Set Of Equations For Amplitudes Research Articles

Dark states and vortex solitary wave with finite energy of Maxwell-Dirac nonlinear equations

Starting from the Maxwell’s equations for media with no-stationary linear and nonlinear polarization, we obtain a set of nonlinear Maxwell amplitude equations in the approximation of the first order of dispersion. After a special kind of complex presentation, the set of amplitude equations is written as a set of nonlinear Dirac equations. For broad-band pulses solitary solutions with half-integer spin and finite energy are found. The solutions correspond to an electromagnetic wave with circular Poynting vector and zero divergence. These invisible for the detectors waves are called dark states and the localized energy is determinates as electromagnetic mass.

Long-wave Marangoni convection in a thin film heated from below

We consider long-wave Marangoni convection in a liquid layer atop a substrate of low thermal conductivity, heated from below. We demonstrate that the critical perturbations are materialized at the wave number K ∼ √Bi, where Bi is the Biot number which characterizes the weak heat flux from the free surface. In addition to the conventional monotonic mode, a novel oscillatory mode is found. Applying the K ∼ √Bi scaling, we derive a new set of amplitude equations. Pattern selection on square and hexagonal lattices shows that supercritical branching is possible. A large variety of stable patterns is found for both modes of instability. Finite-amplitude one-dimensional solutions of the set, corresponding to either steady or traveling rolls, are studied numerically; a complicated sequence of bifurcations is found in the former case. The emergence of an oscillatory mode in the case of heating from below and stable patterns with finite-amplitude surface deformation are shown in this system for the first time.

Three-dimensional oscillatory long-wave Marangoni convection in a binary liquid layer with the Soret effect: Bifurcation analysis

Three-dimensional long-wave oscillatory Marangoni convection in a thin layer of binary mixture with a nondeformable interface is investigated in the presence of the Soret effect. Both thermocapillary and solutocapillary effects are taken into account. A set of amplitude equations is obtained and studied analytically near the critical value of the Marangoni number. It is shown that alternating rolls (either rhombic or square) are selected and they bifurcate supercritically. Subcritical bifurcation takes place only for alternating rolls on a square lattice in a narrow range of parameters.

Modulational instability of two pairs of counter-propagating waves and energy exchange in a two-component system

The modulational instability of two pairs of plane counter-propagating waves in a two-component system is considered within the framework of two coupled Sine-Gordon equations. The emphasis is on the generic case when the system is not integrable, and the group velocities of each pair of waves are arbitrary and usually different from each other. We consider the dynamics of weakly nonlinear wave packets, and using an asymptotic multiple-scales expansion we obtain several sets of evolution equations describing the two components of the system. Depending on the wave packet length scale vis-a-vis the wave amplitude scale, these evolution equations are either four non-dispersive and nonlinearly coupled envelope equations, or four non-locally coupled nonlinear Schrödinger equations. We also consider a set of fully coupled nonlinear Schrödinger equations, even though in general this system contains small dispersive terms which are strictly beyond the leading order of the asymptotic multiple-scales expansion procedure. Although each set of amplitude equations can be used to describe a range of dynamical phenomena, we focus here on stability of plane-wave solutions, and show that they may be modulationally unstable. We study these instabilities in the context of solutions exhibiting an energy exchange between the two physical components of the system. We also perform numerical simulations of the original unapproximated coupled Sine-Gordon equations to assess the accuracy of the instability predictions derived from the various asymptotic models, and consider the range of validity of each asymptotic model. We show that the instabilities can lead to the formation of localized structures, and to a modification of the linear energy exchange, which then continues for some time into the nonlinear regime as an energy exchange between these localized structures. When the system is close to being integrable, the time evolution is distinguished by a remarkable almost periodic sequence of energy exchange scenarios, with spatial patterns alternating between approximately uniform wavetrains and localized structures, which invites further theoretical study of this special case.

Monotonic and oscillatory free-convective instability of solution in the space between two plane horizontal electrodes: Solutions containing three types of ions

The conditions for the onset of natural convection (convective instability) in the electrochemical system consisting of two plane horizontal electrodes and solution containing three types of ions between them were analyzed theoretically. The equations of flow of incompressible viscous liquid to the Boussinesq approximation and the equations of material balance with the electroneutrality condition were used as the mathematical model. The set of equations for amplitudes of perturbations was solved using the Galerkin method. The dependences of critical Rayleigh number on the system parameters are obtained for monotonic and oscillatory instabilities in the case that the electrochemical reaction proceeds under the conditions of limiting current. The examples of instability diagrams are presented. In contrast to the binary electrolyte, for a solution containing three types of ions, the critical Rayleigh number depends on the charges of ions and the ratio between the diffusion coefficients of ions in the solution.

Hexagonal patterns in multistep optical parametric processes

The existence and competition of a novel class of hexagonal patterns in a nonlinear optical system are reported. These states are found in a mean-field model of a doubly resonant frequency divide-by-3 optical parametric oscillator (3omega?2omega+omega) in which the multistep parametric process, 2omega=omega+omega , is weakly phase matched. A generalized Swift-Hohenberg equation and a set of amplitude equations are derived to describe the coexistence of hexagonal patterns formed by the superposition of either three or six phase-locked tilted waves.

Chaotic alternation of waves in ring lasers.

We intend to give an analytic description of the mechanisms involved in the periodic and chaotic wave alternation frequently observed in ring lasers. A set of amplitude equations is derived from the Maxwell-Bloch equations. These equations are studied analytically and numerically.

Resonant Interactions of Dean Vortices in a Curved Channel Flow

Stability of the flow in a curved channel is investigated with the Dean approximation by using the weakly nonlinear stability theory. A set of amplitude equations for the Dean vortices of wavenumber ratio 1:2 is derived. The amplitude equations describe the resonant interactions between the fundamental mode and the higher harmonic. The equilibrium amplitudes of the Dean vortices are evaluated from the set of amplitude equations numerically. It is concluded that the phenomena of the coalescence of two vortices and the splitting of one large vortex into two are explained from the 1:2 resonance.