Time-dependent Evolution Equations Research Articles

Evolution of fermion resonance in thick brane

Abstract In this work, we investigate the numerical evolution of massive Kaluza–Klein (KK) modes of a Dirac field on a thick brane. We deduce the Dirac equation in five-dimensional spacetime, and obtain the time-dependent evolution equation and Schrödinger-like equation of the extra-dimensional component. We use the Dirac KK resonances as the initial data and study the corresponding dynamics. By monitoring the decay law of the left- and right-chiral KK resonances, we compute the corresponding lifetimes and find that there could exist long-lived KK modes on the brane. Especially, for the lightest KK resonance with a large coupling parameter and a large three momentum, it will have an extremely long lifetime.

Analysis of dynamic disturbance and multistage shear creep damage evolution law of the weak intercalated layers in slope under the influence of coupled damage effect

Based on the damage characteristics of multistage shear creep in weak intercalated layers (carbonaceous mud shale) of slopes under the influence of dynamic disturbance, the effective bearing area method was used. A new coupled damage equation (dynamic disturbance damage, shear creep damage, and initial damage) was established through further derivation, and its applicability was demonstrated. The calculation method for the relevant coupled damage degrees was also provided. Furthermore, by targeting the three coupled damage factors and extending the Kachanov damage law, a time-dependent damage evolution equation for weak intercalated layers under the influence of the three coupled damage effects was established. The influence of different dynamic disturbance intensities on the evolution of multistage shear creep damage in weak intercalated layers of slopes under the influence of coupled damage effects was analysed. The results show that the damage to the rock mass caused by dynamic disturbance mainly occurs in the low-frequency stage (40–80 Hz). The instantaneous damage caused by dynamic disturbance to the shear plane of weak intercalated layers is not only affected by the intensity of the dynamic disturbance but also limited by the magnitude of the shear creep load. The influence of the dynamic disturbance intensity on the entire process of multistage shear creep damage of weak intercalated layers was analysed. With increasing of dynamic disturbance intensity, the cumulative coupled damage at the end of shear creep at all levels gradually exhibits linear evolution. The time-dependent coupling damage evolution process of weak intercalated layers was quantitatively characterized.

Periodic solutions to a class of quasi-variational evolution equations

We study a time-periodicity problem for a class of abstract nonlinear evolution equations associated with subdifferential operators depending on both time and the unknown. Assuming time-periodicity for the subdifferential operator, we prove the existence of a periodic solution using the abstract theory of time-dependent subdifferential evolution equations and its generalization. We provide application examples for quasilinear parabolic variational inequalities with time-periodic constraints of non-local time-delay or memory type depending on the unknown.

An Averaging Principle for the Time-Dependent Abstract Stochastic Evolution Equations with Infinite Delay and Wiener Process

In this paper, averaging principle for the time-dependent stochastic evolution equations (TDSEEs) with infinite delay is investigated. In proper non-Lipschitz conditions, we prove that the mild solution of the averaged stochastic evolution equations (ASEEs), governed by time-independent family of linear operators, converges to that of the original one in $$L^{2}$$ sense. At last, an application example is presented to prove the obtained theory.

Quantum Zeno effect generalized

The quantum Zeno effect, in its original form, uses frequent projective measurements to freeze the evolution of a quantum system that is initially governed by a fixed Hamiltonian. We generalize this effect simultaneously in three directions by allowing open system dynamics, time-dependent evolution equations, and general quantum operations in place of projective measurements. More precisely, we study Markovian master equations with bounded generators whose time dependence is Lipschitz continuous. Under a spectral gap condition on the quantum operation, we show how frequent measurements again freeze the evolution outside an invariant subspace. Inside this space, the evolution is described by a modified master equation.

Almost periodic solutions of evolution equations

We state existence theorems for almost periodic solutions of evolution problems, namely, quasi-autonomous problems and more generally, time dependent evolution equations. We apply these theorems firstly, to a boundary value quasilinear hyperbolic equation of first order, and secondly, to a boundary value quasi-parabolic equation.

The fundamental properties of quasi-semigroups

ABSTRACTIn this paper we focus on the abstract Cauchy problems of the time-dependent evolution equation ẋ = A(t)x(t). If the operator A(t) is time-independent, we can use the C0-semigroups theory to solve the abstract Cauchy problem. In this case A is a infinitesimal generator of the C0-semigroups. However, if A(t) is time-dependent, we can not apply directly the C0-semigroups theory to solve the problem. In this situation we can use the quasi semigroups theory as development of the two parameters semigroups. This semigroups is induced by bounded evolution operators U(t, s) that satisfy some assumptions. In this paper we determine the fundamental properties of the quasi semigroups included its generator related to the time-dependent evolution equation.

The implicit midpoint rule for nonexpansive mappings

Abstract The implicit midpoint rule (IMR) for nonexpansive mappings is established. The IMR generates a sequence by an implicit algorithm. Weak convergence of this algorithm is proved in a Hilbert space. Applications to the periodic solution of a nonlinear time-dependent evolution equation and to a Fredholm integral equation are included. MSC:47J25, 47N20, 34G20, 65J15.

On time-dependent stochastic evolution equations driven by fractional Brownian motion in a Hilbert space with finite delay

In this paper, we show the existence and uniqueness of the mild solution for a class of time-dependent stochastic evolution equations with finite delay driven by a standard cylindrical Wiener process and an independent cylindrical fractional Brownian motion with Hurst parameter H ∈ (1 / 2,1). An example is provided to illustrate the theory. Copyright © 2013 John Wiley & Sons, Ltd.

Temporal behavior of low-amplitude dark spatial solitons in two-photon centrosymmetric photorefractive media

We investigate the temporal behavior of low-amplitude dark spatial solitons in biased centrosymmetric crystals with two-photon photorefractive effects. The expressions of time-dependent space-charge field and dynamical evolution equation are derived. The temporal behaviors of the intensity profiles and the intensity full width at half maximum (FWHM) of dark solitons are investigated by numerical method. The results indicate that a broad soliton is generated at the beginning, and as time evolves, the intensity width of solitons decrease monotonously to a minimum value toward steady state. Within the same propagation time, the higher the ratio of soliton peak intensity to the dark irradiation intensity is, the shorter the FWHM of dark solitons is.

Existence, Uniqueness and Stability of Mild Solutions for Time-Dependent Stochastic Evolution Equations with Poisson Jumps and Infinite Delay

In this paper, we study a class of time-dependent stochastic evolution equations with Poisson jumps and infinite delay. We establish the existence, uniqueness and stability of mild solutions for these equations under non-Lipschitz condition with Lipschitz condition being considered as a special case. An application to the stochastic nonlinear wave equation, with Poisson jumps and infinite delay, is given to illustrate the obtained theory.

Critical conditions of control light for a silicon-based photonic crystal bistable switching

In this paper, we investigate the critical conditions of control light for a silicon-based photonic crystal bistable switching. By establishing a time-dependent evolution equation, the critical pump power and pump time of the control light are derived, respectively. It is found that with the increase of the frequency detuning of the incident light, the critical power of the control light will rise, while the corresponding critical time will be shortened. It is also revealed that under the same conditions, the critical total power of the multiple-beam control light is less than the one of the single-beam control light. The theoretical predictions show perfect agreement with the simulation results.

A maximum dissipation thermodynamic multi-scale model for the dynamic response of the arterial elastin–water system

The mechanical response of the elastin–water system in an artery wall is viscoelastic. Elastin in vivo must operate in the rubbery region, i.e., above the glass transition, that depends on moisture content. A dynamic multi-scale time-dependent evolution equation is presented for the mechanical response of the elastin–water system that captures the effect of moisture content on the glass transition of the elastin. To define non-equilibrium evolution processes, the construction requires only a hyperelastic strain energy density function describing the long-term behavior and the thermodynamic relaxation modulus that describes the relaxation speed of non-equilibrium processes. The thermodynamic relaxation modulus also relates spatial scales, the molecular scale including moisture bonding to the bulk material scale. The model reproduces published experimental data on the elastin glass transition behavior with respect to load frequency and to ambient relative humidity but is not merely empirical in the sense of being a fit to such data because it predicts dynamic responses such as non-physiological creep and physiological rate-dependent stress–stretch relations. The new viscoelastic model predicts the influence of moisture content and the glass transition of the elastin on the time-dependent response of the circumferential stretch and the change in radius of a hydrated arterial cylindrical elastin lamella under cyclic radial pressure loads in the hemodynamic range. Such an elastin cylinder approximates the behavior of the elastin substructure in an elastic artery wall.

A Thermo-Elasto-Viscoplastic Model for Soft Sedimentary Rock

ABSTRACT In this paper, a new thermo-elasto-viscoplastic model is proposed, which can characterize thermodynamic behaviours of soft sedimentary rocks. Firstly, as in the Cam-clay model, plastic volumetric strain which consists of two parts, stress-induced part and thermodynamic part, is used as hardening parameter. Both parts of the plastic volumetric strain can be derived from an extended e-ln p relation in which the thermodynamic part is deduced based on a concept of ‘equivalent stress’. Secondly, regarding soft sedimentary rocks as a heavily overconsolidated soil in the same way as the model proposed by Zhang et al. (2005), an extended subloading yield surface (Hashiguchi and Ueno, 1977; Hashiguchi, 1980; Hashiguchi and Chen, 1998) and an extended void ratio difference are proposed based on the concept of the equivalent stress. Furthermore, a time-dependent evolution equation for the extended void ratio difference is formularized, which considers both the influences of temperature and stresses. Finally, it is proved that the proposed model satisfies thermodynamic theorems in the framework of non-equilibrium thermodynamics.

A mathematical approach to HIV infection dynamics

In order to obtain a comprehensive form of mathematical models describing nonlinear phenomena such as HIV infection process and AIDS disease progression, it is efficient to introduce a general class of time-dependent evolution equations in such a way that the associated nonlinear operator is decomposed into the sum of a differential operator and a perturbation which is nonlinear in general and also satisfies no global continuity condition. An attempt is then made to combine the implicit approach (usually adapted for convective diffusion operators) and explicit approach (more suited to treat continuous-type operators representing various physiological interactions), resulting in a semi-implicit product formula. Decomposing the operators in this way and considering their individual properties, it is seen that approximation–solvability of the original model is verified under suitable conditions. Once appropriate terms are formulated to describe treatment by antiretroviral therapy, the time-dependence of the reaction terms appears, and such product formula is useful for generating approximate numerical solutions to the governing equations. With this knowledge, a continuous model for HIV disease progression is formulated and physiological interpretations are provided. The abstract theory is then applied to show existence of unique solutions to the continuous model describing the behavior of the HIV virus in the human body and its reaction to treatment by antiretroviral therapy. The product formula suggests appropriate discrete models describing the dynamics of host pathogen interactions with HIV1 and is applied to perform numerical simulations based on the model of the HIV infection process and disease progression. Finally, the results of our numerical simulations are visualized and it is observed that our results agree with medical and physiological aspects.

The Surface Density Distribution in the Solar Nebula

The commonly used minimum mass power law representation of the pre-solar nebula is reanalyzed using a new cumulative-mass-model. This model predicts a smoother surface density approximation compared with methods based on direct computation of surface density. The density is quantified using two independent analytical formulations. First, a best-fit transcendental function is applied directly to the basic planetary data. Next a solution to the time-dependent disk evolution equation is parametrically adapted to the solar nebula data. The latter model is shown to be a good approximation to the finite-size early Solar Nebula, and by extension to other extra solar protoplanetary disks.

Exact solutions of space–time dependent non‐linear Schrödinger equations

AbstractUsing a general symmetry approach we establish transformations between different non‐linear space–time dependent evolution equations of Schrödinger type and their respective solutions. As a special case we study the transformation of the standard non‐linear Schrödinger equation (NLS)‐equation to a NLS‐equation with a dispersion coefficient which decreases exponentially with increasing distance along the fiber. By this transformation we construct from well known solutions of the standard NLS‐equation some new exact solutions of the NLS‐equation with dispersion. Copyright 2004 John Wiley & Sons, Ltd.

Dynamic interaction between suspended particles and defects in a nematic liquid crystal.

Insertion of spherical particles into a uniform nematic liquid crystal gives rise to the formation of topological defects. In the present work, we investigate how a spherical particle accompanied by its topological defects interacts with neighboring disclination lines. We perform two- and three-dimensional dynamic simulations to analyze the effect of a particle on the annihilation process of two disclination lines. The dynamics of the liquid crystal is described by a time-dependent evolution equation on the symmetric traceless order parameter that includes some of the salient features of liquid crystalline materials: excluded volume effects, or equivalently, short-range order elasticity and long-range order elasticity. At the surface of the particle, the liquid crystal is assumed to exhibit strong homeotropic anchoring. The particle is located between two disclination lines of topological charges +1/2 and -1/2. Two-dimensional simulations indicate that the topological defects bound to the particle mediate an interaction between the two disclination lines which increases the attraction between them. This result is confirmed by three-dimensional simulations that provide a complete description of the director field and of the order parameter around the particle. These simulations indicate that a spherical particle between two disclination lines can be surrounded by a Saturn ring, and suggest that the dynamic behavior of disclination lines could be used to report the structure of a defect around the particle.

Time-Dependent Recursion Operators and Symmetries

The recursion operators and symmetries of nonautonomous, (1+1) dimensional integrable evolution equations are considered. It has been previously observed that the symmetries of the integrable evolution equations obtained through their recursion operators do not satisfy the symmetry equations. There have been several attempts to resolve this problem. It is shown that in the case of time-dependent evolution equations or time-dependent recursion operators associativity is lost. Due to this fact such recursion operators need modification. A general formula is given for the missing term of the recursion operators. Apart from the recursion operators a method is introduced to calculate the correct symmetries. For illustrations several examples of scalar and coupled system of equations are considered.

Spatiotemporal model of the LIGO interferometer

We develop a detailed spatiotemporal model in the adiabatic approximation of the optical response of the Laser Interferometer Gravitational-Wave Observatory (LIGO) optical system to environmental perturbations. We begin by deriving a first-order linear time-dependent evolution equation that models the electromagnetic field as a function of position within a Fabry–Perot interferometer. This model allows both the length of the resonator and the misalignment angles of the end mirrors to vary in time and describes both resonant and nonresonant phenomena. After defining a biorthogonality relation that must be satisfied by general unperturbed spatial eigenfunctions of the Fabry–Perot interferometer, we expand the intracavity field as a linear combination of these functions and convert the spatiotemporal evolution equation into a linear system of coupled time-dependent iteration and/or differential equations. We then calculate the adiabatic connection equations that link the two LIGO Fabry–Perot interferometers through the power recycling cavity, which comprises two mirrors (the power recycling input mirror and the beam splitter) that have the same mechanical degrees of freedom as those in the Fabry–Perot arm cavities. We develop a detailed instance of this model for the evolution of the intracavity field within a resonator with sufficiently small misalignment angles that a Hermite–Gauss basis set can be used. We develop a detailed general approach to signal demodulation for simulation of servo-control systems and describe its implementation in the Hermite–Gauss approximation for Cartesian split-plane detectors. Finally, we demonstrate the use of this small-angle model to simulate the effect of angular misalignment on the longitudinal response function.