General Asymptotic Solution Research Articles

Asymptotic Analysis of the Forced Oscillations of Double-Layered Plates with Viscous Resistance

The three-dimensional dynamic problem of forced oscillations of an orthotropic two-layered plate in the presence of internal friction in the layers is solved by the asymptotic method. It is assumed that the friction is proportional to the velocity of the points. The values of stress set on the front faces of the double-layered packet harmonically vary in time. The general asymptotic solution of the interior problem is found. It is shown that, due to friction (dissipation), the oscillation amplitudes always remain finite, while, in the absence of internal friction, there are frequencies at which the oscillation amplitude becomes infinite.

Asymptotic structure of Einstein-Maxwell-dilaton theory and its five dimensional origin

We consider Einstein-Maxwell-dilaton theory in four dimensions including the Kaluza-Klein theory and obtain the general asymptotic solutions in Bondi gauge. We find that there are three different types of news functions representing gravitational, electromagnetic, and scalar radiations. The mass density at any angle of the system can only decrease whenever there is any type of news function. The solution space of the Kaluza-Klein theory is also lifted to five dimensions. We also compute the asymptotic symmetries in both four dimensional Einstein-Maxwell-dilaton theory and five dimensional pure Einstein theory. We find that the symmetry algebras of the two theories are the same.

(A)dS4 in Bondi gauge

We obtain the general asymptotic solutions of Einstein gravity with or without cosmological constant in Bondi gauge. The Bondi gauge was originally introduced in the context of gravitational radiation in asymptotically flat gravity. In the original work, initial conditions were prescribed at a null hypersurface and the Einstein equations were shown to take a nested form, which may be used to explicitly integrate them asymptotically. We streamline the derivation of the general asymptotic solution in the asymptotically flat case, and derive the most general asymptotic solutions for the case of non-zero cosmological constant of either sign (asymptotically locally AdS and dS solutions). With non-zero cosmological constant, we present integration schemes which rely on either prescribing data on the conformal boundary or on a null hypersurface and part of the conformal boundary. We explicitly work out the transformation to Fefferman–Graham gauge and identity how to extract the holographic data directly in Bondi coordinates. We illustrate the discussion with a number of examples and show that for asymptotically AdS4 spacetimes the Bondi mass is constant.

Asymptotically simple spacetimes and mass loss due to gravitational waves

The cosmological constant [Formula: see text] used to be a freedom in Einstein’s theory of general relativity (GR), where one had a proclivity to set it to zero purely for convenience. The signs of [Formula: see text] or [Formula: see text] being zero would describe universes with different properties. For instance, the conformal structure of spacetime directly depends on [Formula: see text]: null infinity [Formula: see text] is a spacelike, null, or timelike hypersurface, if [Formula: see text], [Formula: see text], or [Formula: see text], respectively. Recent observations of distant supernovae have taught us that our universe expands at an accelerated rate, and this can be accounted for by choosing [Formula: see text] in Einstein’s theory of GR. A quantity that depends on the conformal structure of spacetime, especially on the nature of [Formula: see text], is the Bondi mass which in turn dictates the mass loss of an isolated gravitating system due to energy carried away by gravitational waves. This problem of extending the Bondi mass to a universe with [Formula: see text] has spawned intense research activity over the past several years. Some aspects include a closer inspection on the conformal properties, working with linearization, attempts using a Hamiltonian formulation based on “linearized” asymptotic symmetries, as well as obtaining the general asymptotic solutions of de Sitter-like spacetimes. We consolidate on the progress thus far from the various approaches that have been undertaken, as well as discuss the current open problems and possible directions in this area.

On the internal structure of the current sheet in the pulsar wind

We investigate the internal structure of the current sheet in the pulsar wind within force-free and two-fluid MHD approximations. Within the force-free approximation we obtain general asymptotic solution of the Grad-Shafranov equation for quasi-spherical pulsar wind up to the second order in small parameter $\varepsilon = (\Omega r/c)^{-1}$. The solution allows an arbitrary latitudinal structure of the radial magnetic field, including that obtained in the numerical simulations of oblique rotators. It is also shown that the shape of the current sheet does not depend on the latitudinal structure. For the internal region of the current sheet outside the fast magnetosonic surface where the force-free approximation is not valid we use two-fluid MHD approximation. Carrying out calculations in the comoving reference frame we succeed in determining intrinsic electric and magnetic fields of a sheet. It allows us to analyze time-dependent effects which were not investigated up to now. In particular, we estimate the efficiency of the particle acceleration inside the sheet. Finally, after investigating the motion of individual particles in the time-dependent current sheet, we find self-consistently the width of the sheet and its time evolution.

Internal shear layers from librating objects

In this work, we analyse the internal shear layer structures generated by the libration of an axisymmetric object in an unbounded fluid rotating at a rotation rate $\unicode[STIX]{x1D6FA}^{\ast }$ using direct numerical simulation and small Ekman number asymptotic analysis. We consider weak libration amplitude and libration frequency $\unicode[STIX]{x1D714}^{\ast }$ within the inertial wave interval $(0,2\unicode[STIX]{x1D6FA}^{\ast })$ such that the fluid dynamics is mainly described by a linear axisymmetric harmonic solution. The internal shear layer structures appear along the characteristic cones of angle $\unicode[STIX]{x1D703}_{c}=\text{acos}(\unicode[STIX]{x1D714}^{\ast }/(2\unicode[STIX]{x1D6FA}^{\ast }))$ which are tangent to the librating object at so-called critical latitudes. These layers correspond to thin viscous regions where the singularities of the inviscid solution are smoothed. We assume that the velocity field in these layers is described by the class of similarity solutions introduced by Moore & Saffman (Phil. Trans. R. Soc. Lond. A, vol. 264, 1969, pp. 597–634). These solutions are characterized by two parameters only: a real parameter $m$, which measures the strength of the underlying singularity, and a complex amplitude coefficient $C_{0}$. We first analyse the case of a disk for which a general asymptotic solution for small Ekman numbers is known when the disk is in a solid plane. We demonstrate that the numerical solutions obtained for a free disk and for a disk in a solid plane are both well described by the asymptotic solution and by its similarity form within the internal shear layers. For the disk, we obtain a parameter $m=1$ corresponding to a Dirac source at the edge of the disk and a coefficient $C_{0}\propto E^{1/6}$ where $E$ is the Ekman number. The case of a smoothed librating object such as a spheroid is found to be different. By asymptotically matching the boundary layer solution to similarity solutions close to a critical latitude on the surface, we show that the adequate parameter $m$ for the similarity solution is $m=5/4$, leading to a coefficient $C_{0}\propto E^{1/12}$, that is larger than for the case of a disk for small Ekman numbers. A simple general expression for $C_{0}$ valid for any axisymmetric object is obtained as a function of the local curvature radius at the critical latitude in agreement with this change of scaling. This result is tested and validated against direct numerical simulations.

Grain Boundary Diffusion in Nanocrystalline Materials Produced by Severe Plastic Deformation

The model of diffusional mass transfer in the medium with a strong spatial dependence of diffusivity and its application to the grain boundary (GB) diffusion problem is presented. A significant decrease of diffusion activation energy is shown to take place takes place in the vicinity of non-equilibrium grain boundary, which leads to the formation of a region of enhanced diffusion. The generalization of grain boundary diffusion theory is given which accounts for the diffusion enhancement near GB. An original mathematical approach is developed and general asymptotic solutions of the one-and two-dimensional diffusion problems are derived for two types of diffusant source — constant and instant. The applicability domain of presented model is discussed.

Generalized dilatation operator method for non-relativistic holography

We present a general algorithm for constructing the holographic dictionary for Lifshitz and hyperscaling violating Lifshitz backgrounds for any value of the dynamical exponent z and any value of the hyperscaling violation parameter θ compatible with the null energy condition. The objective of the algorithm is the construction of the general asymptotic solution of the radial Hamilton–Jacobi equation subject to the desired boundary conditions, from which the full dictionary can be subsequently derived. Contrary to the relativistic case, we find that a fully covariant construction of the asymptotic solution for running non-relativistic theories necessitates an expansion in the eigenfunctions of two commuting operators instead of one. This provides a covariant but non-relativistic grading of the expansion, according to the number of time derivatives.

Amplification limit of weak measurements: A variational approach

Postselected weak measurement has been widely used in experiments to observe weak effects in various physical systems. However, it is still unclear how large the amplification ability of a weak measurement can be and what determines the limit of this ability, which is fundamental to understanding and applying weak measurements. The limitation of the conventional weak-value formalism for this problem is the divergence of weak values when the pre- and postselections are nearly orthogonal. In this paper, we study this problem by a variational approach for a general Hamiltonian ${H}_{\mathrm{int}}=gA\ensuremath{\bigotimes}\ensuremath{\Omega}\ensuremath{\delta}(t\ensuremath{-}{t}_{0}),\phantom{\rule{0.16em}{0ex}}g\ensuremath{\ll}1$. We derive a general asymptotic solution and show that the amplification limit is essentially independent of $g$ and is determined by only the initial state of the detector and the number of distinct eigenvalues of $A$ or $\ensuremath{\Omega}$. An example of spin-$\frac{1}{2}$ particles with a pair of Stern-Gerlach devices is given to illustrate the results. The limiting case of continuous-variable systems is also investigated to demonstrate the influence of system dimension on the amplification limit.

Heat Conduction of Walls with a Monotone Temperature Change. Asymptotics and Quasi-Stationarity

Systematizing the partial solutions of the nonstationarity heat conduction problem of a flat wall in comparison with the general asymptotic solution of this problem, we have found the transverse temperature distributions with any monotone change in the ambient conditions and elucidated the heat conduction properties of the wall under these conditions. The asymptotic solution is given by semiconvergent series and definite integrals and has been investigated for power time dependences with an exponent of 0–2, which has enabled us to justify the concept of quasi-stationarity of the thermal parameters of the wall and obtain asymptotic errors and corrections defining the deviations of these parameters from their stationary values. The features of the average heat flows most resistant to thermal disturbances as to both time and amplitude have been considered.

Asymptotic Solutions of Pressure and Heat Flux of Highly Rarefied Gas Flows in Vacuum Packaged Micro-Electro-Mechanical System Devices

This study analyzed a highly rarefied gas flow fields inside an enclosure. The enclosure is formed by four plates with two different temperatures. The plate surfaces are assumed completely diffuse; at the final steady flow stage for any point inside the enclosure or at the surface, the velocity distribution function was determined and all the macroscopic properties was determined by integrating the velocity distribution function. First, the collisionless flow solution was considered as the base solution, and then based on the Bhatnagar-Gross-Krook model, the characteristic relaxation time scale τ was used as a small parameter, and obtained general asymptotic analytical solutions for the flow fields and heat transfer expressions. To validate these asymptotic solutions, a series of numerical simulations with the direct simulation Monte Carlo method were performed and compared with the analytical results. In general, as Knudsen number decreases, the solutions for the pressure, temperature, and density fields deteriorate; the temperature fields had relatively better agreement than the density and pressure fields.

On flow in weakly precessing cylinders: the general asymptotic solution

AbstractWe investigate, through both asymptotic and numerical analysis, precessionally driven flow of a homogeneous fluid confined in a fluid-filled circular cylinder that rotates rapidly about its symmetry axis and precesses slowly about a different axis that is fixed in space. After demonstrating that the inviscid approximation is always divergent even far away from resonance, we derive a general asymptotic solution for an asymptotically small Ekman number in the rotating frame of reference describing the weakly precessing flow that satisfies the no-slip boundary condition and that is valid at or near or away from resonance. Numerical analysis of the same problem using the Galerkin method in terms of a Chebyshev polynomial expansion is also carried out, showing satisfactory agreement between the general asymptotic solution and the corresponding numerical solution at or near or away from resonance.

Resonance in Low-Temperature Oxidation Waves for Porous Media

This paper analyzes traveling wave profiles possessing an internal resonance point for a class of systems of partial differential equations describing oxidation and vaporization of liquid fuel in a porous medium when an oxidizer (air) is injected. It is shown that the resonance is characterized by a saddle point of an associated vector field defined on a folded surface in state space. We prove existence and uniqueness of this singularity for an open set of parameters. The singularity yields an extra restriction on wave parameters. This restriction is found explicitly in the physically relevant case of a small ratio of reaction/vaporization rates. We find the general large time asymptotic solutions of the problem as a sequence of waves and show that resonance waves play a key role in determining these solutions. A numerical example is presented.

Holographic renormalization as a canonical transformation

The gauge/string dualities have drawn attention to a class of variational problems on a boundary at infinity, which are not well defined unless a certain boundary term is added to the classical action. In the context of supergravity in asymptotically AdS spaces these problems are systematically addressed by the method of holographic renormalization. We argue that this class of a priori ill defined variational problems extends far beyond the realm of holographic dualities. As we show, exactly the same issues arise in gravity in non asymptotically AdS spaces, in point particles with certain unbounded from below potentials, and even fundamental strings in flat or AdS backgrounds. We show that the variational problem in all such cases can be made well defined by the following procedure, which is intrinsic to the system in question and does not rely on the existence of a holographically dual theory: (i) The first step is the construction of the space of the most general asymptotic solutions of the classical equations of motion that inherits a well defined symplectic form from that on phase space. The requirement of a well defined symplectic form is essential and often leads to a necessary repackaging of the degrees of freedom. (ii) Once the space of asymptotic solutions has been constructed in terms of the correct degrees of freedom, then there exists a boundary term that is obtained as a certain solution of the Hamilton-Jacobi equation which simultaneously makes the variational problem well defined and preserves the symplectic form. This procedure is identical to holographic renormalization in the case of asymptotically AdS gravity, but it is applicable to any Hamiltonian system.

Y-system, TBA and Quasi-Classical Strings in AdS 4 × CP3

We study the exact spectrum of the AdS4/CFT3 duality put forward by Aharony, Bergman, Jafferis and Maldacena (ABJM). We derive thermodynamic Bethe ansatz (TBA) equations for the planar ABJM theory, starting from "mirror" asymptotic Bethe equations which we conjecture. We also propose generalization of the TBA equations for excited states. The recently proposed Y-system is completely consistent with the TBA equations for a large subsector of the theory, but should be modified in general. We find the general asymptotic infinite length solution of the Y-system, and also several solutions to all wrapping orders in the strong coupling scaling limit. To make a comparison with results obtained from string theory, we assume that the all-loop Bethe ansatz of N.G. and P. Vieira is the valid worldsheet theory description in the asymptotic regime. In this case we find complete agreement, to all orders in wrappings, between the solution of our Y-system and generic quasi-classical string spectrum in AdS3 x S.

Radiative energy transfer in disordered photonic crystals

The difficulty of description of the radiative transfer in disordered photonic crystals arisesfrom the necessity to consider on an equal footing the wave scattering by periodicmodulations of the dielectric function and by its random inhomogeneities. We resolve thisdifficulty by approaching this problem from the standpoint of the general multiplescattering theory in media with an arbitrary regular profile of the dielectric function. Weuse the general asymptotic solution of the Bethe–Salpeter equation in order to show thatfor a sufficiently weak disorder the diffusion limit in disordered photonic crystals ispresented by incoherent superpositions of the modes of the ideal structure with weightsinversely proportional to the respective group velocities. The radiative transfer and thediffusion equations are derived as a relaxation of long scale deviations from this limitingdistribution. In particular, it is shown that in general the diffusion is anisotropic unless thecrystal has sufficiently rich symmetry, say, the square lattice in 2D or the cubic lattice in3D. In this case, the diffusion is isotropic and only in this case can the effect of thedisorder be characterized by a single mean free path depending on frequency.

Riffle shuffles of a deck with repeated cards

We study the Gilbert-Shannon-Reeds model for riffle shuffles and ask 'How many times must a deck of cards be shuffled for the deck to be in close to random order?'. In 1992, Bayer and Diaconis gave a solution which gives exact and asymptotic results for all decks of practical interest, e.g. a deck of 52 cards. But what if one only cares about the colors of the cards or disregards the suits focusing solely on the ranks? More generally, how does the rate of convergence of a Markov chain change if we are interested in only certain features? Our exploration of this problem takes us through random walks on groups and their cosets, discovering along the way exact formulas leading to interesting combinatorics, an 'amazing matrix', and new analytic methods which produce a completely general asymptotic solution that is remarkable accurate.

Precision holography for non-conformal branes

We set up precision holography for the non-conformal branes preserving 16 supersymmetries. The near-horizon limit of all such p-brane solutions with p \leq 4, including the case of fundamental string solutions, is conformal to AdS_{p+2} x S^{8-p} with a linear dilaton. We develop holographic renormalization for all these cases. In particular, we obtain the most general asymptotic solutions with appropriate Dirichlet boundary conditions, find the corresponding counterterms and compute the holographic 1-point functions, all in complete generality and at the full non-linear level. The result for the stress energy tensor properly defines the notion of mass for backgrounds with such asymptotics. The analysis is done both in the original formulation of the method and also using a radial Hamiltonian analysis. The latter formulation exhibits most clearly the existence of an underlying generalized conformal structure. In the cases of Dp-branes, the corresponding dual boundary theory, the maximally supersymmetric Yang-Mills theory SYM_{p+1}, indeed exhibits the generalized conformal structure found at strong coupling. We compute the holographic 2-point functions of the stress energy tensor and gluon operator and show they satisfy the expected Ward identities and the constraints of generalized conformal structure. The holographic results are also manifestly compatible with the M-theory uplift, with the asymptotic solutions, counterterms, one and two point functions etc of the IIA F1 and D4 appropriately descending from those of M2 and M5 branes, respectively. We present a few applications including the computation of condensates in Witten's model of holographic YM_4 theory.

A unified Krylov-Bogoliubov-Mitropolskii method for solving hyperbolic-type nonlinear partial differential systems

A general asymptotic solution is presented for investigating the transient response of non-linear systems modeled by hyperbolic-type partial differential equations with small nonlinearities. The method covers all the cases when eigen-values of the corresponding unperturbed systems are real, complex conjugate, or purely imaginary. It is shown that by suitable substitution for the eigen-values in the general result that the solution corresponding to each of the three cases can be obtained. The method is an extension of the unified Krylov-Bogoliubov-Mitropolskii method, which was initially developed for un-darnped, under-clamped and over-clamped cases of the second order ordinary differential equation. The methods also cover a special condition of the over-damped case in which the general solution is useless.

Analytical self-similar solutions of Ginzburg-Landau equation for the dispersion decreasing fiber

Using the method based on the technique of symmetry reduction, we find the general analytical parabolic asymptotic self-similar solutions for the varying coefficient of Ginzburg-Landau equation that take consideration of the influence of the doped fiber retarding time. The parabolic asymptotic amplitude function, change of strict linear phase chirp and the effective temporal pulse width of self-similar pulse with gain dispersion are given for the dispersion decreasing fibers with longitudinal exponential distribution and hyperbolic distribution. And these theoretical results have been confirmed by numerical simulation in this paper.