Exact Nonstationary Solutions Research Articles

Quantum reversibility in cooperative interaction of the atom system with bi-modal cavity field in Raman conversion

The restoration of the initial state in the Raman interaction of the two atoms with the quantified pump and Stokes (anti-Stokes) cavity modes is analytically obtained. For this the su (2) symmetry representation of the bimodal field and a system of two atoms in scattering interaction is proposed for the simplification of the problem. Using the proprieties of the generators su (2) algebra the exact non-stationary solution of the system of two atoms in interaction with the bimodal field is found, taking into consideration the initial disentangled state between the atomic and field superposition. It is demonstrated that after this collapse and revivals of the inversion of flying atoms in the processes of absorption and emission of the photons from Stokes and pump modes the system atoms and field becomes entangled. The conditions of the restoration of the initial state of the atom-field system were established as a function of the flying time through the resonator possible. A set of discrete flying time intervals for the restoration of the initial disentangled state was found.

Non-stationary Antonov self-gravitating layer: analytics and numerics

ABSTRACTLarge-scale instability of gravitating systems plays a key role in collisionless relaxation and in reaching a quasi-stationary state at the early stage of evolution. Advanced high-resolution methods and permanently increasing performance of computational systems allow this phenomenon to be studied by means of computer simulations at a new level. In this paper, an approach to verification and validation of computer codes implementing high-resolution methods is proposed. The approach is based on comparisons of the simulation results with exact non-stationary solutions of the Vlasov–Poisson equations. The evolution of the gravitating layer model is considered as an example of implementation of this approach. A one-parameter family of exact models of a non-stationary gravitating layer is described, and their stability to large-scale disturbances in the linear approximation is analytically studied. Non-linear instability development is computed with the use of the fifth-order conservative semi-Lagrangian WENO scheme.

Nonstationary self-gravitating configurations of scalar and electromagnetic fields

Mathematical modeling of gravitating configurations of physical fields is one of the priority directions of the modern theory of gravity. Most of the exact solutions constructed within the framework of the general relativity are static or stationary configurations. This is due to the objective complexity of solving the Einstein equations under the assumption of nonstationarity. We present an approach to constructing nonstationary configurations of a spherically symmetric nonlinear real scalar field and the electromagnetic field, which are assumed both to be minimally coupled to gravity. It is based on the isolation of one invariant equation written in terms of the characteristic function and scalar field potential. Using the proposed method, an exact nonstationary solution with a nontrivial topology of space-time will be constructed.

Nonlinear density collapse in one-dimensional cold atomic gas

The nonlinear collective dynamics of a one-dimensional cold atomic cloud is investigated using a Lagrangian fluid description. An exact non-stationary nonlinear solution with a nontrivial space and time dependence is obtained. The results demonstrate the formation of nonlinear localized soliton-like density structures when nonlinearity is being balanced by the wave dispersion. Moreover, the solution predicts the catastrophic density collapse in the cloud of cold atomic gas when nonlinearity overpowers the dispersion.

An Exact Non-stationary Vortex Solution of the Plane Flow and Brownian Motion

An Exact Non-stationary Vortex Solution of the Plane Flow and Brownian Motion

On exact nonstationary solutions of equations of vibrational convection

On exact nonstationary solutions of equations of vibrational convection

Exact Stationary and Non-stationary Solutions to Inelastic Maxwell Model with Infinite Energy

The one-dimensional inelastic Boltzmann equation with a constant collision rate (the Maxwell model) is considered. It is shown that for special values of restitution parameter there exists a stationary solution with the characteristic function in the form \(e^{-P(\log (z))z},\) where P is a periodic function. The corresponding distribution function belongs to a one special class of stochastic processes termed as a generalized stable in the probability theory. The Fourier transform of the non-stationary equation has the solution \(\bigl (1+P(\log (z))z\bigr )e^{-Q(\log (z))z}\). It is proved that this solution is a characteristic function if periodic functions P, Q satisfy some not very restrictive conditions. The stationary and non-stationary solutions correspond to a gas with infinite temperature.

Convoluted solutions in supergravity

Inspired by the convoluted solutions for two intersecting M2 branes in elevendimensional supergravity, in which one brane in the system is completely localized along the overall and relative transverse coordinates while the other brane in the system is localized only along the overall transverse coordinates, we construct two classes of exact solutions to Einstein-Maxwell theory in six and higher dimensions. We show that the membrane configuration preserves four supersymmetries and upon dimensional reduction, the solutions provide intersecting configurations of three D-branes in type IIA supergravity. Moreover, we show that the metric functions in six and higher dimensions can be written as convolution-like integrals of two special functions. The solutions are regular everywhere and show a bolt structure on a single point in any dimensionality. Also, we find the exact nonstationary solutions to the Einstein-Maxwell theory with positive cosmological constant. We show that the cosmological solutions are expanding patches in asymptotically de Sitter spacetime.

Nonstationary laminar thermal and solutal Marangoni convection of a viscous fluid

We have determined and analyzed exact stationary and nonstationary solutions to the laminar Marangoni convection problem, which is an overdetermined boundary value problem. The numerical solution of this problem belongs to the class of Birikh solutions. The overdetermination of the resolving system of equations results from the zeroness of the velocity parallel to the applicate axis. Cases of thermal and solutal convection of a viscous incompressible fluid are considered. To make the boundary value problem solvable, the class of exact solutions is proposed for use, where velocities are one-dimensional in coordinates, the pressure and temperature fields are three-dimensional. Identical equality to zero of the convective derivative in the impulse conservation equation is typical of the class presented. The convective derivative remains in the caloric equation. The discussed boundary value problem is shown to be irreducible to the one-dimensional problem when the temperature gradient is specified on both boundaries of the fluid layer, as distinct from the classical Birikh solution and its numerous generalizations. The obtained stationary and nonstationary solutions have a stagnation point for velocities, thus suggesting the presence of counter flows in the moving fluid. It is demonstrated by localization of the polynomial roots of the stationary solutions that there exists such a value of layer thickness that the tangential stress can become zero on the lower boundary of the fluid layer only under thermal Marangoni convection. The nonstationary solutions obtained by the boundary element method, which can be treated as an exact method, tend to become stationary. The application of the boundary element method extends the class of exact nonstationary solutions considerably, since this method enables one to study not only invariant exact solutions.

Exact convoluted solutions in higher-dimensional Einstein-Maxwell theory

We construct two classes of exact solutions to six and higher dimensional Einstein-Maxwell theory in which the metric functions can be written as convolution-like integrals of two special functions. The solutions are regular everywhere and show a bolt structure on a single point in any dimensionality. Moreover, we find the exact nonstationary solutions to the Einstein-Maxwell theory with positive cosmological constant. We show that the cosmological solutions are expanding patches in asymptotically de Sitter spacetime.

Wave-Breaking Phenomena in a Relativistic Magnetized Plasma

We study the wave-breaking phenomenon of relativistic upper-hybrid (UH) oscillations in a cold magnetoplasma. For our purposes, we use the electron continuity and relativistic electron momentum equations, together with Maxwell's equations, as well as introduce Lagrangian coordinates to obtain an exact nonstationary solution of the governing nonlinear equations. It is found that bursts in the electron density appear in a finite time as a result of relativistic electron mass variations in the UH electric field, indicating a phase mixing or breaking of relativistic UH oscillations. We highlight the relevance of our investigation of the UH wave phase-mixing or UH wave-breaking process to electron energization and plasma particle heating.

Dipole pulse theory: Maximizing the field amplitude from4πfocused laser pulses

We present a class of exact nonstationary solutions of Maxwell equations in vacuum from dipole pulse theory: electric and magnetic dipole pulses. These solutions can provide for a very efficient focusing of electromagnetic field and can be generated by $4\ensuremath{\pi}$ focusing systems, such as parabolic mirrors, by using radially polarized laser pulses with a suitable amplitude profile. The particular cases of a monochromatic dipole wave and a short dipole pulse with either quasi-Gaussian or Gaussian envelopes in the far-field region are analyzed and compared in detail. As a result, we propose how to increase the maximum field amplitude in the focus by properly shaping the temporal profile of the input laser pulses with given main wavelength and peak power.

Construction of exact solutions to the Muskat problem

A modification of Howison’s method is proposed to construct exact nonstationary solutions of the unsteady two-phase Hele-Shaw problem (Muskat problem). The effectiveness of the modified method is demonstrated by reproducing D. Crowdy’s exact solution for the evolution of an elliptic bubble of a viscous fluid surrounded by another fluid in an unbounded Hele-Shaw cell. This solution is found to be directly related to the explicit solution of the single-phase Hele-Shaw problem. A comparison of these solutions shows that allowance for the second phase leads to a weak regularizing effect: the single-phase solution exists over a finite time, while the two-phase solution exists over an infinite time under the same initial conditions.

Dynamic analysis of the nature of nonlinearity of the lighting characteristics of the active pixels of a CMOS image detector

The dynamics of the information-signal readout in active photosensors of modern CMOS image detectors is studied by the methods of mathematical simulation. Under the assumption of a specific (quasicylindrical) shape of the pixel photodiode and with the appropriete application of the exact nonstationary solutions of the 2D diffusion equation, an universal curve of relaxation of the photodiode potential at the signal preset stage and at the readout stage is constructed. An algorithm allowing calculation of the pixel lighting characteristics at any ratio of frame duration and the exposure and readout times is proposed. The corresponding numerical calculation yields the shape of the lighting characteristic close to that observed experimentally. The proposed model explains the anomalously high photosensitivity at small signals by incomplete erasing of the frame exposure at the potential preset stage before the exposure.

Exact Nonstationary Solutions of a Two-Component Bose–Einstein Condensate

We investigate the exact nonstationary solutions of a two-component Bose–Einstein condensate which compose of two species having different atomic masses. We also consider the interesting behavior of the atomic velocity and the flow density. It is shown that the motion of the two components can be controlled by the experimental parameters.

Formation of density singularities in ideal hydrodynamics of freely cooling inelastic gases: A family of exact solutions

We employ granular hydrodynamics to investigate a paradigmatic problem of clustering of particles in a freely cooling dilute granular gas. We consider large-scale hydrodynamic motions where the viscosity and heat conduction can be neglected, and one arrives at the equations of ideal gas dynamics with an additional term describing bulk energy losses due to inelastic collisions. We employ Lagrangian coordinates and derive a broad family of exact nonstationary analytical solutions that depend only on one spatial coordinate. These solutions exhibit a new type of singularity, where the gas density blows up in a finite time when starting from smooth initial conditions. The density blowups signal formation of close-packed clusters of particles. As the density blow-up time tc is approached, the maximum density exhibits a power law ∼(tc−t)−2. The velocity gradient blows up as ∼−(tc−t)−1 while the velocity itself remains continuous and develops a cusp (rather than a shock discontinuity) at the singularity. The gas temperature vanishes at the singularity, and the singularity follows the isobaric scenario: the gas pressure remains finite and approximately uniform in space and constant in time close to the singularity. An additional exact solution shows that the density blowup, of the same type, may coexist with an “ordinary” shock, at which the hydrodynamic fields are discontinuous but finite. We confirm stability of the exact solutions with respect to small one-dimensional perturbations by solving the ideal hydrodynamic equations numerically. Furthermore, numerical solutions show that the local features of the density blowup hold universally, independently of details of the initial and boundary conditions.

On Self-Oscillations during the Outflow of a Swirled Jet

It is shown that the core of a swirled helical flow can be described using a novel exact nonstationary solution of the hydrodynamic equations for a viscous incompressible fluid, which generalizes the rigid-body asymptotics for the Burgers and Sullivan vortices in the form of rigid-body rotation with a finite helicity. An estimate of the pressure fluctuations corresponding to this nonstationary vortex regime, which is proportional to the frequency of the swirled-jet core rotation as a rigid body and also depends on the parameters of the initial velocity field structure, is obtained. It is noted that this frequency may correspond to the frequency observed in the pressure fluctuation spectrum, which is almost proportional to the swirled flow rate in vortex acoustic emitters.

Exact nonstationary solutions to the mean-field equations of motion for two-component Bose–Einstein condensates in periodic potentials

We study the dynamics of two-component Bose-Einstein condensates in periodic potentials in one dimension. Elliptic potentials which have the sinusoidal optical potential as a special case are considered. We construct exact nonstationary solutions to the mean-field equations of motion. Among the solutions are two types of temporally-periodic solutions -- in one type there are condensate oscillations between neighboring potential wells, while in the other the condensates oscillate from side to side within the wells. Our numerical studies of the stability of these solutions suggests the existence of one-parameter families of stable nonstationary solutions.

Breakdown of waves described by exact solutions of the Thomas-Fermi model

The dynamics of large localized repulsive clouds is examined by means of exact non-stationary solutions of the one-dimensional Thomas-Fermi model. The nonlinear flattening of the cloud peak, the wave breakdown at the cloud peripheries, and the condensate velocity distributions are thus described. Our solutions, which can contain an arbitrary amount of free parameters, show the nonlinear evolution of an arbitrary initial wave form. A unique procedure for analyzing these solutions is presented. The difference between our breakdown matter wave solutions and the well known Riemann shock waves is stressed.

An exact non-stationary solution of simple shear flow in a Bingham fluid

The flow of Bingham fluids can have yield surfaces that separate unyielded regions from yielded regions. The analytical solution for the Bingham flow equation that contains a moving yield surface is presented. An initially homogeneous simple shear flow in the semi-infinite bulk bounded by a planar boundary is assumed, and consideration is given to what disturbance occurs upon the reduction of the applied shear stress on the boundary after time t = 0. The solution is of the ‘similarity’ type in which all length scales grow as ∼ t 1 2 and the unyielded region in particular grows from the boundary such that its thickness is proportional to t 1 2 .