Higher-order Dirac solitons in binary waveguide arrays

We study optical analogues of higher-order Dirac solitons (HODSs) in binary waveguide arrays. Like higher-order solitons obtained from the well-known nonlinear Schrödinger equation governing the pulse propagation in an optical fiber, these HODSs have amplitude profiles which are numerically shown to be periodic over large propagation distances. At the same time, HODSs possess some unique features. Firstly, the period of a HODS depends on its order parameter. Secondly, the discrete nature in binary waveguide arrays imposes the upper limit on the order parameter of HODSs. Thirdly, the order parameter of HODSs can vary continuously in a certain range.     

Pseudospin symmetric solutions of the Dirac equation with the modified Rosen—Morse potential using Nikiforov—Uvarov method and supersymmetric quantum mechanics approach

Employing the Pekeris-type approximation to deal with the pseudo-centrifugal term, we analytically study the pseudospin symmetry of a Dirac nucleon subjected to equal scalar and vector modified Rosen-Morse potential including the spin-orbit coupling term by using the Nikiforov-Uvarov method and supersymmetric quantum mechanics approach. The complex eigenvalue equation and the total normalized wave functions expressed in terms of Jacobi polynomial with arbitrary spin-orbit coupling quantum number k are presented under the condition of pseudospin symmetry. The eigenvalue equations for both methods reproduce the same result to affirm the mathematical accuracy of analytical calculations. The numerical solutions obtained for different adjustable parameters produce degeneracies for some quantum number.     

Dirac quasinormal modes of <Emphasis Type="Italic">D</Emphasis>-dimensional de Sitter spacetime

We find exact solutions to the Dirac equation in D-dimensional de Sitter spacetime. Using these solutions we analytically calculate the de Sitter quasinormal (QN) frequencies of the Dirac field. For the massive Dirac field this computation is similar to that previously published for massive fields of half-integer spin moving in four dimensions. However to calculate the QN frequencies of the massless Dirac field we must use distinct methods in odd and even dimensions, therefore the computation is different from that already known for other massless fields of integer spin.     

Nonperturbative effects of the minimal length uncertainty on the relativistic quantum mechanics

We study the nonperturbative effects of the minimal length on the energy spectrum of a relativistic particle in the context of the generalized uncertainty principle (GUP). This form of GUP is consistent with various candidates of quantum gravity such as string theory, loop quantum gravity, and black-hole physics and predicts a minimum measurable length proportional to the Planck length. Using a recently proposed formally self-adjoint representation, we solve the generalized Dirac and Klein–Gordon equations in various situations and find the corresponding exact energy eigenvalues and eigenfunctions. We show that for the Dirac particle in a box, the number of the solutions renders to be finite as a manifestation of both the minimal length and the theory of relativity. For the case of the Dirac oscillator and the wave equations with scalar and vector linear potentials, we indicate that the solutions can be obtained in a more simpler manner through the self-adjoint representation. It is also shown that, in the ultrahigh frequency regime, the partition function and the thermodynamical variables of the Dirac oscillator can be expressed in a closed analytical form. The Lorentz violating nature of the GUP-corrected relativistic wave equations is discussed finally.     

Coherent quantum states of a relativistic particle in an electromagnetic plane wave and a parallel magnetic field

We analyze the solutions of the Klein–Gordon and Dirac equations describing a charged particle in an electromagnetic plane wave combined with a magnetic field parallel to the direction of propagation of the wave. It is shown that the Klein–Gordon equation admits coherent states as solutions, while the corresponding solutions of the Dirac equation are superpositions of coherent and displaced-number states. Particular attention is paid to the resonant case in which the motion of the particle is unbounded.     

Some extensions of the Einstein–Dirac equation

We considered an extension of the standard functional for the Einstein–Dirac equation where the Dirac operator is replaced by the square of the Dirac operator and a real parameter controlling the length of spinors is introduced. For one distinguished value of the parameter, the resulting Euler–Lagrange equations provide a new type of Einstein–Dirac coupling. We establish a special method for constructing global smooth solutions of a newly derived Einstein–Dirac system called the CL-Einstein–Dirac equation of type II (see Definition 3.1).     

一类相对论性非球谐振子系统的束缚态

给出了具有形式为12r²+A2r²的非球谐振子型标量势和矢量势 的相对论系 统在两种势相等的条件下三维Klein-Gordon方程，二维和三维Dirac方程的s波束缚态解. 关键词： 三维非球谐振子势 Klein-Gordon方程 Dirac方程 束缚态     

Supersymmetry in quantum mechanics

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable. In this lecture I review the theoretical formulation of supersymmetric quantum mechanics and discuss many of its applications. I show that the well-known exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials and shape invariance. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Further, it is pointed out that the connection between the solutions of the Dirac equation and the Schrödinger equation is exactly same as between the solutions of the MKdV and the KdV equations.     

Analytical Investigation of Soliton Solutions to Three Quantum Zakharov-Kuznetsov Equations

In the present paper, the two-dimensional quantum Zakharov-Kuznetsov (QZK) equation, three-dimensional quantum Zakharov-Kuznetsov equation and the three-dimensional modified quantum Zakharov-Kuznetsov equation are analytically investigated for exact solutions using the modified extended tanh-expansion based method. A variety of new and important soliton solutions are obtained including the dark soliton solution, singular soliton solution, combined dark-singular soliton solution and many other trigonometric function solutions. The used method is implemented on the Mathematica software for the computations as well as the graphical illustrations.     

Analytical Investigation of Soliton Solutions to Three Quantum Zakharov-Kuznetsov Equations

In the present paper, the two-dimensional quantum Zakharov-Kuznetsov(QZK) equation, three-dimensional quantum Zakharov-Kuznetsov equation and the three-dimensional modified quantum Zakharov-Kuznetsov equation are analytically investigated for exact solutions using the modified extended tanh-expansion based method. A variety of new and important soliton solutions are obtained including the dark soliton solution, singular soliton solution, combined dark-singular soliton solution and many other trigonometric function solutions. The used method is implemented on the Mathematica software for the computations as well as the graphical illustrations.     

Level spacing statistics for two-dimensional massless Dirac billiards

Classical-quantum correspondence has been an intriguing issue ever since quantum theory was proposed. The searching for signatures of classically nonintegrable dynamics in quantum systems comprises the interesting field of quantum chaos. In this short review, we shall go over recent efforts of extending the understanding of quantum chaos to relativistic cases. We shall focus on the level spacing statistics for two-dimensional massless Dirac billiards, i.e., particles confined in a closed region. We shall discuss the works for both the particle described by the massless Dirac equation(or Weyl equation)and the quasiparticle from graphene. Although the equations are the same, the boundary conditions are typically different,rendering distinct level spacing statistics.     

A new generalized fractional Dirac soliton hierarchy and its fractional Hamiltonian structure

Based on the differential forms and exterior derivatives of fractional orders,Wu first presented the generalized Tu formula to construct the generalized Hamiltonian structure of the fractional soliton equation.We apply the generalized Tu formula to calculate the fractional Dirac soliton equation hierarchy and its Hamiltonian structure.The method can be generalized to the other fractional soliton hierarchy.     

The Invariant-Relat ed Unitary Transformation Method and the Evolution of a Dirac Field in Friedmann-Robertson-Walker Flat Spacetimes

On the basis of the generalized invariant formulation, the invariant-related unitary transformation method is used to study the evolution of a quantum Dirac field in Friedmann-Robertson-Walker spatially flat spacetime. We first solve the functional Schrodinger equation for a free Dirac field and obtain the exact solutions. We then investigate the way of extending the method to treat the case in which there is an interaction between the Dirac field and a scalar field.     

On the connection between the solutions to the Dirac and Weyl equations and the corresponding electromagnetic four-potentials

In this work we study in detail the connection between the solutions to the Dirac and Weyl equations and the associated electromagnetic four-potentials.First,it is proven that all solutions to the Weyl equation are degenerate,in the sense that they correspond to an infinite number of electromagnetic four-potentials.As far as the solutions to the Dirac equation are concerned,it is shown that they can be classified into two classes.The elements of the first class correspond to one and only one four-potential,and are called non-degenerate Dirac solutions.On the other hand,the elements of the second class correspond to an infinite number of four-potentials,and are called degenerate Dirac solutions.Further,it is proven that at least two of these fourpotentials are gauge-inequivalent,corresponding to different electromagnetic fields.In order to illustrate this particularly important result we have studied the degenerate solutions to the forcefree Dirac equation and shown that they correspond to massless particles.We have also provided explicit examples regarding solutions to the force-free Weyl equation and the Weyl equation for a constant magnetic field.In all cases we have calculated the infinite number of different electromagnetic fields corresponding to these solutions.Finally,we have discussed potential applications of our results in cosmology,materials science and nanoelectronics.     

Solution of Dirac equation for Eckart potential and trigonometric Manning Rosen potential using asymptotic iteration method

The Dirac equation for Eckart potential and trigonometric Manning Rosen potential with exact spin symmetry is obtained using an asymptotic iteration method. The combination of the two potentials is substituted into the Dirac equation,then the variables are separated into radial and angular parts. The Dirac equation is solved by using an asymptotic iteration method that can reduce the second order differential equation into a differential equation with substitution variables of hypergeometry type. The relativistic energy is calculated using Matlab 2011. This study is limited to the case of spin symmetry. With the asymptotic iteration method, the energy spectra of the relativistic equations and equations of orbital quantum number l can be obtained, where both are interrelated between quantum numbers. The energy spectrum is also numerically solved using the Matlab software, where the increase in the radial quantum number nr causes the energy to decrease. The radial part and the angular part of the wave function are defined as hypergeometry functions and visualized with Matlab 2011. The results show that the disturbance of a combination of the Eckart potential and trigonometric Manning Rosen potential can change the radial part and the angular part of the wave function.     

Dirac equation for the Hulthén potential within the Yukawa-type tensor interaction

Using the Nikiforov-Uvarov (NU) method, pseudospin and spin symmetric solutions of the Dirac equation for the scalar and vector Hulthén potentials with the Yukawa-type tensor potential are obtained for an arbitrary spin-orbit coupling quantum number κ. We deduce the energy eigenvalue equations and corresponding upper- and lower-spinor wave functions in both the pseudospin and spin symmetry cases. Numerical results of the energy eigenvalue equations and the upper- and lower-spinor wave functions are presented to show the effects of the external potential and particle mass parameters as well as pseudospin and spin symmetric constants on the bound-state energies and wave functions in the absence and presence of the tensor interaction.     

Spin symmetric solutions of Dirac equation with Pöschlben Teller potential

The approximate analytical solutions of the Dirac equation with the Pöschl—Teller potential is presented for arbitrary spin-orbit quantum number kappa within the framework of the spin symmetry concept. The energy eigenvalues and the corresponding two Dirac spinors are obtained approximately in closed forms. The limiting cases of the energy eigenvalues and the two Dirac spinors are briefly discussed.     

Soliton solutions,travelling wave solutions and conserved quantities for a three-dimensional soliton equation in plasma physics

Many physical systems can be successfully modelled using equations that admit the soliton solutions. In addition, equations with soliton solutions have a significant mathematical structure. In this paper, we study and analyze a three-dimensional soliton equation, which has applications in plasma physics and other nonlinear sciences such as fluid mechanics, atomic physics, biophysics, nonlinear optics, classical and quantum fields theories. Indeed, solitons and solitary waves have been observed in numerous situations and often dominate long-time behaviour. We perform symmetry reductions of the equation via the use of Lie group theory and then obtain analytic solutions through this technique for the very first time. Direct integration of the resulting ordinary differential equation is done which gives new analytic travelling wave solutions that consist of rational function, elliptic functions, elementary trigonometric and hyperbolic functions solutions of the equation. Besides, various solitonic solutions are secured with the use of a polynomial complete discriminant system and elementary integral technique. These solutions comprise dark soliton, doubly-periodic soliton, trigonometric soliton, explosive/blowup and singular solitons. We further exhibit the dynamics of the solutions with pictorial representations and discuss them. In conclusion, we contemplate conserved quantities for the equation under study via the standard multiplier approach in conjunction with the homotopy integral formula. We state here categorically and emphatically that all results found in this study as far as we know have not been earlier obtained and so are new.     

q\bar q pair creation in a flux tube with confinement

We calculate the effect of radial confinement on the Schwinger pair production rate by solving the Dirac equation in a flux-tube cylinder containing a constant chromoelectric field in the longitudinal direction. We show how the Dirac equation separates into radial and longitudinal equations for a mass term which has an arbitrary radial dependence and introduce radial confinement by having a finite mass inside the cylinder and an infinitely large mass outside. The resulting boundary conditions are equivalent to the MIT boundary condition. The equations are solved analytically for a constant quark mass inside the flux-tube, which acts like a waveguide. The discretization of the transverse wave vector which has a continuous spectrum in the non-confined case leads to a large suppression of the Schwinger pair-production rate for small radii. The minimal radius where pairs are created decreases with increasing field strength. The suppression turns out to be larger for heavier quarks than for light quarks.     

Solution of the Dirac Equation on the Bertotti–Robinson Metric

It is shown that each component of the Dirac field satisfies a decoupled equation, which admits separable solutions, when the background spacetime is the Bertotti–Robinson metric, which is a solution of the Einstein vacuum field equations with a cosmological constant. Furthermore, the seperated functions appearing in the solutions are shown to obey identities of the Teukolsky–Starobinsky type and the separable solutions are shown to be eigenfunctions of a certain differential operator.