field equations, we expect that our study will lead to new modelling and computational perspectives for interfacial transport and phase transformations in strongly heterogeneous environments. [Copyright &y& Elsevier]