System Of Equations For Amplitudes Research Articles

New vector type optical vortex structures in isotropic media

In the present work, the formation of vector-type vortex structures in isotropic media is studied. These vortices exist in the electric field components and generate a depolarization in the vector field of the laser spot. Their behavior is governed by a system of amplitude equations. Additionally, new solutions have been found that describe similar types of vortex structures in an optical fiber. The optical vortices admit only amplitude-type singularities. The vortex structures observed within the vector amplitude function have ring-like patterns. The variation of the intensities of the x and y components of the amplitude function along the axis of the fiber is investigated.

Calibration of Siberian Radioheliograph antenna gains using redundancy

The paper describes application of standard gain calibration using redundancy for a 48-antenna prototype of Siberian Radioheliograph. Traditionally, for calibration, the visibilities were measured only between adjacent antennas since they have the highest signal-to-noise ratio and are sufficient for phase calibration. We have shown that this limited set of visibilities did not allow using the antenna array redundancy potential and obtaining images with a high dynamic range on a permanent basis. Images without amplitude calibration contain many artifacts and require special care when analyzed. The inclusion of visibility measurement between antennas with a double step made it possible to significantly increase the accuracy of solving the system of equations for amplitudes. Images constructed using both phase and amplitude calibrations do not have visible artifacts and are more reliable.

Калибровка амплитуд коэффициентов передачи антенн Сибирского радиогелиографа с использованием избыточности

The paper describes application of standard gain calibration using redundancy for a 48-antenna prototype of Siberian Radioheliograph. Traditionally, for calibration, the visibilities were measured only between adjacent antennas since they have the highest signal-to-noise ratio and are sufficient for phase calibration. We have shown that this limited set of visibilities did not allow using the antenna array redundancy potential and obtaining images with a high dynamic range on a permanent basis. Images without amplitude calibration contain many artifacts and require special care when analyzed. The inclusion of visibility measurement between antennas with a double step made it possible to significantly increase the accuracy of solving the system of equations for amplitudes. Images constructed using both phase and amplitude calibrations do not have visible artifacts and are more reliable.

Influence of the interface internal energy on monotone disturbances of a creeping stationary flow with a velocity field of the Hiemenz type

The stability of a creeping stationary flow, which arises as a result of the interface internal energy, is studied. It is shown that the system of amplitude equations allows exact integration. Complex decrement as a solution to the transcendental equation is found. It was found that long-wave perturbations decay with increasing time. Neutral curves for the transformer oil - formic acid system were constructed.

Radiation of a Charge Intersecting a Boundary Between a Bilayer Area and a Homogeneous One in a Circular Waveguide

This paper is devoted to the analysis of electromagnetic field excited by a charge moving in a circular waveguide, which consists of two semi-infinite parts. The charge moves with a constant velocity along the waveguide axis from the partially dielectric section into the homogeneous one. It is assumed that Cherenkov radiation is generated in the bilayer section. The total field is represented as a sum of a known forced field that is the field of the charge in the infinite regular waveguide and a free field resulting from the presence of the transverse boundary. The infinite system of algebraic equations for amplitudes of the free-field modes is derived. The further analysis is conducted using numerical calculations. Special attention is given to the study of the so-called Cherenkov-transition radiation (CTR) within the homogeneous part of the waveguide volume. Typical distributions of CTR over frequencies and modes are presented for the case when the channel and the homogeneous area are free of medium. Analytical results obtained for the point charge are generalized for the charged bunch with an arbitrary profile and verified using direct simulations for the case of Gaussian bunch.

On the Nonlinear Dynamical System of Amplitude Equations Corresponding to Intersections of Bifurcations in the Flow Between Permeable Cylinders with Radial and Axial Flows

In the present work, we study various modes that arise after a circular Couette flow loses its stability in the presence of an axial flow due to a constant axial pressure gradient and a radial flow through the permeable walls of the cylinders. The basic object of our investigation is a nonlinear system of amplitude equations describing multiple flow bifurcations between permeable cylinders. Different bifurcations will be investigated theoretically with application of numerical methods.

Non-one-dimensional optical solitons and ideal fluid dynamics

Influence of transverse effects on the propagation of light beams in gradient glass fiber and on the pulsed mode of the second harmonic generation is investigated. Within the approaches under consideration the equations for the pulse or beam field envelopes are reduced to a system of hydrodynamictype equations for amplitudes and eikonals. These equations are used to describe the vortex and non-vortex beam channeling modes and the propagation of light bullets.

On the theory of longitudinally inhomogeneous waveguides with impedance walls

The Maxwell equations for longitudinally inhomogeneous (irregular) waveguides with impedance walls are reduced equivalently to a system of ordinary differential equations by applying transverse expansions. The obtained system of equations for amplitudes of the coupled modes is used to analyze gyrotron cavities. Specifically, the effect of mode conversion and ohmic losses on the eigenfrequencies, quality factors, and field distributions in the gyrotron cavities operating in the submillimeter range is investigated.

The effect of spin fluctuations on phase transitions in magnetic systems

In this paper we study the problem of admitting an exact solution of the effect of spin fluctuations on phase transition when modeling a multiferroic superconducting system in a strong magnetic field. New results are obtained for phase portraits of the system of equations for amplitudes of spin density and temperature waves. The possibility is justified for the transition of the system to a phase in which superconductivity and antiferromagnetic ordering coexist, particularly via slowly fluctuating spin-density waves.

Large-scale Marangoni convection in a liquid layer with insoluble surfactant of low concentration

We derive a system of amplitude equations describing the evolution of a large-scale Marangoni patterns in a liquid layer with poorly conducting boundaries in the presence of a small amount of an insoluble surfactant on the free flat interface. The presence of quadratic nonlinear terms in the amplitude equation leads to the selection of hexagonal patterns. The type of hexagons bifurcating into the subcritical region, depends on the parameters of the system.

Stabilization of a linearly unstable wave participating in a three-wave resonant interaction

An analytical study of the influence of three-wave resonant interactions on the evolution of unstable wave disturbances is presented in the Kelvin-Helmholtz model. These results may be of interest in analyzing the dynamics of disturbances at the ocean-atmosphere interface and in two-layer flows which arise in the ocean and are characterized by large gradients of flow velocity at the boundary of layers. In the case under consideration, the instability arises when eigenfrequencies coincide in the framework of a single mode and the instability is algebraic. The amplitudes of the two other interacting stable waves are assumed to be small compared to the amplitude of the third, unstable, mode. The system of amplitude equations for this case is investigated using the WKB method. As a result, we obtain the formulas coupling the solutions for the time before and after a transition through a singular point, where the amplitude of the linearly unstable wave has a local minimum. These formulas give the rule of transformation of the parameter that characterizes a phase shift between fast and slow modes and determines the behavior of the system. It is shown that, in a transition through a singular point, this parameter changes randomly. As long as the parameter is positive, the amplitude of the linearly unstable wave remains limited and oscillates stochastically. In a transition of the parameter through zero, we exit the stabilization region and have an infinite growth of amplitude. The transition into the instability region is random. However, if the time interval where the amplitude remains limited is large enough, the scenario of the behavior of the system we have obtained can be treated as the partial stabilization of instability. The results make it possible for us to investigate the stochasticity caused by the nonlinear interaction of gravity-capillary waves in a two-layer model of a shear flow. These results are also of interest in analyzing secondary flows in laboratory facilities modeling the ocean and atmospheric processes.

Resonance regimes in the neighborhood of bifurcation points of codimension 2 in the Couette-Taylor problem

The bifurcation in a dynamical system with cylindrical symmetry dependent on several parameters is studied with reference to the Couette-Taylor problem. Points at which two neutral curves intersect (bifurcation points of codimension 2) corresponding to several independent neutral modes are found. In the neighborhood of the bifurcation points of codimension 2 the interaction of these modes can be described by a system of amplitude equations on the central manifold. If the neutral modes are nonrotationally symmetrical, there exist seven different resonance states that influence the cubic terms of the amplitude system. For the resonances Res 0 and Res 3 the results of calculating the intersection points are presented and the conditions under which stationary regimes exist and are stable are analyzed.

Effect of spin fluctuations on the superconducting phase of Hubbard fermions in the t-t′-t″-J* model

The effect of spin-fluctuation scattering processes on the region of the superconducting phase in strongly correlated electrons (Hubbard fermions) is investigated by the diagram technique for Hubbard operators. Modified Gor’kov equations in the form of an infinitely large system of integral equations are derived taking into account contributions of anomalous components \( P_{0\sigma ,\bar \sigma 0} \) of strength operator \( \hat P \). It is shown that spinfluctuation scattering processes in the one-loop approximation for the t-t′-t″-J* model taking into account long-range hoppings and three-center interactions are reflected by normal (P0σ, 0σ) and anomalous (\( P_{0\sigma ,\bar \sigma 0} \)) components of the strength operator. Three-center interactions result in different renormalizations of the kernels of the integral equations for the superconducting d phase in the expressions for the self-energy and strength operators. In this approximation for the d-type symmetry of the order parameter for the superconducting phase, the system of integral equations is reduced to a system of nonhomogeneous equations for amplitudes. The resultant dependences of critical temperature on the electron concentrations show that joint effect of long-range hoppings, three-center interactions, and spin-fluctuation processes leads to strong renormalization of the superconducting phase region.

Observation of intermittent states and nonlinear wave–wave interaction in neon glow discharges

In the plasma of a neon glow discharge under certain conditions two waves (s- and p-wave) are known to exist simultaneously. The transition region between these two kinds of waves is experimentally investigated in the space of plasma parameters. An intermittent behaviour of s- and p-waves dynamics is proved by the analysis of the recorded spatio-temporal patterns of light emitted by plasma. The experimental data are analyzed with the biorthogonal decomposition (BOD) and the empirical mode decomposition (EMD), respectively. The wave states can be described by means of a coupled system of amplitude equations resulting from a hydrodynamic plasma model. Comparison of experimental data and theoretical predictions shows satisfactory agreement. The discrepancies are discussed.

Instability and localisation of patterns due to a conserved quantity

We describe the influence of a conserved quantity on the stability properties of roll, square and hexagonal patterns near stationary onset. The appropriate systems of amplitude equations are analysed: they comprise equations for the amplitudes of the pattern modes, together with an additional coupled equation governing the evolution of a large-scale mode associated with the conserved quantity. We find that the presence of this large-scale mode may result in the destabilisation of all regular roll, square or hexagonal patterns, leading to amplitude modulation and strong localisation of the pattern. Our analytical results are complemented throughout by numerical simulations of a model partial differential equation, to illustrate the modulational instabilities and their nonlinear development.

Criteria of rigid instability on a flat surface of dielectric fluid in external electric field

Subcritical behavior of a free surface of the ideal dielectric liquid in an external electric field has been theoretically studied. A system of amplitude equations is obtained, which describe nonlinear interaction of three standing waves forming a hexagonal structure. Within the framework of a model described by these equations, a sufficient integral criteria are formulated for the rigid instability excitation on the flat surface of the liquid medium.

LARGE-SCALE DISTORTIONS OF SQUARE PATTERNS

Stationary square patterns are typical in several instability problems. Near the instability threshold, the evolution of long-wave disturbances can be described by a system of amplitude equations resembling the Newell-Whitehead-Segel equations. These equations are used for the linear stability analysis and the investigation of the defects.

Nonlinear stability of a viscous axisymmetric jet

The nonlinear temporal stability of a viscous axisymmetric jet is studied with respect to axisymmetric disturbances. The weakly nonlinear Stuart Watson perturbation expansion is used to study the self interaction of the fundamental in a neighborhood of the critical Reynolds number (Rc). The Landau constant is positive indicating there is a supercritical Hopf bifurcation. In order to extend these results pastR∼Rc, a severely truncated Fourier modal expansion using the Stuart Watson functions to represent ther dependence, exponentials inz and unknown amplitudes in time is substituted into the Navier-Stokes equation. A projection onto an appropriate subspace leads to a low dimensional (five) system of amplitude equations for the disturbance of the fundamental, harmonic and distortion to the mean flow.

Double-diffusive convection in a circular cylinder

A theoretical study is made of the onset of steady double-diffusive convection in a circular cylinder of small to moderate aspect ratio. An eigenfunction expansion method is used to derive systems of amplitude equations for weakly nonlinear evolution of critical disturbances. It is shown that the nature of the convective solution near criticality depends strongly on the cylinder's aspect ratio; this is particularly the case at or near aspect ratios where two modes become unstable at the same Rayleigh number. There is also a strong dependence on Prandtl number, which is discussed.