Exact Time-dependent Solutions Research Articles

New Lie Symmetries and Exact Solutions of a Mathematical Model Describing Solute Transport in Poroelastic Materials

A one-dimensional model for fluid and solute transport in poroelastic materials (PEMs) is studied. Although the model was recently derived and some exact solutions, in particular steady-state solutions and their applications, were studied, special cases occurring when some parameters vanish were not analysed earlier. Since the governing equations are nonintegrable in nonstationary cases, the Lie symmetry method and modern tools for solving ODE systems are applied in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with new Lie symmetries are identified. Some of them have a highly nontrivial structure that cannot be predicted from a physical point of view or using Lie symmetries of other real-world models. Applying the symmetries obtained, multiparameter families of exact solutions are constructed, including those in terms of elementary and special functions (hypergeometric, Whittaker, Bessel and modified Bessel functions). A possible application of the solutions obtained is demonstrated, and it is shown that some exact solutions can describe (at least qualitatively) the solute transport in PEM. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems.

Generation and Life Cycle of Solar Spicules

The physical mechanism for the creation of solar spicules is proposed with three stages of their life cycle. It is assumed that at stage I the density hump is formed locally in the x-y plane in the lower chromosphere in the presence of temperature gradients of electrons and ions along the z-axis (the vertical direction). In this region, the density structure of quasi-neutral (n i ≃ n e = n) plasma after taking birth is accelerated in the vertical direction owing to the thermal force F th ∝ ∇n(x, y, t) × (∇T e + ∇T i ). The exact time-dependent analytical solution of two-fluid plasma equations is presented assuming that density is maximum at the center of the density structure and decays away from it gradually. The 2D density structure is created as a step function H(t) in time at the bottom of the chromosphere, and consequently, the vertical plasma velocity turns out to be the ramp function of time R(t) = tH(t), whereas the source term S(x, y, t) for the density follows the delta function δ(t) form. The upward acceleration a=a(x,y)zˆ produced in this density structure is greater than the downward constant solar acceleration − g ⊙ in the chromosphere. In the transition region, the temperature gradients are steeper; therefore, the upward acceleration increases in magnitude g ⊙ ≪ a and the density hump spends less time there. This is stage II of its life cycle. In stage III, the density structure enters into the corona, where the gradients of temperatures vanish and the structure decelerates to zero velocity under the action of the solar gravitational force.

Microscopic state of BHs and an exact one body method for binary dynamics in general relativity

In gravitational collapses, the horizon and singularity’s realisation in the finite future of the proper time used co-moving observer happens in the future of infinitely far away future of the normal time used outside probe. To the latter the horizon and singularity defined in the singularity theorem are physical realities only in the sense of uncertainty principle and ensemble interpretation. We provide two exact time dependent solution families to the Einstein equation and show that they form a pair of complementary description for the microscopic state of black holes by showing that the Bekenstein–Hawking entropy formula follows properly from their canonical wave function’s degeneracy. We also develop an eXact One Body method for general relativity two-body dynamics whose conservative part calls no post newtonian approximation as input and applies to the full three stages of black hole binary merger events. By this method, we analytically calculate the gravitational wave forms following from such merger processes. In the case black holes carry exact and apriori horizon and singularity our wave forms agree with those following from conventional effective one body method but exhibit more consistent late time behaviour. In the case black holes carry only asymptotic horizon and extended inner structure thus experiencing banana shape deformation as the merger occurs, our wave forms exhibit all features especially the late time quasi-normal mode type oscillation seen in real observations.

Solving the time-dependent protein distributions for autoregulated bursty gene expression using spectral decomposition.

In this study, we obtain an exact time-dependent solution of the chemical master equation (CME) of an extension of the two-state telegraph model describing bursty or non-bursty protein expression in the presence of positive or negative autoregulation. Using the method of spectral decomposition, we show that the eigenfunctions of the generating function solution of the CME are Heun functions, while the eigenvalues can be determined by solving a continued fraction equation. Our solution generalizes and corrects a previous time-dependent solution for the CME of a gene circuit describing non-bursty protein expression in the presence of negative autoregulation [Ramos et al., Phys. Rev. E 83, 062902 (2011)]. In particular, we clarify that the eigenvalues are generally not real as previously claimed. We also investigate the relationship between different types of dynamic behavior and the type of feedback, the protein burst size, and the gene switching rate.

A mathematical model for two solutes transport in a poroelastic material and its applications

Using well-known mathematical foundations of the elasticity theory, a mathematical model for two solutes transport in a poroelastic material (soft tissue is a typical example) is suggested. It is assumed that molecules of essentially different sizes dissolved in fluid and are transported through pores of different sizes. The stress tensor, the main force leading to the material deformation, is taken not only in the standard linear form but also with an additional nonlinear part. The model is constructed in 1D space and consists of five nonlinear equations. It is shown that the governing equations are integrable in stationary case, therefore all steady-state solutions are constructed. The obtained solutions are used in an example for healthy and tumour tissue, in particular, tissue displacements are calculated and compared for parameters taken from experimental data in cases of the linear and nonlinear stress tensors. Since the governing equations are non-integrable in non-stationary case, the Lie symmetry analysis is used in order to construct time-dependent exact solutions. Depending on parameters arising in the governing equations, several special cases with non-trivial Lie symmetries are identified. As a result, multiparameter families of exact solutions are constructed including those in terms of special functions(hypergeometric and Bessel functions). A possible application of the solutions obtained is demonstrated.

On the role of numerical diffusivity in MHD simulations of global accretion disc dynamos

Observations, mainly of outbursts in dwarf novae, imply that the anomalous viscosity in highly ionized accretion discs is magnetic in origin and requires that the plasma ${\beta \sim 1}$ . Until now, most simulations of the magnetic dynamo in accretion discs have used a local approximation (known as the shearing box). While these simulations demonstrate the possibility of a self-sustaining dynamo, the magnetic activity generated in these models saturates at $\beta \gg 1$ . This long-standing discrepancy has previously been attributed to the local approximation itself. There have been recent attempts at simulating magnetic activity in global accretion discs with parameters relevant to the dwarf novae. These too find values of $\beta \gg 1$ . We speculate that the tension between these models and the observations may be caused by numerical magnetic diffusivity. As a pedagogical example, we present exact time-dependent solutions for the evolution of weak magnetic fields in an incompressible fluid subject to linear shear and magnetic diffusivity. We find that the maximum factor by which the initial magnetic energy can be increased depends on the magnetic Reynolds number as ${\mathcal {R}}_{m}^{2/3}$ . We estimate that current global numerical simulations of dwarf nova discs have numerical magnetic Reynolds numbers around six orders of magnitude less than the physical value found in dwarf nova discs of ${\mathcal {R}}_{m} \sim 10^{10}$ . We suggest that, given the current limitations on computing power, expecting to be able to compute realistic dynamo action in observable accretion discs using numerical MHD is, for the time being, a step too far.

A family of nonlinear diffusion equations related to the [formula omitted]-error function

We investigate a bi-parametric family of nonlinear diffusion equations that includes, as special members, the power-law nonlinear diffusion equation, and an evolution equation akin to Ubriaco’s nonlinear diffusion equation. It is well-known that the power-law nonlinear diffusion equation is closely related to the Sq-thermostatistics. We show that the Ubriaco-like member of the above mentioned bi-parametric family is also related to the Sq-theory, because it admits exact time-dependent solutions that exhibit the q-error-function shape. These solutions constitute a generalization, within the Sq-thermostatistical framework, of a celebrated class of solutions of the linear diffusion equation, that have the standard error-function form. Besides exploring the main aspects of a family of non-standard diffusion processes, our present developments constitute a physical-mathematical application of the q-error-function. This function, in spite of its potential relevance for q-statistics and its applications, has not yet attracted the attention it deserves from the Sq-thermostatistics research community.

Solitonic self-sustained charge and energy transport on the superconducting cylinder

We present an exact time-dependent solution for a charged scalar field on a two-dimensional cylinder, that can be interpreted as representing a long-standing excitation on a $S$-wave superconducting state, which propagates along a nanotube constructed out of twisted bilayer graphene. The solution has a topological charge characterized by an integer number, which counts the winding of the Higgs phase winds around the cylinder. The resulting electric current generates its own electromagnetic field in a self-consistent way, without the need of any external fields to keep it alive.

Exact time-dependent solution of the Schrödinger equation, its generalization to the phase space and relation to the Gibbs distribution

Using the simplest but fundamental example, the problem of the infinite potential well, this paper makes an ideological attempt (supported by rigorous mathematical proofs) to approach the issue of ≪understanding≫ the mechanism of quantum mechanics processes, despite the well-known examples of the EPR paradox type. The new exact solution of the Schrödinger equation is analyzed from the perspective of quantum mechanics in the phase space. It is the phase space, which has been extensively used recently in quantum computing, quantum informatics and communications, that is the bridge towards classical physics, where understanding of physical reality is still possible. In this paper, an interpretation of time-dependent processes of energy redistribution in a quantum system, probability waves, the temperature and entropy of a quantum system, and the transition to a time-independent ≪frozen state≫ is obtained, which is understandable from the point of view of classical physics. The material of the paper clearly illustrates the solution of the problem from the standpoint of continuum mechanics, statistical physics and, of course, quantum mechanics in the phase space.

Alternative for black hole paradoxes

In this paper, we investigate the exact time-dependent black hole solution on a warped five-dimensional Randall–Sundrum space–time in conformal dilaton gravity. The zeroes of the model are described by a meromorphic quintic polynomial, which has no essential singularities. The quintic equation can be reduced to the Brioschi form by means of the Weierstrass elliptic curve over [Formula: see text]. The model fits the antipodal boundary condition, i.e. antipodal points in the projected space are identified using the embedding of a Klein surface in [Formula: see text], using the [Formula: see text] symmetry on the two sides of the brane. If one writes [Formula: see text], [Formula: see text], with [Formula: see text] the normal to the brane and [Formula: see text] the dilaton field, then [Formula: see text] is conformally flat. It is the contribution from the bulk which determines the real pole on the effective four-dimensional space–time. There is no objection applying ’t Hooft’s back reaction method in constructing the unitary S-matrix for the Hawking radiation. Again, there is no “inside” of the black hole. The zeroes can also be analyzed by the icosahedron equation and by the Hopf-fibration of the Klein surface.

Nonlinear diffusion of fermions and bosons

A nonlinear diffusion equation is proposed to account for thermalization in fermionic and bosonic systems through analytical solutions. For constant transport coefficients, exact time-dependent solutions are derived through nonlinear transformations, and the corresponding local equilibration times are deduced. Fermi-Dirac and Bose-Einstein distributions emerge as stationary solutions of the nonlinear equation. As examples, local thermalization of quarks and gluons in relativistic heavy-ion collisions, and of ultracold atoms including time-dependent Bose-Einstein condensate formation are discussed.

Exact time-dependent dynamics of discrete binary choice models

We provide a generic method to find full dynamical solutions to binary decision models with interactions. In these models, agents follow a stochastic evolution where they must choose between two possible choices by taking into account the choices of their peers. We illustrate our method by solving Kirman and Föllmer’s ant recruitment model for any number N of discrete agents and for any choice of parameters, recovering past results found in the limit N → ∞. We then solve extensions of the ant recruitment model for increasing asymmetry between the two choices. Finally, we provide an analytical time-dependent solution to the standard voter model and a semi-analytical solution to the vacillating voter model. Our results show that exact analytical time-dependent solutions can be achieved for discrete choice models without invoking that the number of agents N are continuous or that both choices are symmetric, and additionally show how to practically use the analytics for fast evaluation of the resulting probability distributions.

Exact model for evaporating primordial black holes in a cosmological spacetime

Primordial black holes (PBHs) in the mass range ${10}^{17}--{10}^{23}\text{ }\text{ }\mathrm{gm}$ are considered as possible dark matter candidates as they are not subject to big-bang nucleosynthesis constraints and behave like cold dark matter. If PBHs are indeed dark matter, they cannot be treated as isolated objects in asymptotic flat space-time. Furthermore, when compared to stellar-mass black holes, the rate at which the Hawking particles radiate out from PBHs is significantly faster. In this work, we obtain an exact time-dependent solution that models evaporating black holes in the cosmological background. As a result, the solution considers all three aspects of PBHs---mass-loss due to Hawking radiation, black hole surrounded by mass distribution, and cosmological background. Furthermore, our model predicts that the decay of PBHs occurs faster for larger masses; however, the decay rate reduces for lower mass. Finally, we discuss the implications of theoretical constraints on PBHs as dark matter.

Exact time-dependent analytical solutions for entropy production rate in a system operating in a heat bath in which temperature varies linearly in space.

The nonequilibrium thermodynamics feature of a Brownian motor is investigated by obtaining exact time-dependent solutions. This in turn enables us to investigate not only the long time property (steady state) but also the short time the behavior of the system. The general expressions for the free energy, entropy production e[over ̇]_{p}(t) as well as entropy extraction h[over ̇]_{d}(t) rates are derived for a system that is genuinely driven out of equilibrium by time-independent force as well as by spatially varying thermal background. We show that for a system that operates between hot and cold reservoirs, most of the thermodynamics quantities approach a nonequilibrium steady state in the long time limit. The change in free energy becomes minimal at a steady state. However, for a system that operates in a heat bath where its temperature varies linearly in space, the entropy production and extraction rates approach a nonequilibrium steady state while the change in free energy varies linearly in space. This reveals that unlike systems at equilibrium, when systems are driven out of equilibrium, their free energy may not be minimized. The thermodynamic properties of a system that operates between the hot and cold baths are further compared and contrasted with a system that operates in a heat bath where its temperature varies linearly in space along with the reaction coordinate. We show that the entropy, entropy production, and extraction rates are considerably larger for the linearly varying temperature case than a system that operates between the hot and cold baths revealing such systems are inherently irreversible. For both cases, in the presence of load or when a distinct temperature difference is retained, the entropy S(t) monotonously increases with time and saturates to a constant value as t further steps up. The entropy production rate e[over ̇]_{p} decreases in time and at steady state, e[over ̇]_{p}=h[over ̇]_{d}>0, which agrees with the results shown in M. Asfaw's [Phys. Rev. E 89, 012143 (2014)1539-375510.1103/PhysRevE.89.012143; Phys. Rev. E 92, 032126 (2015)10.1103/PhysRevE.92.032126]. Moreover, the velocity, as well as the efficiency of the system that operates between the hot and cold baths, are also collated and contrasted with a system that operates in a heat bath where its temperature varies linearly in space along with the reaction coordinate. A system that operates between the hot and cold baths has significantly lower velocity but a higher efficiency in comparison with a linearly varying temperature case.

Exact time-dependent solutions of a Fisher–KPP-like equation obtained with nonclassical symmetry analysis

We consider a family of exact solutions to a nonlinear reaction–diffusion model, constructed using nonclassical symmetry analysis. In a particular limit, the mathematical model approaches the well-known Fisher–KPP model, which means that it is related to various applications including cancer progression, wound healing and ecological invasion. The exact solution is mathematically interesting since exact solutions of the Fisher–KPP model are rare, and often restricted to long-time travelling wave solutions for special values of the travelling wave speed.

Integrable nonlinear reaction-diffusion population models for fisheries

Sustainable natural resource management strategies are required to maintain a fishery. One common device is to designate a No-Take Area (NTA) or a less restrictive Marine Protected Area (MPA) within a fishery. In this work, exact time-dependent solutions to 2-D nonlinear reaction-diffusion equations of Fisher type are constructed to model the fish population density of a single species within an NTA. The spillover rate from an NTA, important for harvesters, has previously been calculated without allowing for spatial variation within the NTA. Here, that spatial dependence is incorporated in the calculation. Spatially-dependent and density-dependent diffusivity and birth rates are considered for linear, rectangular and circular geometries, the latter being the most efficient with minimal critical domain area. The effect of the spatial dependence of diffusivity and birth rates, common in fish populations where reproduction and protection are favoured towards the centre, on the minimum size of an exclusion zone that ensures species survival, is studied. It is shown that when any one of these factors is taken into account, the size of the critical domain decreases by a small amount.

Solving the chemical master equation for monomolecular reaction systems and beyond: a Doi-Peliti path integral view.

The chemical master equation (CME) is a fundamental description of interacting molecules commonly used to model chemical kinetics and noisy gene regulatory networks. Exact time-dependent solutions of the CME-which typically consists of infinitely many coupled differential equations-are rare, and are valuable for numerical benchmarking and getting intuition for the behavior of more complicated systems. Jahnke and Huisinga's landmark calculation of the exact time-dependent solution of the CME for monomolecular reaction systems is one of the most general analytic results known; however, it is hard to generalize, because it relies crucially on special properties of monomolecular reactions. In this paper, we rederive Jahnke and Huisinga's result on the time-dependent probability distribution and moments of monomolecular reaction systems using the Doi-Peliti path integral approach, which reduces solving the CME to evaluating many integrals. While the Doi-Peliti approach is less intuitive, it is also more mechanical, and hence easier to generalize. To illustrate how the Doi-Peliti approach can go beyond the method of Jahnke and Huisinga, we also find an explicit and exact time-dependent solution to a problem involving an autocatalytic reaction that Jahnke and Huisinga identified as not solvable using their method. Most interestingly, we are able to find a formal exact time-dependent solution for any CME whose list of reactions involves only zero and first order reactions, which may be the most general result currently known. This formal solution also yields a useful algorithm for efficiently computing numerical solutions to CMEs of this type.

Conjugate points in [formula omitted

Rossby-Haurwitz waves on the sphere S2 form a set of exact time-dependent solutions to the Euler equations of hydrodynamics and generate a family of non-stationary geodesics of the L2 metric in the volume preserving diffeomorphism group of S2. Restricting to a particular subset of Rossby-Haurwitz waves, this article shows that under certain conditions on the physical characteristics of the waves each corresponding geodesic contains conjugate points. In addition, a physical interpretation of conjugate points is given and links the result to the stability analysis of meteorological Rossby-Haurwitz waves.

A New Black Hole Solution in Conformal Dilaton Gravity on a Warped Spacetime

An exact time-dependent solution of a black hole is found in a conformally invariant gravity model on a warped Randall-Sundrum spacetime, by writing the metric . Here, represents the “un-physical” spacetime and ω the dilaton field, which will be treated on equal footing as any renormalizable scalar field. In the case of a five-dimensional warped spacetime, we thereafter write . The dilaton field can be used to describe the different notion the in-going and outside observers have of the Hawking radiation by using different conformal gauge freedom. The disagreement about the interior of the black hole is explained by the antipodal map of points on the horizon. The free parameters of the solution can be chosen in such a way that is singular-free and topologically regular, even for ω → 0 . It is remarkable that the 5D and 4D effective field equations for the metric components and dilaton fields can be written in general dimension n = 4,5. From the exact energy-momentum tensor in Eddington-Finkelstein coordinates, we are able to determine the gravitational wave contribution in the process of evaporation of the black hole. It is conjectured that, in context of quantization procedures in the vicinity of the horizon, unitarity problems only occur in the bulk at large extra-dimension scale. The subtraction point in an effective theory will be in the UV only in the bulk, because the use of a large extra dimension results in a fundamental Planck scale comparable with the electroweak scale.

Mean field dynamo action in shear flows. I: fixed kinetic helicity

ABSTRACT We study mean field dynamo action in a background linear shear flow by employing pulsed renewing flows with fixed kinetic helicity and non-zero correlation time (τ). We use plane shearing waves in terms of time-dependent exact solutions to the Navier–Stokes equation as derived by Singh & Sridhar (2017). This allows us to self-consistently include the anisotropic effects of shear on the stochastic flow. We determine the average response tensor governing the evolution of mean magnetic field, and study the properties of its eigenvalues that yield the growth rate (γ) and the cycle period (Pcyc) of the mean magnetic field. Both, γ and the wavenumber corresponding to the fastest growing axisymmetric mode vary non-monotonically with shear rate S when τ is comparable to the eddy turnover time T, in which case, we also find quenching of dynamo when shear becomes too strong. When $\tau /T\sim {\cal O}(1)$, the cycle period (Pcyc) of growing dynamo wave scales with shear as Pcyc ∝ |S|−1 at small shear, and it becomes nearly independent of shear as shear becomes too strong. This asymptotic behaviour at weak and strong shear has implications for magnetic activity cycles of stars in recent observations. Our study thus essentially generalizes the standard αΩ (or α2Ω) dynamo as also the α effect is affected by shear and the modelled random flow has a finite memory.