Nonstationary Solutions Research Articles

Periodic solutions for the N-vortex problem via a superposition principle

We examine the $N$-vortex problem on general domains $\Omega\subset\mathbb{R}^2$ concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the form $$ \Gamma_k\dot{z}_k=J\nabla_{z_k}H(z_1,\ldots,z_N),\quad k=1,\ldots,N, $$ where $\Gamma_k\in\mathbb{R}\setminus\{0\}$ is the strength of the $k$th vortex at position $z_k(t)\in\Omega$, $J\in\mathbb{R}^{2\times 2}$ is the standard symplectic matrix and $$ H(z_1,\ldots,z_N)=-\frac{1}{2\pi}\sum_{\underset{k\neq j}{k,j=1}}^N\Gamma_j\Gamma_k\log|z_k-z_j|-\sum_{k,j=1}^N\Gamma_j\Gamma_k g(z_k,z_j) $$ with some regular and symmetric, but in general not explicitely known function $g:\Omega\times\Omega\rightarrow \mathbb{R}$. The investigation relies on the idea to superpose a stationary solution of a system of less than $N$ vortices and several clusters of vortices that are close to rigidly rotating configurations of the whole-plane system. We establish general conditions on both, the stationary solution and the configurations, under which multiple $T$-periodic solutions are shown to exist for every $T>0$ small enough. The crucial condition holds in generic bounded domains and is explicitely verified for an example in the unit disc $\Omega=B_1(0)$. In particular we therefore obtain various examples of periodic solutions in $B_1(0)$ that are not rigidly rotating configurations.

Periodic Solutions of N-Vortex Type Hamiltonian Systems near the Domain Boundary

The paper deals with the existence of nonstationary collision-free periodic solutions of singular first order Hamiltonian systems of $N$-vortex type in a domain $\Omega\subset\mathbb{C}$. These are solutions $z(t)=(z_1(t),\dots,z_N(t))$ of \[ \dot{z}_j(t)=-i\nabla_{z_j} H_\Omega\big(z(t)\big),\quad j=1,\dots,N, \tag{HS} \] where the Hamiltonian $H_\Omega$ has the form \[ H_\Omega(z_1,\dots,z_N) = -\sum_{{j,k=1}\over{j\ne k}}^N \frac{1}{2\pi}\log|z_j-z_k| -\sum_{j,k=1}^N g(z_j,z_k). \] The function $g:\Omega\times\Omega\to\mathbb{R}$ is required to be of class $C^3$ and symmetric, the regular part of a hydrodynamic Green function being our model. The Hamiltonian is unbounded from above and below, and the associated action integral is not defined on an open subset of the space of periodic $H^{1/2}$ functions. Given a closed connected component $\Gamma\subset\partial\Omega$ of class $C^3$ we are interested in periodic solutions of (HS) near $\Gamma$. We present quite general conditions on the behavior of $g$ near $\Gamma$ which imply that there exists a family of periodic solutions $z^{(r)}(t)$, $0<r<\overline{r}$, with arbitrarily small minimal period $T_r\to0$ as $r\to0$, and such that the "point vortices" $z_j^{(r)}(t)$ approach $\Gamma$ as $r\to0$. The solutions are choreographies, i.e.\ $z_j^{(r)}(t)$ moves on the same trajectory as $z_1^{(r)}(t)$ with a phase shift. We can also relate the speed of each vortex with the curvature of $\Gamma$.

Couette Flow of a Viscoelastic Maxwell-Type Medium with Two Relaxation Times

A Couette flow of a viscoelastic medium is considered that is described by the Johnson–Segalman–Oldroyd model with two relaxation times. The development of singularities related to the appearance of internal discontinuities is studied both analytically and numerically within one-dimensional nonstationary hyperbolic models of viscoelastic Maxwell-type media. A numerical model for calculating nonstationary one-dimensional discontinuous solutions is constructed, discontinuous solutions are studied, and the hysteresis phenomenon, i.e., the dependence of the structure of a steady Couette flow on the prehistory of its formation, is analyzed.

Market Dynamics: On Directional Information Derived from (Time, Execution Price, Shares Traded) Transaction Sequences

A new approach to obtaining market-directional information, based on a non-stationary solution to the dynamic equation future price tends to the value that maximizes the number of shares traded per unit time is presented. In our previous work, we established that it is the share execution flow (𝐼=d𝑉/d𝑡) and not the share trading volume (𝑉) that is the driving force of the market, and that asset prices are much more sensitive to the execution flow 𝐼 (the dynamic impact) than to the traded volume 𝑉 (the regular impact). In this paper, an important advancement is achieved: we define the 𝓟 as the sum of only those price moves that are relevant to dynamics; the criterion of relevance is a high 𝐼. Thus, only follow the market (and not little bounce) events are included in 𝓟. Changes in the scalp-price defined this way indicate a trend change – not a bear rally or a bull sell-off; the approach can be further extended to non-local price change. The software calculating the scalp-price given observations triples (time, execution price, shares traded) is available from the authors.

Stationary and non-stationary solutions of the evolution equation for neutrino in matter

We study solutions of the equation which describes the evolution of a neutrino propagating in dense homogeneous medium in the framework of the quantum field theory. In the two-flavor model the explicit form of Green function is obtained, and as a consequence the dispersion law for a neutrino in matter is derived. Both the solutions describing the stationary states and the spin-flavor coherent states of the neutrino are found. It is shown that the stationary states of the neutrino are different from the mass states, and the wave function of a state with a definite flavor should be constructed as a linear combination of the wave functions of the stationary states with coefficients, which depend on the mixing angle in matter. In the ultra-relativistic limit the wave functions of the spin-flavor coherent states coincide with the solutions of the quasi-classical evolution equation. Quasi-classical approximation of the wave functions of spin-flavor coherent states is used to calculate the probabilities of transitions between neutrino states with definite flavor and helicity.

Boundary stabilisation to non-stationary solutions for deterministic and stochastic parabolic-type equations

ABSTRACTIn this work, we design explicit, finite-dimensional boundary feedback laws for stabilisation to trajectories for parabolic-type equations. The simple form of the feedback allows to write the solution of the corresponding closed-loop equation in a mild formulation via a kernel; then, taking advantage of this, the stability is shown. As an application, null stabilisation for stochastic parabolic-type equations is deduced as well. As far as we know, the present work is the first result on boundary feedback stabilisation to trajectories and for stochastic parabolic-type equations, with stability guaranteed independent of how large the level of the noise is.

Semi-inverse method in nonlinear mechanics: application to couple shell- and beam-type oscillations of single-walled carbon nanotubes

We demonstrate the application of the efficient semi-inverse asymptotic method to resonant interaction of the nonlinear normal modes belonging to different branches of the CNT vibration spectrum. Under condition of the 1:1 resonance of the beam and circumferential flexure modes we obtain the dynamical equations, the solutions of which describe the coupled stationary states. The latter are characterized by the non-uniform distribution of the energy along the circumferential coordinate. The non-stationary solutions for obtained equations correspond to the slow change of the energy distribution. It is shown that adequate description of considered resonance processes can be achieved in the domain variables. They are the linear combinations of the shell- and beam-type normal modes. Using such variables we have analyzed not only nonlinear normal modes but also the limiting phase trajectories describing the strongly non-stationary dynamics. The evolution of the considered resonance processes with the oscillation amplitude growth is analyzed by the phase portrait method and verified by the numerical integration of the respective dynamical equations.

Modeling of Electrochemical Copying in a Finite-Width Cell

The problem of modeling of electrochemical machining is reduced to the solution of the Schwartz problem on a parametrical rectangle with the use of theta-functions. Various conditions (non-equipotentiality of electrodes and inconstancy of current efficiency) at the boundary of a processed surface are considered. Nonstationary, quasistationary, stationary, and limit solutions are studied. Results of machining of surfaces by tool electrodes of various shapes are given. It is shown that machining mode parameters significantly affect the dissolved layer size necessary for obtaining high-precision copying.

Invariant operators and separation of residual stresses

We consider the equations of linear theory of elasticity in stresses for the threedimensional space. Solutions are decomposed into sums of stationary solutions not satisfying the compatibility condition (residual stresses) and nonstationary solutions satisfying the compatibility condition and hence represented through the displacements. The construction of this decomposition is reduced to solving a series of Poisson equations.

Analytical and exact solutions of the spherical and cylindrical diodes of Langmuir–Blodgett law

This paper discloses the exact solutions of a mathematical model that describes the cylindrical and spherical electron current emissions within the context of a physics approximation method. The solution involves analyzing the 1D nonlinear Poisson equation, for the radial component. Although an asymptotic solution has been previously obtained, we present a theoretical solution that satisfies arbitrary boundary conditions. The solution is found in its parametric form (i.e., φ(r)=φ(r(τ))) and is valid when the electric field at the cathode surface is non-zero. Furthermore, the non-stationary spatial solution of the electric potential between the anode and the cathode is also presented. In this work, the particle–beam interface is considered to be at the end of the plasma sheath as described by Sutherland et al. [Phys. Plasmas 12, 033103 2005]. Three regimes of space charge effects—no space charge saturation, space charge limited, and space charge saturation—are also considered.

OpenMP GNU and Intel Fortran programs for solving the time-dependent Gross–Pitaevskii equation

We present Open Multi-Processing (OpenMP) version of Fortran 90 programs for solving the Gross-Pitaevskii (GP) equation for a Bose-Einstein condensate in one, two, and three spatial dimensions, optimized for use with GNU and Intel compilers. We use the split-step Crank-Nicolson algorithm for imaginary- and real-time propagation, which enables efficient calculation of stationary and non-stationary solutions, respectively. The present OpenMP programs are designed for computers with multi-core processors and optimized for compiling with both commercially-licensed Intel Fortran and popular free open-source GNU Fortran compiler. The programs are easy to use and are elaborated with helpful comments for the users. All input parameters are listed at the beginning of each program. Different output files provide physical quantities such as energy, chemical potential, root-mean-square sizes, densities, etc. We also present speedup test results for new versions of the programs.

Paracontrolled calculus and Funaki–Quastel approximation for the KPZ equation

In this paper, we consider the approximating KPZ equation introduced by Funaki and Quastel (2015), which is suitable for studying invariant measures. They showed that the stationary solution of the approximating equation converges to the Cole–Hopf solution of the KPZ equation with extra term 124t. On the other hand, Gubinelli and Perkowski (2017) gave a pathwise meaning to the KPZ equation as an application of the paracontrolled calculus. We show that Funaki and Quastel’s result is extended to nonstationary solutions by using the paracontrolled calculus.

Stable Vortex Clusters in Two-Dimensional Bose-Einstein Condensates with Spatially Modulated Defocusing Nonlinearity

We investigate the vortex dynamic behavior of Bose-Einstein condensate in a spatially two-dimensional harmonic potential with spatially modulated defocusing nonlinearity. Kinds of exact vortex solutions of the Gross-Pitaevskii equation are obtained by designing various laser potentials which can be experimentally realized. In particular, we demonstrate that the spatially modulated defocusing nonlinearity supports stable vortex clusters. For the stationary vortex solutions, the vortex clusters are dynamically stable, and the circulation direction of each vortex and vortex core positions remain unchanged. However, for the non-stationary vortex solutions, stable and unstable vortex clusters exist. For the stable vortex clusters, the vortex core positions are constant during dynamical evolution, while the phase increase directions vary periodically. Additionally, corresponding to the unstable vortex clusters, vortices disappear and appear periodically in the condensate.

Exact helicoidal and catenoidal solutions in five- and higher-dimensional Einstein-Maxwell theory

We present several new exact solutions in five and higher dimensional Einstein-Maxwell theory by embedding the Nutku instanton. The metric functions for the five-dimensional solutions depend only on a radial coordinate and on two spatial coordinates for the six and higher dimensional solutions. The six and higher dimensional metric functions are convoluted-like integrals of two special functions. We find that the solutions are regular almost everywhere and some spatial sections of the solution describe wormhole handles. We also find a class of exact and nonstationary convoluted-like solutions to the Einstein-Maxwell theory with a cosmological constant.

Exact solutions to Einstein–Maxwell theory on Eguchi–Hanson space

In this paper, we construct explicit analytical exact solutions to the six and higher-dimensional Einstein–Maxwell theory. In all solutions, a subspace of the metric is the Eguchi–Hanson space where the metric functions are completely determined in terms of known analytical functions. Moreover, we find the solutions can be extended from nonstationary exact solutions to Einstein–Maxwell theory with cosmological constant. We show that the solutions are asymptotically expanding patches of de Sitter space–time.

Exact Solutions for Stationary and Unsteady Layered Convection of a Viscous Incompressible Fluid with the Specified Velocities at the Bottom

The layered convective flow of a viscous incompressible fluid is considered with the specified velocities at the bottom of an infinite layer. A new exact stationary and nonstationary solution of the Oberbeck-Boussinesq system is presented. The account of fluid velocity at the bottom is characterized by the presence of two stagnant points, this being indicative of the nonmonotonic kinetic energy profile with two local extrema.

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

Parametric Excitation of a Microbeam-String With Asymmetric Electrodes: Multimode Dynamics and the Effect of Nonlinear Damping

The dynamic motion of a parametrically excited microbeam-string affected by nonlinear damping is considered asymptotically and numerically. It is assumed that the geometrically nonlinear beam-string, subject to only modulated alternating current voltage, is closer to one of the electrodes, thus resulting in an asymmetric dual gap configuration. A consequence of these novel assumptions is a combined parametric and hard excitation in the derived continuum-based model that incorporates both linear viscous and nonlinear viscoelastic damping terms. To understand how these assumptions influence the beam's performance, the conditions that lead to both principal parametric resonance and a three-to-one internal resonance are investigated. Such conditions are derived analytically from a reduced-order nonlinear model for the first three modes of the microbeam-string using the asymptotic multiple-scales method which requires reconstitution of the slow-scale evolution equations to deduce an approximate spatio-temporal solution. The response is investigated analytically and numerically and reveals a bifurcation structure that includes coexisting in-phase and out-of-phase solutions, Hopf bifurcations, and conditions for the loss of orbital stability culminating with nonstationary quasi-periodic solutions and chaotic strange attractors.

Extremal charged Vaidya and its near horizon geometry

Recently [Formula: see text]-dimensional spherically symmetric charged Vaidya black hole solution has been constructed. We observe that this nonstationary solution admits extremal limit and study its near horizon geometry. We show that the symmetry of the near horizon geometry is [Formula: see text]. Our analysis shows that the theorems for the near horizon geometry of stationary extremal black holes, may be extended to nonstationary cases.

Initial boundary value problem for the radiative transfer equation with diffusion matching conditions

Under study is the Cauchy problemfor the nonstationary radiative transfer equation with generalized matching conditions that describes the diffuse reflection and refraction on the interface. The solvability of the initial-boundary value problem is proved. Some stabilization conditions for the nonstationary solution are obtained.