Nonstationary Solutions Research Articles

C programs for solving the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap

We present C programming language versions of earlier published Fortran programs (Muruganandam and Adhikari, Comput. Phys. Commun. 180 (2009) 1888) for calculating both stationary and non-stationary solutions of the time-dependent Gross-Pitaevskii (GP) equation. The GP equation describes the properties of dilute Bose-Einstein condensates at ultra-cold temperatures. C versions of programs use the same algorithms as the Fortran ones, involving real- and imaginary-time propagation based on a split-step Crank-Nicolson method. In a one-space-variable form of the GP equation, we consider the one-dimensional, two-dimensional, circularly-symmetric, and the three-dimensional spherically-symmetric harmonic-oscillator traps. In the two-space-variable form, we consider the GP equation in two-dimensional anisotropic and three-dimensional axially-symmetric traps. The fully-anisotropic three-dimensional GP equation is also considered. In addition to these twelve programs, for six algorithms that involve two and three space variables, we have also developed threaded (OpenMP parallelized) programs, which allow numerical simulations to use all available CPU cores on a computer. All 18 programs are optimized and accompanied by makefiles for several popular C compilers. We present typical results for scalability of threaded codes and demonstrate almost linear speedup obtained with the new programs, allowing a decrease in execution times by an order of magnitude on modern multi-core computers.

Nonstationary poiseuille-type solutions for the second-grade fluid flow

We study solutions to the nonstationary equations describing the motion of the second-grade fluid in a twodimensional infinite channel and in a three-dimensional rotationally symmetric pipe. We prove the existence of a unique Poiseuille-type solution having a given flow rate (flux) and investigate the behavior of this solution as t → ∞.

Solvability of the initial-boundary value problem for an integrodifferential equation

Under study is the well-posedness of the Cauchy problem for the nonstationary radiation transfer equation with generalized matching conditions at the interface between the media. We prove the existence of a unique strongly continuous semigroup of resolvents, estimate its order of growth, and consider the question of stabilization of the nonstationary solution.

A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space

Abstract Considering the features of the fractional Klein-Kramers equation (FKKE) in phase space, only the unilateral boundary condition in position direction is needed, which is different from the bilateral boundary conditions in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. In the paper, a finite difference scheme is constructed, where temporal fractional derivatives are approximated using L1 discretization. The advantages of the scheme are: for every temporal level it can be dealt with from one side to the other one in position direction, and for any fixed position only a tri-diagonal system of linear algebraic equations needs to be solved. The computational amount reduces compared with the ADI scheme in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and the five-point scheme in [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. The stability and convergence are proved and two examples are included to show the accuracy and effectiveness of the method.

A new mode and its interaction through ponderomotive force in electron-positron-ion plasmas

A new mode is found in e-p-i plasma in the presence of density and temperature difference of lighter particles. The electron beam induced Cherenkov instability condition for the excitation of positron sound wave is obtained for the system under consideration. Zakharov’s equation with sign modification due to negative Ponderomotive pressure is obtained. Nonlinear Schrödinger wave equation for the envelope type solitary waves is derived. Both stationary and nonstationary solutions are found and the subsonic and supersonic limits are also discussed. In the stationary case, rarefactive type solitary solution is obtained, whereas the nonstationary case yields the ion acoustic shock like structure solution which is very interesting. The importance of the study with relevance to both laboratory and astrophysical plasmas is pointed out.

Model of nonequilibrium crystallization for the numerical solution to the problem of semiconductor crystal growth from melts

A model combining both non-stationary Stefan problem solution and classical ideas of constitutional supercooling is proposed in this paper. Together with Navier-Stokes equations in melt and heat equation in crystal this model is used for numerical solution of semiconductor crystal growth problem. Simulation of formation of fundamental and technological striations is performed. Numerical simulation results are compared with experimental data.

Nonlinear viscous fluid patterns in a thin rotating spherical domain and applications

We study the nonlinear incompressible fluid flows within a thin rotating spherical shell. The model uses the two-dimensional Navier-Stokes equations on a rotating three-dimensional spherical surface and serves as a simple mathematical descriptor of a general atmospheric circulation caused by the difference in temperature between the equator and the poles. Coriolis effects are generated by pseudoforces, which support the stable west-to-east flows providing the achievable meteorological flows rotating around the poles. This work addresses exact stationary and non-stationary solutions associated with the nonlinear Navier-Stokes. The exact solutions in terms of elementary functions for the associated Euler equations (zero viscosity) found in our earlier work are extended to the exact solutions of the Navier-Stokes equations (non-zero viscosity). The obtained solutions are expressed in terms of elementary functions, analyzed, and visualized.

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.

Synchronous exploration for the control of real-time external noise suppression in a three-dimensional subdomain

A mathematical (difference) model is proposed for a real-time active shielding device that shields an acoustic field in a given subdomain from the influence of sound sources located in an additional subdomain. An algorithm for computing the current control ensuring a prescribed process is based on information produced by the author’s technique of synchronous weak noise exploration. This information can be measured in real time. Active control problems for nonstationary solutions of linear difference equations in a three-dimensional domain consisting of two subdomains are studied using the difference potential method. The shape of the domain and the boundary conditions may depend on time, while the coefficients may depend on time and spatial coordinates. If the difference problem is a mathematical model of sound propagation, the goal of control is to change the acoustic field in the given subdomains, for example, to shield the acoustic field in one subdomain from the undesirable influence (noise) of sources located in the other subdomain.

Phase-Controlled Atom-Photon Entanglement in a Three-Level V-Type Atomic System via Spontaneously Generated Coherence

We investigate the dynamical behavior of the atom-photon entanglement in a V-type three-level quantum system using the atomic reduced entropy. It is shown that an atom and photons are entangled at the steady-state; however disentanglement can also be achieved in an special condition. It is demonstrated that in the presence of quantum interference induced by spontaneous emission, the reduced entropy and the atom-photon entanglement are phase-dependent. A non-stationary solution is also obtained when the quantum interference due to the spontaneous emission is completely included.

A stable-focus pattern in quantum geometrodynamics

A nonstationary solution to the gravitational analog of the Schrodinger equation in quantum geometrodynamics is analyzed. It is found that the evolution trajectory in functional space can appear as a stable-focus pattern.

Non-vacuum conformally flat space-times: dark energy

In this paper we propose a class of exact solutions (stationary and non-stationary) of Einstein’s field equations. We find that the space-time geometries of the solutions are non-vacuum and conformally flat, whose energy-momentum tensors possess dark energy with negative pressure and the energy equation of state parameter w=−1/2. We also find that the time-like vector fields of the matter distributions of the solutions are expanding, shearing with acceleration and zero-twist. It is also found that, due to the negative pressure, the energy-momentum tensors violate the strong energy conditions leading to the repulsive gravitational fields of the space-time geometries. Energy-momentum tensors for the solutions also obey the energy conservation equations. From these physical properties of the matter distribution we may refer the space-times to as examples of exact solutions of the Einstein’s field equations admitting dark energy with negative pressure. It is to note that the approximate sizes of the masses of the (stationary and non-stationary) solutions are less than (1/2)×10−60 in Bousso’s length scale r>1060. We also find that the surface gravities on the horizons are directly proportional to their respective masses.

Nonergodic solutions of the generalized Langevin equation

It is known that in the regime of superlinear diffusion, characterized by zero integral friction (vanishing integral of the memory function), the generalized Langevin equation may have nonergodic solutions that do not relax to equilibrium values. It is shown that the equation may have nonergodic (nonstationary) solutions even if the integral of the memory function is finite and diffusion is normal.

Determination of the exact solutions to the inhomogeneous burgers equation with the use of the darboux transformation

A method of determining the exact solutions to the Burgers equation on the basis of the Darboux transformation is described. It is shown that a single application of the Darboux transformation to the homogeneous Burgers equation transforms the latter into the inhomogeneous equation describing acoustic wave propagation against transonic flow in the de Laval nozzle. In this case, the contraction ratio of the nozzle is fixed and determined by the viscosity coefficient of the medium. Based on the exact solution of the homogeneous Burgers equation, for the aforementioned problem of the flow in the nozzle, all the possible regular steady-state solutions are presented and the evolution of nonstationary solutions is investigated. The algorithm of a multiple Darboux transformation, which allows an increase in the strength of inhomogeneity, i.e., in the contraction ratio of the nozzle, is determined. This approach leads to a discrete set of possible contraction ratios at which exact solutions can be obtained. The Crum’s theorem is used to derive a formula that allows determination of the exact solutions to the inhomogeneous Burgers equation from the solutions to the homogeneous heat transfer equation. It is noted that, in fact, the proposed algorithm of the multiple Darboux transformation makes it possible to decrease the viscosity coefficient of the medium in a discrete way.

Three-dimensional spatiotemporal vector solitary waves

We investigate three-dimensional (3D) spatiotemporal vector solitary waves in spherical coordinates. The exact 3D analytical nonstationary (slowly expanding) solutions are obtained by the separation of variables and the Hirota bilinear method. Novel 3D spatiotemporal vector solitary waves are built with the help of spherical harmonics include multipole and necklace rings.

The dynamics of nonstationary solutions in one-dimensional two-component Bose—Einstein condensates

This paper investigates the dynamical properties of nonstationary solutions in one-dimensional two-component Bose—Einstein condensates. It gives three kinds of stationary solutions to this model and develops a general method of constructing nonstationary solutions. It obtains the unique features about general evolution and soliton evolution of nonstationary solutions in this model.

Gas Flow in Fan-Shaped Networks: Classical Solutions and Feedback Stabilization

We analyze the subcritical gas flow through fan-shaped networks of pipes, that is, through tree-shaped networks with exactly one node where more than two pipes meet. The gas flow in pipe networks is modeled by the isothermal Euler equations, a hyperbolic PDE system of balance laws. For this system we analyze stationary states and classical nonstationary solutions locally around a stationary state on a finite time interval. Furthermore, we present a Lyapunov function and boundary feedback laws to stabilize a fan-shaped network around a given stationary state.

Internal Exponential Stabilization to a Nonstationary Solution for 3D Navier–Stokes Equations

We consider the Navier–Stokes system in a bounded domain with a smooth boundary. Given a sufficiently regular time-dependent global solution, we construct a finite-dimensional feedback control that is supported by a given open set and stabilizes the linearized equation. The proof of this fact is based on a truncated observability inequality, the regularizing property for the linearized equation, and some standard techniques of the optimal control theory. We then show that the control constructed for the linear problem stabilizes locally also the full Navier–Stokes system.

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.

Cosmological solutions in five-dimensional minimal supergravity

We construct new non-stationary cosmological solutions to five-dimensional minimal supergravity that do not have any tri-holomorphic U(1) isometries. Our new solutions, in part, contain some of the previously constructed solutions to the minimal supergravity. The c-function of solutions shows monotonic increasing/decreasing behavior in time, in agreement with the expected behavior of the c-function in spacetimes with a positive cosmological constant.