Escape times for rigid Brownian rotators in a bistable potential from the time evolution of the Green function and the characteristic time of the probability evolution

The greatest relaxation time for an assembly of three- dimensional rigid rotators in an axially symmetric bistable potential is obtained exactly in terms of continued fractions as a sum of the zero frequency decay functions (averages of the Legendre polynomials) of the system. This is accomplished by studying the entire time evolution of the Green function (transition probability) by expanding the time dependent distribution as a Fourier series and proceeding to the zero frequency limit of the Laplace transform of that distribution. The procedure is entirely analogous to the calculation of the characteristic time of the probability evolution (the integral of the configuration space probability density function with respect to the position co-ordinate) for a particle undergoing translational diffusion in a potential; a concept originally used by Malakhov and Pankratov (Physica A 229 (1996) 109). This procedure allowed them to obtain exact solutions of the Kramers one-dimensional translational escape rate problem for piecewise parabolic potentials. The solution was accomplished by posing the problem in terms of the appropriate Sturm-Liouville equation which could be solved in terms of the parabolic cylinder functions. The method (as applied to rotational problems and posed in terms of recurrence relations for the decay functions, i.e., the Brinkman approach c.f. Blomberg, Physica A 86 (1977) 49, as opposed to the Sturm-Liouville one) demonstrates clearly that the greatest relaxation time unlike the integral relaxation time which is governed by a single decay function (albeit coupled to all the others in non-linear fashion via the underlying recurrence relation) is governed by a sum of decay functions. The method is easily generalized to multidimensional state spaces by matrix continued fraction methods allowing one to treat non-axially symmetric potentials, where the distribution function is governed by two state variables. (C) 2001 Elsevier Science B.V. All rights reserved.

Instability and evolution of nonlinearly interacting water waves

We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schrodinger equations. We derive a nonlinear dispersion relation. The latter is numerically analyzed to obtain the regions and the associated growth rates of the modulational instability. Furthermore, we follow the long term evolution of the latter by means of computer simulations of the governing nonlinear equations and demonstrate the formation of localized coherent wave envelopes. Our results should be useful for understanding the formation and nonlinear propagation characteristics of large-amplitude freak waves in deep water.

Evolution of elastic x-ray scattering in laser-shocked warm dense lithium

We have studied the dynamics of warm dense Li with near-elastic x-ray scattering. Li foils were heated and compressed using shock waves driven by 4-ns-long laser pulses. Separate 1-ns-long laser pulses were used to generate a bright source of 2.96 keV Cl Ly-alpha photons for x-ray scattering, and the spectrum of scattered photons was recorded at a scattering angle of 120 degrees using a highly oriented pyrolytic graphite crystal operated in the von Hamos geometry. A variable delay between the heater and backlighter laser beams measured the scattering time evolution. Comparison with radiation-hydrodynamics simulations shows that the plasma is highly coupled during the first several nanoseconds, then relaxes to a moderate coupling state at later times. Near-elastic scattering amplitudes have been successfully simulated using the screened one-component plasma model. Our main finding is that the near-elastic scattering amplitudes are quite sensitive to the mean ionization state Z and by extension to the choice of ionization model in the radiation-hydrodynamics simulations used to predict plasma properties within the shocked Li.

Weakly nonlinear vertical dust grain oscillations in dusty plasma crystals in the presence of a magnetic field

The weakly nonlinear regime of transverse paramagnetic dust grain oscillations in dusty (complex) plasma crystals is discussed. The nonlinearity, which is related to the sheath electric/magnetic field(s) and to the intergrain (electrostatic/magnetic dipole) interactions, is shown to lead to the generation of phase harmonics and, in the case of propagating transverse dust-lattice modes, to the modulational instability of the carrier wave due to self-interaction. The stability profile depends explicitly on the form of the electric and magnetic fields in the plasma sheath. The long term evolution of the modulated wave packet, which is described by a nonlinear Schrodinger-type equation, may lead to propagating localized envelope structures whose exact forms are presented and discussed. Explicit suggestions for experimental investigations are put forward. (C) 2004 American Institute of Physics.

Complete nonlinear theory of longitudinal-to-transverse dust lattice mode coupling in a single-layer dusty plasma crystal

A comprehensive nonlinear model is put forward for coupled longitudinal to transverse displacements in a horizontal dust mono-layer, levitated under the combined influence of gravity and an electric and/or magnetic sheath field. A set of coupled nonlinear evolution equations are obtained in a discrete description, and a pair of coupled (Boussinesq-like) PDEs are obtained in the continuum approximation. Finally, the amplitude modulation of the coupled modes is discussed, pointing out the importance of the coupling. All these results are generic, i.e. valid for any assumed form of the inter-grain interaction potential U and the sheath potential Phi.

Evolution of linearly polarized electromagnetic pulses in laser plasmas

An analytical and numerical investigation is presented of the behavior of a linearly polarized electromagnetic pulse as it propagates through a plasma. Considering a weakly relativistic regime, the system of one-dimensional fluid-Maxwell equations is reduced to a generalized nonlinear Schrodinger type equation, which is solved numerically using a split step Fourier method. The spatio-temporal evolution of an electromagnetic pulse is investigated. The evolution of the envelope amplitude of density harmonics is also studied. An electromagnetic pulse propagating through the plasma tends to broaden due to dispersion, while the nonlinear frequency shift is observed to slow down the pulse at a speed lower than the group velocity. Such nonlinear effects are more important for higher density plasmas. The pulse broadening factor is calculated numerically, and is shown to be related to the background plasma density. In particular, the broadening effect appears to be stronger for dense plasmas. The relation to existing results on electromagnetic pulses in laser plasmas is discussed. (c) 2008 American Institute of Physics.

Spatiotemporal evolution of high-power relativistic laser pulses in electron-positron-ion plasmas

The spatiotemporal pulse dynamics of a high-power relativistic laser pulse interacting with an electron-positron-ion plasmas is investigated theoretically and numerically. The occurrence of pulse compression is studied. The dependence of the mechanism on the concentration of the background ions in electron positron plasma is emphasized.

Nonlinear Dynamics of Rotating Multi-Component Pair Plasmas and e-p-i Plasmas

The propagation of small amplitude stationary profile nonlinear electrostatic excitations in a pair plasma is investigated, mainly drawing inspiration from experiments on fullerene pair-ion plasmas. Two distinct pair ion species are considered of opposite polarity and same mass, in addition to a massive charged background species, which is assumed to be stationary, given the frequency scale of interest. In the pair-ion context, the third species is thought of as a background defect (e.g. charged dust) component. On the other hand, the model also applies formally to electron-positron-ion (e-p-i) plasmas, if one neglects electron-positron annihilation. A two-fluid plasma model is employed, incorporating both Lorentz and Coriolis forces, thus taking into account the interplay between the gyroscopic (Larmor) frequency ?c and the (intrinsic) plasma rotation frequency O0. By employing a multi-dimensional reductive perturbation technique, a Zakharov-Kuznetsov (ZK) type equation is derived for the evolution of the electric potential perturbation. Assuming an arbitrary direction of propagation, with respect to the magnetic field, we derive the exact form of nonlinear solutions, and study their characteristics. A parametric analysis is carried out, as regards the effect of the dusty plasma composition (background number density), species temperature(s) and the relative strength of rotation to Larmor frequencies. It is shown that the Larmor and mechanical rotation affect the pulse dynamics via a parallel-to-transverse mode coupling diffusion term, which in fact diverges at ?c ? ±2O0. Pulses collapse at this limit, as nonlinearity fails to balance dispersion. The analysis is complemented by investigating critical plasma compositions, in fact near-symmetric (T- ˜ T+) “pure” (n- ˜ n+) pair plasmas, i.e. when the concentration of the 3rd background species is negligible, case in which the (quadratic) nonlinearity vanishes, so one needs to resort to higher order nonlinear theory. A modified ZK equation is derived and analyzed. Our results are of relevance in pair-ion (fullerene) experiments and also potentially in astrophysical environments, e.g. in pulsars.

Nonlinear electrostatic excitations of charged dust in degenerate ultra-dense quantum dusty plasmas

The linear and nonlinear properties of low-frequency electrostatic excitations of charged dust particles (or defects) in a dense collisionless, unmagnetized Thomas-Fermi plasma are investigated. A fully ionized three-component model plasma consisting of electrons, ions, and negatively charged massive dust grains is considered. Electrons and ions are assumed to be in a degenerate quantum state, obeying the Thomas-Fermi density distribution, whereas the inertial dust component is described by a set of classical fluid equations. Considering large-amplitude stationary profile travelling-waves in a moving reference frame, the fluid evolution equations are reduced to a pseudo-energy-balance equation, involving a Sagdeev-type potential function. The analysis describes the dynamics of supersonic dust-acoustic solitary waves in Thomas-Fermi plasmas, and provides exact predictions for their dynamical characteristics, whose dependence on relevant parameters (namely, the ion-to-electron Fermi temperature ratio, and the dust concentration) is investigated. An alternative route is also adopted, by assuming weakly varying small-amplitude disturbances off equilibrium, and then adopting a multiscale perturbation technique to derive a Korteweg–de Vries equation for the electrostatic potential, and finally solving in terms for electric potential pulses (electrostatic solitons). A critical comparison between the two methods reveals that they agree exactly in the small-amplitude, weakly superacoustic limit. The dust concentration (Havnes) parameter h = Zd0nd0/ne0 affects the propagation characteristics by modifying the phase speed, as well as the electron/ion Fermi temperatures. Our results aim at elucidating the characteristics of electrostatic excitations in dust-contaminated dense plasmas, e.g., in metallic electronic devices, and also arguably in supernova environments, where charged dust defects may occur in the quantum plasma regime.

Nonlinear wavepackets in pair-ion and electron-positron-ion plasmas

The occurrence of amplitude-modulated electrostatic and electromagnetic
wavepackets in pair plasmas is investigated. A static additional charged background species is considered, accounting for dust defects or for heavy ion
presence in the background. Relying on a two-fluid description, a nonlinear
Schrodinger type evolution equation is obtained and analyzed, in terms of the
slow dynamics of the wave amplitude. Exact envelope excitations are obtained,
modelling envelope pulses or holes, and their characteristics are discussed.

Nonlinear Modulated Envelope Electrostatic Wavepacket Propagation in Plasmas

A brief review of the occurrence of amplitude modulated structures in space and laboratory plasmas is provided, followed by a theoretical analysis of the mechanism of carrier wave (self-) interaction, with respect to electrostatic plasma modes. A generic collisionless unmagnetized fluid model is employed. Both cold-(zero-temperature) and warm-(finite temperature) fluid descriptions are considered and compared. The weakly nonlinear oscillation regime is investigated by applying a multiple scale (reductive perturbation) technique and a Nonlinear Schrödinger Equation (NLSE) is obtained, describing the evolution of the slowly varying wave amplitude in time and space. The amplitude’s stability profile reveals the possibility of modulational instability to occur under the influence of external perturbations. The NLSE admits exact localized envelope (solitary wave) solutions of bright (pulses) or dark (holes, voids) type, whose characteristics depend on intrinsic plasma parameters. The role of perturbation obliqueness (with respect to the propagation direction), finite temperature and — possibly — defect (dust) concentration is explicitly considered. The relevance of this description with respect to known electron-ion (e-i) as well as dusty (complex) plasma modes is briefly discussed. © 2004 American Institute of Physics

Nonlinear dynamics of multidimensional electrostatic excitations in nonthermal plasmas

Electrostatic solitary waves in plasmas are the focus of many current studies of localized electrostatic disturbances in both laboratory and astrophysical plasmas. Motivated by recent experimental observations, in which electrostatic solitary structures were detected in laser-plasma experiments, we have undertaken an investigation of the nonlinear dynamics of plasma evolving in two dimensions, in the presence of excess superthermal background electrons. We investigate the effect of a magnetic field on weakly nonlinear ion-acoustic waves. Deviation from the Maxwellian distribution is effectively modelled by the kappa model. A linear dispersion relation is derived, and a decrease in frequency and phase speed in both parallel and perpendicular modes can be seen, which is due to excess superthermal electrons, and which is stronger in the upper mode, and hardly noticeable in the lower (acoustic) mode. We show that ion-acoustic solitary waves can be generated during the nonlinear evolution of a plasma fluid, and their nonlinear propagation is governed by a Zakharov-Kuznetsov (ZK) type equation. A multiple scales perturbation technique is used to derive the ZK equation. Shock excitations can be produced if we allow for dissipation in the model, resulting in a Zakharov-Kuznetsov Burgers type equation. Different types of shock solutions and solitary waves are obtained, depending on the relation between the system parameters, and the effect of these on electrostatic shock structures is investigated numerically. A parametric investigation is conducted into the role of plasma nonthermality and magnetic field strength. © 2013 IOP Publishing Ltd.

Effect of magnetic bias on Gaussian pulse scattering by stacked nonlinear semiconductor layers

The nonlinear scattering of pulses by periodic stacks of semiconductor layers with magnetic bias has been studied in the self-consistent problem formulation, taking into account mobility of carriers. The three-wave mixing technique has been applied to the analysis of the waveform evolution in the stacks illuminated by two Gaussian pulses with different central frequencies and lengths. The effects of external magnetic bias, and stack physical and geometrical parameters on the properties of the scattered waveforms are discussed. © 2013 IEEE.

Modeling relativistic soliton interactions in overdense plasmas: A perturbed nonlinear Schrodinger equation framework

We investigate the dynamics of localized solutions of the relativistic cold-fluid plasma model in the small but finite amplitude limit, for slightly overcritical plasma density. Adopting a multiple scale analysis, we derive a perturbed nonlinear Schrodinger equation that describes the evolution of the envelope of circularly polarized electromagnetic field. Retaining terms up to fifth order in the small perturbation parameter, we derive a self-consistent framework for the description of the plasma response in the presence of localized electromagnetic field. The formalism is applied to standing electromagnetic soliton interactions and the results are validated by simulations of the full cold-fluid model. To lowest order, a cubic nonlinear Schrodinger equation with a focusing nonlinearity is recovered. Classical quasiparticle theory is used to obtain analytical estimates for the collision time and minimum distance of approach between solitons. For larger soliton amplitudes the inclusion of the fifth-order terms is essential for a qualitatively correct description of soliton interactions. The defocusing quintic nonlinearity leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude

Differential evolution combined with clonal selection for dynamic economic dispatch

Dynamic economic load dispatch (DELD) is one of the most important steps in power system operation. Various optimisation algorithms for solving the problem have been developed; however, due to the non-convex characteristics and large dimensionality of the problem, it is necessary to explore new methods to further improve the dispatch results and minimise the costs. This article proposes a hybrid differential evolution (DE) algorithm, namely clonal selection-based differential evolution (CSDE), to solve the problem. CSDE is an artificial intelligence technique that can be applied to complex optimisation problems which are for example nonlinear, large scale, non-convex and discontinuous. This hybrid algorithm combines the clonal selection algorithm (CSA) as the local search technique to update the best individual in the population, which enhances the diversity of the solutions and prevents premature convergence in DE. Furthermore, we investigate four mutation operations which are used in CSA as the hyper-mutation operations. Finally, an efficient solution repair method is designed for DELD to satisfy the complicated equality and inequality constraints of the power system to guarantee the feasibility of the solutions. Two benchmark power systems are used to evaluate the performance of the proposed method. The experimental results show that the proposed CSDE/best/1 approach significantly outperforms nine other variants of CSDE and DE, as well as most other published methods, in terms of the quality of the solution and the convergence characteristics.