Nonplanar Electron - Acoustic Shock Waves with Superthermal Hot Electrons

Nonplanar electron-acoustic shock waves having superthermal hot electrons are investigated with two temperature electrons model in unmagnetized plasma. Using reductive perturbation method, Korteweg-de Vries-Burgers (KdVB) equation is obtained in the cylindrical/spherical coordinates. Dissipation effect is introduced in the model by means of kinematic viscosity term. On the basis of the solutions of KdVB equation, variation of shock waves features (amplitude, velocity and width) with different plasma parameters are analysed. KdV-Burgers equation always leads to monotonic solitons and no oscillatory peak may appear. The combined effect of particle density (α), superthermal parameter (κ), electron temperature ratio (??) and kinetic viscosity (η₀) is numerically studied, and it is observed that these parameters significantly change the properties of the shock waves in nonplanar geometry especially in spherical coordinates. Results could be helpful to analyse the soliton features in laboratory as well as in the space environments.     

Thermodynamics and relativistic kinetic theory for <Emphasis Type="Italic">q</Emphasis>-generalized Bose–Einstein and Fermi–Dirac systems

The thermodynamics and covariant kinetic theory are elaborately investigated in a non-extensive environment considering the non-extensive generalization of Bose–Einstein (BE) and Fermi–Dirac (FD) statistics. Starting with Tsallis’ entropy formula, the fundamental principles of thermostatistics are established for a grand canonical system having q-generalized BE/FD degrees of freedom. Many particle kinetic theory is set up in terms of the relativistic transport equation with q-generalized Uehling–Uhlenbeck collision term. The conservation laws are realized in terms of appropriate moments of the transport equation. The thermodynamic quantities are obtained in a weak non-extensive environment for a massive pion–nucleon and a massless quark–gluon system with non-zero baryon chemical potential. In order to get an estimate of the impact of non-extensivity on the system dynamics, the q-modified Debye mass and hence the q-modified effective coupling are estimated for a quark–gluon system.     

Modulational instability,rogue waves,and envelope solitons in opposite polarity dusty plasmas

Dust-acoustic (DA) waves (DAWs) and their modulational instability (MI) have been investigated theoretically in a plasma system consisting of inertial opposite polarity (positively and negatively) warm adiabatic charged dust grains as well as inertialess non-extensive $q?$distributed electrons and non-thermal ions. A nonlinear Schrödinger equation (NLSE) is derived by using the reductive perturbation method. It has been observed from the analysis of NLSE that the modulationally stable solitary DAWs give rise to the existence of dark envelope solitons, and that the modulationally unstable solitary DAWs give rise to the existence of bright envelope solitons or rogue structures. It is also observed for the fast mode of DAWs that the basic features (viz. stability of the DAWs, MI, growth rate, amplitude, and width of the DA rogue waves, etc.) are significantly modified by the related plasma parameters (viz. dust masses, dust charge state, non-extensive parameter q, and non-thermal parameter α). The results of our present investigation might be useful for understanding different nonlinear electrostatic phenomena in both space (viz. ionosphere and mesosphere) and laboratory plasmas (viz. high intensity laser irradiation and hot cathode discharge).     

Nonlinear structures for extended Korteweg–de Vries equation in multicomponent plasma

Using the fluid hydrodynamic equations of positive and negative ions, as well as q-nonextensive electron density distribution, an extended Korteweg–de Vries (EKdV) equation describing a small but finite amplitude dust ion-acoustic waves (DIAWs) is derived. Extended homogeneous balance method is used to obtain a new class of solutions of the EKdV equation. The effects of different physical parameters on the propagating nonlinear structures and their relevance to particle acceleration in space plasma are reported.     

Solitary waves and rogue waves in a plasma with nonthermal electrons featuring Tsallis distribution

In this Letter, we discuss the electron acoustic (EA) waves in plasmas, which consist of nonthermal hot electrons featuring the Tsallis distribution, and obtain the corresponding governing equation, that is, a nonlinear Schrödinger (NLS) equation. By means of Modulation Instability (MI) analysis of the EA waves, it is found that both electron acoustic solitary wave and rogue wave can exist in such plasmas. Basing on the Darboux transformation method, we derive the analytical expressions of nonlinear solutions of NLS equations, such as single/double solitary wave solutions and single/double rogue wave solutions. The existential regions and amplitude of solitary wave solutions and the rogue wave solutions are influenced by the nonextensive parameter q and nonthermal parameter α. Moreover, the interaction of solitary wave and how to postpone the excitation of rogue wave are also studied.     

Electron gas induced in SrTiO<Subscript>3</Subscript>

This mini-review is dedicated to the 85th birthday of Prof. L.V. Keldysh, from whom we have learned so much. In this paper, we study the potential and electron density depth profiles in surface accumulation layers in crystals with a large and nonlinear dielectric response such as SrTiO₃ (STO) in the cases of planar, spherical, and cylindrical geometries. The electron gas can be created by applying an induction D₀ to the STO surface. We describe the lattice dielectric response of STO using the Landau–Ginzburg free energy expansion and employ the Thomas–Fermi (TF) approximation for the electron gas. For the planar geometry, we arrive at the electron density profile n(x) ∝ (x + d)^–12/7, where d ∝ D₀^–12/7. We extend our results to overlapping electron gases in GTO/STO/GTO heterojunctions and electron gases created by spill-out from NSTO (heavily n-type doped STO) layers into STO. Generalization of our approach to a spherical donor cluster creating a big TF atom with electrons in STO brings us to the problem of supercharged nuclei. It is known that for an atom with a nuclear charge Ze where Z > 170, electrons collapse onto the nucleus, resulting in a net charge Zn < Z. Here, instead of relativistic physics, the collapse is caused by the nonlinear dielectric response. Electrons collapse into the charged spherical donor cluster with radius R when its total charge number Z exceeds the critical value Z_c ≈ R/a, where a is the lattice constant. The net charge eZ_n grows with Z until Z exceeds Z* ≈ (R/a)^9/7. After this point, the charge number of the compact core Z_n remains ≈ Z*, with the rest Z* electrons forming a sparse TF atom with it. We extend our studies of collapse to the case of long cylindrical clusters as well.     

Hamilton’s equations of motion of a vortex filament in the rotating Bose–Einstein condensate and their “soliton” solutions

The equation of motion of a quantized vortex filament in a trapped Bose–Einstein condensate [A. A. Svidzinsky and A. L. Fetter, Phys. Rev. A 62, 063617 (2000)] has been generalized to the case of an arbitrary anharmonic anisotropic rotating trap and presented in the variational form. For condensate density profiles of the form ρ = f(x² + y² + ReΨ(x + iy)) in the presence of the plane of symmetry y = 0, the solutions x(z) describing stationary vortices of U and S types coming to the surface and solitary waves have been found in quadratures. Analogous three-dimensional configurations of the vortex filament uniformly moving along the z axis have also been found in strictly cylindrical geometry. The dependence of solutions on the form of the function f(q) has been analyzed.     

The Non-linear Schrödinger Equation and the Conformal Properties of Non-relativistic Space-Time

The cubic non-linear Schrödinger equation where the coefficient of the nonlinear term is a function F(t,x) only passes the Painlevé test of Weiss, Tabor, and Carnevale only for F=(a+bt)^?1, where a and b are constants. This is explained by transforming the time-dependent system into the constant-coefficient NLS by means of a time-dependent non-linear transformation, related to the conformal properties of non-relativistic space-time. A similar argument explains the integrability of the NLS in a uniform force field or in an oscillator background.     

Strict parabolicity of the multifractal spectrum at the Anderson transition

Using the well-known “algebra of multifractality,” we derive the functional equation for anomalous dimensions Δ _q , whose solution Δ = χq(q–1) corresponds to strict parabolicity of the multifractal spectrum. This result demonstrates clearly that a correspondence of the nonlinear σ-models with the initial disordered systems is not exact.     

The Davey-Stewartson Equation on the Half-Plane

The Davey-Stewartson (DS) equation is a nonlinear integrable evolution equation in two spatial dimensions. It provides a multidimensional generalisation of the celebrated nonlinear Schrödinger (NLS) equation and it appears in several physical situations. The implementation of the Inverse Scattering Transform (IST) to the solution of the initial-value problem of the NLS was presented in 1972, whereas the analogous problem for the DS equation was solved in 1983. These results are based on the formulation and solution of certain classical problems in complex analysis, namely of a Riemann Hilbert problem (RH) and of either a d-bar or a non-local RH problem respectively. A method for solving the mathematically more complicated but physically more relevant case of boundary-value problems for evolution equations in one spatial dimension, like the NLS, was finally presented in 1997, after interjecting several novel ideas to the panoply of the IST methodology. Here, this method is further extended so that it can be applied to evolution equations in two spatial dimensions, like the DS equation. This novel extension involves several new steps, including the formulation of a d-bar problem for a sectionally non-analytic function, i.e. for a function which has different non-analytic representations in different domains of the complex plane. This, in addition to the computation of a d-bar derivative, also requires the computation of the relevant jumps across the different domains. This latter step has certain similarities (but is more complicated) with the corresponding step for those initial-value problems in two dimensions which can be solved via a non-local RH problem, like KPI.     

Modulational instability of dust-ion-acoustic waves in pair-ion plasma having non-thermal non-extensive electrons

The modulational instability (MI) criteria of dust-ion-acoustic (DIA) waves (DIAWs) have been investigated in a four-component pair-ion plasma having inertial pair ions, inertialess non-thermal non-extensive electrons, and immobile negatively charged massive dust grains. A nonlinear Schrödinger equation (NLSE) is derived by using reductive perturbation method. The nonlinear and dispersive coefficients of the NLSE can predict the modulationally stable and unstable parametric regimes of DIAWs and associated first and second-order DIA rogue waves (DIARWs). The MI growth rate and the configuration of the DIARWs are examined, and it is found that the MI growth rate increases (decreases) with increasing the number density of the negatively charged dust grains in the presence (absence) of the negative ions. It is also observed that the amplitude and width of the DIARWs increase (decrease) with the negative (positive) ion mass. The implications of the results to laboratory and space plasmas are briefly discussed.     

Nonplanar dust ion-acoustic solitary and shock waves in a dusty plasma with electrons following a vortex-like distribution

Properties of nonplanar (viz. cylindrical and spherical) dust ion-acoustic (DIA) solitary and shock waves propagating in a dusty plasma containing charge fluctuating stationary dust, inertial warm ions, and non-isothermal electrons following a vortex-like distribution, are investigated by the reductive perturbation method. It has been shown that all the basic features of the DIA solitary and shock waves are significantly modified by the effects of vortex-like electron distribution, dust charge fluctuation, and nonplanar cylindrical and spherical geometries. The implications of our results in some space and laboratory dusty plasma environments are briefly discussed.     

Spin waves on chains of YIG particles: dispersion relations,Faraday rotation,and power transmission

We calculate the dispersion relations for spin waves on a periodic chain of spherical or cylindrical Yttrium Iron Garnet (YIG) particles. We use the quasistatic approximation, appropriate when kd ? 1, where k is the wave number and d the interparticle spacing. In this regime, because of the magnetic dipole-dipole interaction between the localized magnetic excitations on neighboring particles, dispersive spin waves can propagate along the chain. The waves are analogous to plasmonic waves generated by electric dipole-dipole interactions between plasmons on neighboring metallic particles. The spin waves can be longitudinal (L), transverse (T), or elliptically polarized. We find that a linearly polarized spin wave undergoes a Faraday rotation as it propagates along the chain. The amount of Faraday rotation can be tuned by varying the off-diagonal component of the permeability tensor. We also discuss the possibility of wireless power transmission along the chain using these coupled spin waves.     

Obliquely Propagating Electron Acoustic Shock Waves in Magnetized Plasma

Obliquely propagating electron acoustic shock waves in plasma with stationary ions, cold and superthermal hot electrons are investigated in magnetized plasma. Employing reductive perturbation method, Korteweg-de Vries-Burgers equation (KdVB) is derived in the small amplitude approximation limit. The analytical and numerical calculations of the KdVB equation show the variation of shock waves structure (amplitude, velocity, and width) with different plasma parameters. Particle density (α), superthermal parameter (κ), electron temperature ratio (??), kinetic viscosity (η₀), obliqueness (k_z), and strength of magnetic field (ω_c) significantly modify the properties of the shock waves structures. The present investigation is useful to understand dissipative structures observed in space or laboratory plasma where multielectrons population with superthermal electrons are prevalent.     

Generation of Coupled Modes in an Unmatched Three-Mirror Laser Cavity

An extraordinary phenomenon of generation of coupled types of oscillations with a transverse field distribution corresponding to the ТЕМ _pq eigenmodes with different pairs of indices p and q in the main laser cavity and in an external cavity has been experimentally detected. The excitation of coupled modes depends on the configuration and tuning of partial cavities. To solve the system of integral equations for a three-mirror cavity with mismatched spherical mirrors in the quasioptical approximation, it has been proposed to use modified boundary conditions including the coupling coefficients for eigenmodes in partial cavities.     

The fourth-order nonlinear Schrödinger equation for the envelope of Stokes waves on the surface of a finite-depth fluid

The method of multiple scales is used to derive the fourth-order nonlinear Schrödinger equation (NSEIV) that describes the amplitude modulations of the fundamental harmonic of Stokes waves on the surface of a medium-and large-depth (compared to the wavelength) fluid layer. The new terms of this equation describe the third-order linear dispersion effect and the nonlinearity dispersion effects. As the nonlinearity and the dispersion decrease, the equation uniformly transforms into the nonlinear Schrödinger equation for Stokes waves on the surface of a finite-depth fluid that was first derived by Hasimoto and Ono. The coefficients of the derived equation are given in an explicit form as functions of kh (h is the fluid depth, and k is the wave number). As kh tends to infinity, these coefficients transform into the coefficients of the NSEIV that was first derived by Dysthe for an infinite depth.     

Taylor collocation method for the numerical solution of the nonlinear Schrödinger equation using quintic B-spline basis

In this study, we construct a Taylor collocation method for the numerical solution of the nonlinear Schrödinger (NLS) equation. We use suitable initial and boundary conditions. Taylor series expansion is used for time discretization. The cubic B-spline collocation method is applied to spatial discretization. Test problems concerning the single soliton motion, interaction of two colliding solitons, and the formation and bound states of solitons of the NLS equation are studied to evaluate the method. The L ₂ and L _∞ error norms are calculated to improve the accuracy of the numerical results.     

Exact travelling solutions for some nonlinear physical models by ( <Emphasis Type="Italic">G</Emphasis>′/ <Emphasis Type="Italic">G</Emphasis>)-expansion method

In this paper, we establish exact solutions for some special nonlinear partial differential equations. The (G′/G)-expansion method is used to construct travelling wave solutions of the two-dimensional sine-Gordon equation, Dodd–Bullough–Mikhailov and Schrödinger–KdV equations, which appear in many fields such as, solid-state physics, nonlinear optics, fluid dynamics, fluid flow, quantum field theory, electromagnetic waves and so on. In this method we take the advantage of general solutions of second-order linear ordinary differential equation (LODE) to solve many nonlinear evolution equations effectively. The (G′/G)-expansion method is direct, concise and elementary and can be used with a wider applicability for handling many nonlinear wave equations.     

Schrödinger’s Equation with Gauge Coupling Derived from a Continuity Equation

A quantization procedure without Hamiltonian is reported which starts from a statistical ensemble of particles of mass m and an associated continuity equation. The basic variables of this theory are a probability density ρ, and a scalar field S which defines a probability current j=ρ ? S/m. A first equation for ρ and S is given by the continuity equation. We further assume that this system may be described by a linear differential equation for a complex-valued state variable χ. Using these assumptions and the simplest possible Ansatz χ(ρ,S), for the relation between χ and ρ,S, Schrödinger’s equation for a particle of mass m in a mechanical potential V(q,t) is deduced. For simplicity the calculations are performed for a single spatial dimension (variable q). Using a second Ansatz χ(ρ,S,q,t), which allows for an explicit q,t-dependence of χ, one obtains a generalized Schrödinger equation with an unusual external influence described by a time-dependent Planck constant. All other modifications of Schrödinger’ equation obtained within this Ansatz may be eliminated by means of a gauge transformation. Thus, this second Ansatz may be considered as a generalized gauging procedure. Finally, making a third Ansatz, which allows for a non-unique external q,t-dependence of χ, one obtains Schrödinger’s equation with electrodynamic potentials A,φ in the familiar gauge coupling form. This derivation shows a deep connection between non-uniqueness, quantum mechanics and the form of the gauge coupling. A possible source of the non-uniqueness is pointed out.     

Strong gravitational lensing by regular electrically charged black holes

In nonlinear electrodynamics a photon does not follow null geodesics of background geometry, but moves along null geodesics of a corresponding effective geometry. Therefore, in the strong deflection limit, in order to study the gravitational lensing of the regular electrically charged black holes obtained by coupling general relativity to nonlinear electrodynamics, one should firstly obtain the corresponding effective geometry, which is a necessary and key step. I obtain the deflection angle of the photon in the strong deflection limit, and further calculate the angular position and magnification of relativistic images. It is found that, the electric charge has significant effect on the gravitational lensing of regular black holes. With the increase of the electric charge q, the angular position of the relativistic images \(\theta _{\infty }\) and the relative magnification \(\mathcal {R}_{m}\) as a function of q decrease, while the angular separation between the outermost relativistic image and the others \(\mathcal {S}\) as a function of q increases. I also discuss the measurement of observables for the black hole at the center of our Galaxy in the cases of regular electrically charged black hole effective metrics.