Analysis Of Stationary Solutions Research Articles

Delay epidemic models determined by latency, infection, and immunity duration

We propose new single and two-strain epidemic models represented by systems of delay differential equations and based on the number of newly exposed individuals. Transitions between exposed, infectious, recovered, and back to susceptible compartments are determined by the corresponding time delays. Existence and positiveness of solutions are proved. Reduction of delay differential equations to integral equations allows the analysis of stationary solutions and their stability. In the case of two strains, they compete with each other, and the strain with a larger individual basic reproduction number dominates the other one. However, if the basic reproduction number exceeds some critical values, stationary solution loses its stability resulting in periodic time oscillations. In this case, both strains are present and their dynamics is not completely determined by the basic reproduction numbers but also by other parameters. The results of the work are illustrated by comparison with data on seasonal influenza.

Analytical models of rotating magnetized gas filaments

ABSTRACT Several rotating molecular gas filaments with striking helical morphology have been observed in the Central Molecular Zone, including the Double Helix Nebula (DHN), the Galactic Centre Tornado (GCT), and the Pigtail Nebula (PTN). We present a model for isothermal, rotating, magnetized, pressure-truncated, non-self-gravitating filaments, as a model for these objects. We find a novel analytical stationary solution for these molecular tornadoes by assuming a propagating torsional Alfvén wave resulting in balance between toroidal magnetic stress and centrifugal force, as well as constant flux to mass ratio for the poloidal magnetic field. Our model allows for differential rotation and the density profile for the stationary solution goes asymptotically as r−1 at large radius for a special case where the flux to mass ratios for the poloidal and toroidal fields are constant inside the filament. We subject our stationary solutions to a virial analysis and show that they satisfy the virial equation for cylindrical symmetry $2\mathcal {K}+\mathcal {M}=0$. Finally, we estimate the ratio of gravitational potential energy per unit length, $\mathcal {W}$, to the bulk kinetic energy per unit length, $\mathcal {K}_0$, and calculate $|\mathcal {W}|/\mathcal {K}_0$ for the DHN, GCT, and PTN. Our estimates suggest that self-gravity is negligible in the DHN and the PTN and GCT are at most weakly self-gravitating. We also briefly address an application of our model to filamentary plasma lenses in the interstellar medium.

Propagation of dust-acoustic nonlinear waves in a homogeneous collisional dusty plasma

Nonlinear propagation of dust-acoustic waves DAWs in a weakly collisional dusty plasma comprising warm adiabatic fluid dust particles, isothermal electrons, and ions is investigated. We used the reductive perturbation theory to reduce the basic set of fluid equations to one evolution equation, namely damped Kadomtsev–Petviashivili (DKP). The analytical stationary solutions of the DKP equation are numerically analyzed, and the effect of various dusty plasma parameters on DAWs wave propagation is taken into account. We obtained, blast, anti-kink, periodic cnoidal and cnoidal waves. It is well known that explosive waves are a double edged sword. They can be seen, for example, in the atmosphere, or in engineering applications in metal coating.

Multispecies effects in the equilibrium and out-of-equilibrium thermostatistics of overdamped motion.

Progress has been recently made, both theoretical and experimental, regarding the thermostatistics of complex systems of interacting particles or agents (species) obeying a nonlinear Fokker-Planck dynamics. However, major advances along these lines have been restricted to systems consisting of only one type of species. The aim of the present contribution is to overcome that limitation, going beyond single-species scenarios. We investigate the dynamics of overdamped motion in interacting and confined many-body systems having two or more species that experience different intra- and interspecific forces in a regime where forces arising from standard thermal noise can be neglected. Even though these forces are neglected, the behavior of the system can be analyzed in terms of an appropriate thermostatistical formalism. By recourse to a mean-field treatment, we derive a set of coupled nonlinear Fokker-Planck equations governing the behavior of these systems. We obtain an H theorem for this Fokker-Planck dynamics and discuss in detail an example admitting an exact, analytical stationary solution.

Sparse Elitist Group Lasso Denoising in Frequency Domain for Bearing Fault Diagnosis

The fault-induced impulse responses of localized bearing fault are usually interfered by the background noise and other harmonic components. They are strongly coupled together and are hard to be separated. It is crucial to develop a fast and reliable method to extract the impulse-based feature for online bearing fault diagnosis in the industry application. In this article, we propose a new sparse elitist group lasso denoising (SEGLD) algorithm in frequency domain to detect the incipient impulse-based fault feature, which is free of utilizing the prior knowledge. We first reveal the sparse characteristics of the bearing fault signals in frequency domain. Then, a tailored denoising model is proposed. To obtain a satisfactory analytical stationary solution, the Douglas-Rachford splitting solver is employed for the denoising model. Moreover, we explore the relationship between the best regularization parameters, the periodic information and the normalization estimated noise of the rolling bearing fault signal. A rule of adaptively selecting the best regularization parameters is demonstrated. Finally, the robustness and effectiveness of the proposed SEGLD algorithm are profoundly verified by the numerical simulation and two evaluation experiments under the conditions of early fault stage and low speed scenario. Also, it is demonstrated that the proposed approach outperforms the state-of-the-art method for extracting the weak fault feature.

On the existence of stationary patches

In this paper, we show the existence of the first nontrivial family of analytic stationary patch solutions of the SQG and gSQG equations. This answers an open problem in F. de la Hoz et al. (2016) [13].

Congested shallow water model: roof modeling in free surface flow

We are interested in the modeling and the numerical approximation of flows in the presence of a roof, for example flows in sewers or under an ice floe. A shallow water model with a supplementary congestion constraint describing the roof is derived from the Navier-Stokes equations. The congestion constraint is a challenging problem for the numerical resolution of hyperbolic equations. To overcome this difficulty, we follow a pseudo-compressibility relaxation approach. Eventually, a numerical scheme based on a finite volume method is proposed. The well-balanced property and the dissipation of the mechanical energy, acting as a mathematical entropy, are ensured under a non-restrictive condition on the time step in spite of the large celerity of the potential waves in the congested areas. Simulations in one dimension for transcritical steady flow are carried out and numerical solutions are compared to several analytical (stationary and non-stationary) solutions for validation.

Lax Pair and new exact solutions of the nonlinear Dirac equation

We derive, using two different methods, exact analytic stationary solutions of the massive nonlinear Dirac equation (NLDE) in (1+1) dimensions as described by Thirring and Gross-Neveu models. In the first method, the equations are analyzed and some mathematical and physical properties of the solutions are inferred including continuity and wave equations of the current density. These properties are then used to derive the exact stationary solutions. The same solutions of Thirring model are rederived by finding a Lax pair representing the model and applying the Darboux transformation. The significance of finding the Lax pair lies in the fact that it opens the possibility to find an infinite chain of solutions by successive applications of the Darboux transformation. Within the range of physical interest, the derived solutions are localized. They show solitonic behavior of the spinor fields of a bounded fermions. Basic conserved quantities are calculated from the momentum energy tensor and some conclusions are drawn.

Nonlinear Dynamics of Laminar-Turbulent Transition in Generalized 3D Kolmogorov Problem for Incompressible Viscous Fluid at Symmetric Solution Subset

A three dimensional Kolmogorov problem with extended forcing term for Navier-Stokes equations is considered. The Galerkin-Fourier method is applied and the symmetry preserving subset of solutions is considered. The bifurcation patterns are revealed through the numerical analysis of eigenvalues of the linearized perturbed system from the analytical main stationary solution and through the analysis of phase space trajectories that the system generates. It was found that the initial stage of laminar-turbulent transition undergoes pitchfork bifurcation, through which the system can either go through the series of cycle cascades or through continuous tori bifurcations in accordance with the FShM scenario.

Recursively constructing analytic expressions for equilibrium distributions of stochastic biochemical reaction networks.

Noise is often indispensable to key cellular activities, such as gene expression, necessitating the use of stochastic models to capture its dynamics. The chemical master equation (CME) is a commonly used stochastic model of Kolmogorov forward equations that describe how the probability distribution of a chemically reacting system varies with time. Finding analytic solutions to the CME can have benefits, such as expediting simulations of multiscale biochemical reaction networks and aiding the design of distributional responses. However, analytic solutions are rarely known. A recent method of computing analytic stationary solutions relies on gluing simple state spaces together recursively at one or two states. We explore the capabilities of this method and introduce algorithms to derive analytic stationary solutions to the CME. We first formally characterize state spaces that can be constructed by performing single-state gluing of paths, cycles or both sequentially. We then study stochastic biochemical reaction networks that consist of reversible, elementary reactions with two-dimensional state spaces. We also discuss extending the method to infinite state spaces and designing the stationary behaviour of stochastic biochemical reaction networks. Finally, we illustrate the aforementioned ideas using examples that include two interconnected transcriptional components and biochemical reactions with two-dimensional state spaces.

Faraday wave patterns on a square cell network

We present the experimental observations of the Faraday instability when the vibrated liquid is contained in a network of small square cells for exciting frequencies in the range $$10\le F\le 24$$ Hz. A sweep of the parameter space has been performed to investigate the amplitudes and frequencies of the driving force for which different patterns form over the network. Regular patterns in the form of square lattices are observed for driving frequencies in the range $$10\le F<14$$ Hz, while ordered matrices of oscillons are formed for $$14<F\le 23$$ Hz. At $$F>23$$ Hz, disordered periodic patterns appear within individual cells for a small range of amplitudes. In this case, the wave field is dominated by oscillating blobs that interact on the capillary–gravity scale. A Pearson correlation analysis of the recorded videos shows that for all ordered patterns, the surface waves are periodic and correspond to Faraday waves of dominant frequency equal to half the excitation frequency (i.e., $$f=F/2$$ ). In contrast, the oscillons formed for $$14<F\le 23$$ Hz are at the first subharmonic ( $$f=F/2$$ ) and first harmonic ( $$f=F$$ ) response frequencies, with higher harmonics being negligible or absent as in most cases. The disordered wave fields forming at $$F>23$$ Hz are not subharmonic and correspond to periodic harmonic waves with $$f=nF/2$$ (for $$n=2,4,\ldots $$ ). We find that the experimentally determined minimum forcing necessary to destabilize the rest state and generate surface waves is consistent with a recent stability analysis of stationary solutions as derived from a new dispersion relation for time-periodic waves with nonzero forcing and dissipation.

Hermite–Gaussian stationary solutions in strongly nonlocal nonlinear optical media

Approximate analytical stationary solutions (SSs) of a cluster of Hermite–Gaussian (HG) shape is obtained in strongly nonlocal nonlinear media by the variational approach. The evolution of the HG SSs shows that when the order n⩽3, they propagate stably and form solitons; otherwise, when n⩾4, they always propagate unstably and evolve into self-trapped speckle-like beams. However, all these SSs maintain nearly invariant statistic beam-width during their propagation. Furthermore, when the input power deviates from the so-called critical power, the unstable HG beam will adjust its beam-width to form a new self-trapped beam, unlike the soliton which will turn to be a breather. But the average beam-widths are independent of the stability of the propagation of the HG SSs.

Electron acceleration with improved Stochastic Differential Equation method: Cutoff shape of electron distribution in test-particle limit

We develop a method of stochastic differential equation to simulate electron acceleration at astrophysical shocks. Our method is based on Itô's stochastic differential equations coupled with a particle splitting, employing a skew Brownian motion where an asymmetric shock crossing probability is considered. Using this code, we perform simulations of electron acceleration at stationary plane parallel shock with various parameter sets, and studied how the cutoff shape, which is characterized by cutoff shape parameter a, changes with the momentum dependence of the diffusion coefficient β. In the age-limited cases, we reproduce previous results of other authors, a≈2β. In the cooling-limited cases, the analytical expectation a≈β+1 is roughly reproduced although we recognize deviations to some extent. In the case of escape-limited acceleration, numerical result fits analytical stationary solution well, but deviates from the previous asymptotic analytical formula a≈β.

A Parabolic System Model for the Formation of Porous Silicon: Existence, Uniqueness, and Stability

In this paper, a nonlinear weakly coupled parabolic system is constructed to model porous silicon formation. The model considers physical and chemical factors such as charge carriers transport, ions transport, electrochemical reactions, and material balance. In order to prove the existence of a unique solution for the proposed parabolic model with zero Neumann boundary, we use a monotone method based on the upper and lower solutions. The asymptotic stability of the steady-state solution is also demonstrated, and the behavior of this model is shown for specific cases where the analytical stationary solution is known. Finally, a discussion of the results is carried out, and we conclude that the proposed model reproduces important qualitative features which are observed experimentally.

On Stationary Solutions of the Reduced Gardner–Ostrovsky Equation

The detailed analysis of stationary solutions of the reduced Gardner–Ostrovsky (GO) equation is presented. The GO equation is the popular model for the description of large-amplitude internal oceanic waves affected by Earth’s rotation. Its reduced version in which the small-scale dispersion is neglected is used when very long internal waves are considered. The equation is also applicable to other types of nonlinear waves in various media (plasma, optical media, relaxing media, etc.) when the large-scale dispersion plays a dominant role in comparison with the small-scale dispersion. Balancing the nonlinear effect such dispersion gives rise to existence of stationary waves, both periodic and non-periodic. It is shown that only smooth periodic waves make physical sense. Systematic analysis of stationary solutions to the GO equation and their categorisation is presented.

Domain of oscillatory growth in directional solidification of dilute binary alloys

The oscillatory growth of a dilute binary alloy has recently been described by a nonlinear oscillator equation that applies to small temperature gradients and large growth velocities in the setup of directional solidification. Based on a one-dimensional stability analysis of stationary solutions of this equation, we explore in the present paper the complete region where the solidification front propagates in an oscillatory way. The boundary of this region is calculated exactly, and the nature of the oscillations is evaluated numerically in several segments of the region.

Overdamped motion of interacting particles in general confining potentials: time-dependent and stationary-state analyses

By comparing numerical and analytical results, it is shown that a system of interacting particles under overdamped motion is very well described by a nonlinear Fokker-Planck equation, which can be associated with nonextensive statistical mechanics. The particle-particle interactions considered are repulsive, motivated by three different physical situations: (i) modified Bessel function, commonly used in vortex-vortex interactions, relevant for the flux-front penetration in disordered type-II superconductors; (ii) Yukawa-like forces, useful for charged particles in plasma, or colloidal suspensions; (iii) derived from a Gaussian potential, common in complex fluids, like polymer chains dispersed in a solvent. Moreover, the system is subjected to a general confining potential, φ(x) = (α|x| z )/z (α > 0, z > 1), so that a stationary state is reached after a sufficiently long time. Recent numerical and analytical investigations, considering interactions of type (i) and a harmonic confining potential (z = 2), have shown strong evidence that a q-Gaussian distribution, P(x,t), with q = 0, describes appropriately the particle positions during their time evolution, as well as in their stationary state. Herein we reinforce further the connection with nonextensive statistical mechanics, by presenting numerical evidence showing that: (a) in the case z = 2, different particle-particle interactions only modify the diffusion parameter D of the nonlinear Fokker-Planck equation; (b) for z ≠ 2, all cases investigated fit well the analytical stationary solution P st(x), given in terms of a q-exponential (with the same index q = 0) of the general external potential φ(x). In this later case, we propose an approximate time-dependent P(x,t) (not known analytically for z ≠ 2), which is in very good agreement with the simulations for a large range of times, including the approach to the stationary state. The present work suggests that a wide variety of physical phenomena, characterized by repulsive interacting particles under overdamped motion, present a universal behavior, in the sense that all of them are associated with the same entropic form and nonlinear Fokker-Planck equation.

Pulse vs. optimal stationary fishing: The Northern Stock of Hake

Pulse fishing may be a global optimal strategy in multicohort fisheries. In this article we compare pulse fishing solutions obtained by using global numerical methods with the analytical stationary optimal solution. This allows us to quantify the potential benefits associated with the use of periodic fishing in the Northern Stock of Hake. Results show first that management plans based exclusively on traditional reference targets such as Fmsy may drive economic results for fisheries far away from the optimal; second, that global optimal solutions would imply the cyclical closure of fisheries for some periods; and third, that second best stationary policies with stable employment only reduce the optimal present value of discounted profit by 2%.

New exact solutions of magnetic ellipsoidal figures of equilibrium

Abstract In this Letter we report brand new analytic stationary solutions of constant density stars with magnetic field and self-gravity. These solutions include prolate configurations even for purely poloidal magnetic fields as well as oblate configurations. These new analytic solutions are expressed in very simple forms and can be considered as generalized configurations of uniformly rotating constant density spheroids, i.e. Maclaurin spheroids, and of constant density ellipsoids with constant vorticity, i.e. Dedekind ellipsoids. As the axisymmetric Maclaurin spheroids and the triaxial Dedekind ellipsoids have been widely used for the estimation of the effect of rotations and/or internal motions on the self-gravitating bodies, our new analytic solutions may be used widely to estimate the effect of the magnetic fields semi-quantitatively in various contexts hereafter.

Quantitative comparison of the phase-transformation kinetics in a sharp-interface and a phase-field model

A sharp-interface model, which provides an accurate description of diffusional phase-transformation kinetics [C. Bos, J. Sietsma, Scripta Mater. 57 (2007) 1085–1088], is used to evaluate the transformation kinetics as calculated by a diffuse-interface multi-phase-field model, in particular the effect of the interface thickness will be investigated. This is done for the isothermal austenite-to-ferrite transformation in a C–Mn steel, using the simple geometry of a single ferrite grain growing in the bulk of the austenite matrix. For the sake of computation time, this analysis is limited to one- and two-dimensional space, although it can be extended to three-dimensional space straightforwardly. Small deviations of the numerical phase-field profile from the analytical stationary solution, used to match the phase-field equation to the equation for the sharp-interface kinetics, results in a lower effective interface mobility compared to the input value. This deviation, more pronounced in the early stage of transformation, can be minimised by reducing the grid size, while keeping the number of interface grid points constant. During stationary growth, a good agreement was found between phase-field and sharp-interface solution, although a small deviation of about 7% still remains if six grid points are chosen for the interface width. This deviation could not be reduced further by a better grid resolution, but can be overcome by a higher number of grid points within the diffuse interface.