Schr Dinger Equation Research Articles

Exploration of electron vortices in the photoionization of diatomic molecules in intense laser fields

Based on the approach within the single-electron approximation (SEA) frame, we numerically solve the two-dimensional (2D) time-dependent Schrdinger equation of N2 molecules under co-rotating and counter-rotating circularly polarized attosecond pulses. The vortex structure appears in counter-rotating circularly laser fields rather than the co-rotating case. The calculated vortex structures corresponding to the photoionization are found to be sensitive to the time delays T d of the two pulses, laser ellipticities and carrier envelope phases (CEPs). As the ellipticity increases to 1, the momentum distribution rotates and a vortex structure appears, and the interference is proved to play an important role in momentum distribution. The vortex structures also rotate clockwisely as the relative CEPs increase. The arms of the vortex structure become longer and thinner with the increasing T d . All these phenomena can be well understood by the attosecond perturbation ionization theory and the interference effect. Our findings clarify the applicability of the SEA model of diatomic molecules in attosecond extreme-ultraviolet fields.

Power and Chebyshev Series Transformation Formulas with Applications to Solving Ordinary Differential Equations via the Fröbenius and Taylor’s Methods

In this paper, we present formulas that turn finite power series into series of shifted Chebyshev polynomials of the first kind. Thereafter, we derive formulas for coefficients of economized power series obtained by truncating the resulting Chebyshev series. To illustrate the utility of our formulas, we apply them to the solution of first order ordinary differential equations via Taylor methods and to solving the Schr?dinger equation (SE) for a spherically symmetric hyperbolic potential via the Fr?benius method. In each of the two applications, we show that the use of our formulas makes it possible to reduce the computing time, while preserving the accuracy of the results.

Asymptotic solution of Sturm-Liouville problem with periodic boundary conditions for relativistic finite-difference Schrödinger equation

The quasi-potential approach is very famous in modern relativistic particles physics. This approach is based on the so-called covariant single-time formulation of quantum field theory in which the dynamics of fields and particles is described on a space-like three-dimensional hypersurface in the Minkowski space. Special attention in this approach is paid to methods for constructing various quasi-potentials. The quasipotentials allow to describe the characteristics of relativistic particles interactions in quark models such as amplitudes of hadron elastic scatterings, mass spectra, widths of meson decays and cross sections of deep inelastic scatterings of leptons on hadrons. In this paper SturmLiouville problems with periodic boundary conditions on a segment and a positive half-line for the 2m-order truncated relativistic finite-difference Schrdinger equation (LogunovTavkhelidzeKadyshevsky equation, LTKT-equation) with a small parameter are considered. A method for constructing of asymptotic eigenfunctions and eigenvalues in the form of asymptotic series for singularly perturbed SturmLiouville problems with periodic boundary conditions is proposed. It is assumed that eigenfunctions have regular and boundary-layer components. This method is a generalization of asymptotic methods that were proposed in the works of A. N. Tikhonov, A. B. Vasilyeva, and V. F Butuzov. We present proof of theorems that can be used to evaluate the asymptotic convergence for singularly perturbed problems solutions to solutions of degenerate problems when 0 and the asymptotic convergence of truncation equation solutions in the case m. In addition, the SturmLiouville problem on the positive half-line with a periodic boundary conditions for the quantum harmonic oscillator is considered. Eigenfunctions and eigenvalues
 are constructed for this problem as asymptotic solutions for 4-order LTKT-equation.

Stability Analysis of Magnetohydrodynamics Waves in Compressible Turbulent Plasma

Here, the solitary waves propagation in the cold plasma and their stability conditions are studied. The governing equations are expanded by using the reductive perturbation method with taken under the influence of the magnetic field under consideration. A new nonlinear wave equation is obtained that reconciles the derivative nonlinear Schrdinger equation with a modified form. By considering the magnetic field is constant along the x-direction, the complete set of equations is obtained, and the stable solitary waves are observed. A compariso between the soliton solutions of the modified nonlinear Schrdinger evolution equation (MNLS) and the solutions of the compressible magnetohydrodynamic (MHD) equations has been performed. It is shown that stable solitons can be created in such nonrelativistic fluids in the presence of the magnetic field. The modulation instability for a one-dimensional MNLS equation is carried as well. The regions of stability and instability fo the present system are well determined.

Trajectories of electron vortices in photoionization by bichromatic co-rotating circularly polarized laser fields

By solving the two-dimensional time-dependent Schrdinger equation (2D TDSE), we investigate the electron vortex in photoelectron momentum distribution (PMD) for He+ in bichromatic co-rotating time delayed circularly polarized laser fields. We find that the PMDs exhibit a vortex spiral structure, and the trajectories of the electron vortices are different when the emission sequence of two laser pulses changes, which is demonstrated by the trajectories of the vector potentials of two laser pulses.

Four-dark Soliton Solution of the Coupled Nonlinear Schrdinger Equation

Four-dark Soliton Solution of the Coupled Nonlinear Schrdinger Equation

N-soliton Solution of the Generalized Coupled Nonlinear Schrdinger Equation

N-soliton Solution of the Generalized Coupled Nonlinear Schrdinger Equation

Classification of difference schemes of the maximum possible accuracy on extended symmetric stencils for the Schrdinger equation and the heat transfer equation

Classification of difference schemes of the maximum possible accuracy on extended symmetric stencils for the Schrdinger equation and the heat transfer equation

Existence of nontrivial solution for Schrdinger equations with Hatree and Logarithmic nonlinearities

Existence of nontrivial solution for Schrdinger equations with Hatree and Logarithmic nonlinearities

Soliton solutions of higher order dispersive cubic-quintic nonlinear Schrödinger equation and its applications

The nonlinear dispersive cubic-quintic Schrdinger equation of higher order studies in this paper by employing two analytical schemes. The exp(−ϕ(ξ))-expansion and extended simple equation methods are used and exact solutions are attained in several kinds in which some are novel. According to our best knowledge, some solutions do not exist earlier. These solutions have key applications in engineering and applied physics. The attained results can be utilized to elucidate and know the physical nature of waves spread in the dispersive optical medium. A few attractive shapes are also depicted for the interpretation physically of the achieved solutions. Several such types of other models occurring in physics and other fields of applied science can be solved by using this consistent, influential and effective technique.

Cross-phase modulation instability under delayed response and walk-off effects in the anomalous regime of dispersion

The dynamics of the modulation instability induced by cross phase modulation is studied by considering the influence of the walk-off and noninstantaneous response effects for two copropagating optical fields travelling in the anomalous regime of dispersion. To do so, we make use of extensions of the nonlinear Schrdinger equation jointly with the Debye model for polarization, which is shown to be effective and simplified when compared to the implementation of other methods for the noninstantaneous nonlinear response. In analyzing the sideband formation, two bands are observed with different behaviors with respect to the way the maximum gain and the respective frequency vary with increasing phase mismatch. Further, we also show the manner in which the maximum gain as well as its corresponding frequency scale with the delay parameter τ is substantially different from the cases of both fields experiencing the normal group-velocity dispersion regime and the case of mixed regimes. These facts may give rise to many new possibilities for the evolution of the modulated wave when compared to the nonanomalous cases studied in the literature.

变系数耦合非线性薛定谔方程的二孤子解及其相互作用

变系数耦合非线性薛定谔方程的二孤子解及其相互作用

基于分数薛定谔方程对高斯光束的操控

基于分数薛定谔方程对高斯光束的操控

Three‐body molecular states of the system in the Born–Oppenheimer approximation

AbstractIn this work, we present a quantum mechanical treatment of the three‐body molecular system in the Born–Oppenheimer (BO) approximation, were the nuclei dynamics is evaluated over the potential energy surfaces (PES) induced by the electronic states. The PES corresponding to the two lowest electronic levels are the ones described by Martinazzo et al. (Chem. Phys. 2003, 287, 335), and are used to write the three‐body Schrdinger equation of the three atomic system. We use the generalized Sturmian functions method to expand the wave functions in each (distinguishable) pair of relative coordinates or Jacobi pairs, and analyze the convergence differences between the series. A partial‐wave decomposition of the potential is proposed to simplify the Hamiltonian matrix element calculation. Bound states are considered for the ground and first excited electronic PES, the spreading of energies after sudden electronic transitions studied, and the break‐up probability induced by the sudden change of the PES evidenced.

群速度与相速度对层级结构NSLE一孤子解的影响

群速度与相速度对层级结构NSLE一孤子解的影响

Local non-integer order dynamic problems on time scales revisited

This paper deals with a non-integer order calculus on time scales in a local sense. Taking advantage of a new field structure over real and complex numbers, the researchers have normalized the proposed derivative. Non-integer order mean value theorem on time scales is stated. In addition to a hypothesis on the existence of a H\( \ddot{o} \)lder continuous function h, a given time scale \( \mathbb {T} \) is explored. This consideration provided an extension to a recently proposed local fractional differentiation theory and it paved the ground to a local fractional integration on time scales. Also, making use of the presented non-integer order calculus, a class of dynamic initial value problems of fractional orders on an arbitrary time scale is studied. Eventually, an approximate solution to the potential-free Schrdinger equation of time-fractional order is obtained and illustrated.

Pt-Symmetric Nonlinear Directional Fiber Couplers with Gain and Loss for Ultrashort Optical Pulses

In this paper, we study the higher-order nonlinear Schrdinger (HNLS) equations for a directional fiber optic coupler with the third-order dispersion (TOD), self-steeping (SS) and stimulated Raman scattering (SRS) effects and gain in one waveguide and loss in the other. We have reduced the system equations to an uncoupled HNLS equation which has bright and dark soliton solutions. It is shown that the Hamiltonian for this equation has a PT-symmetric form. Furthermore the equilibrium points are found and the discussion about their stability is presented.

Buffer layer influence on light absorption of electron intersubband transition in binary energy level systems of quantum wells

Due to the restriction of the ZnO buffer layer on the left barrier in a wurtzite asymmetric ZnO/MgxZn1-xO single quantum well (QWs) structure with finite barriers, the other factors such as the size of the well and right barrier, and Mg component, etc. will influence the critical value of the left barrier width to form a binary level energy system. By adopting a finite difference method to solve the Schrdinger equation, the eigenstates and eigenenergies of electrons in a two-dimensional electron gas are obtained, and the influences of buffer layer ZnO, size and ternary mixed crystal effects on the formation of binary energy level system in QW are discussed. In our computation, the influences of energy band bending, material doping and built-in electric fields on a realistic heterostructure potential are considered. Furthermore, based on the Fermi's golden rule, the optical absorption coefficient of electronic intersubband transition in QW and the influences of buffer layer thickness, the widths of left barrier, well and right barrier and ternary mixed crystal effects are discussed. Our results indicate that the critical width of left barrier increases with the increases of the right barrier width and buffer layer thickness for a binary energy level system of ZnO/MgxZn1-xO single quantum well with a ZnO buffer layer on the left side. However, the critical width of left barrier decreases with the increase of well width and Mg component. Besides, the buffer layer thickness, the widths of left barrier, well and right barrier and ternary mixed crystal also affect the light absorption induced by the electronic intersubband transitions. The increases of Mg component, the widths of right barrier and left barrier will increase the absorption peak and produce its blue-shift. Whereas, increasing well width will reduce the absorption peak and produce its red-shift. The above conclusions are expected to give theoretical guidance in improving the opto-electronic properties of materials and devices made of these heterostructures.

All-fiber spectral compression of femtosecond pulse for coherent anti-Stokes Raman scattering excitation source

Coherent anti-Stokes Raman scattering (CARS) imaging of femtosecond pulses has been a research hotspot in recent years, but the wide spectrum of the femtosecond pulse limits the spectral resolution of CARS imaging. Spectral compression is considered as an effective method to solve this problem. In this work, an all-fiber chirp spectral compression method of graded-index multi-mode fiber/single-mode fiber (GI-MMF/SMF) structure based on fiber pre-chirp and self-phase modulation is presented. It can be used as a CARS excitation source to increase the spectral resolution of CARS imaging. In the section of numerical simulation, the mean group velocity dispersion value of GI-MMF is used as a numerical parameter of the chirp analysis, which is estimated by analyzing modes of GI-MMF. On one hand, the mode field distributions in GI-MMF are simulated numerically by the finite-difference time-domain method, and these different modes are divided into eight mode groups. On the other hand, the energy proportion of each mode group is regarded as a weight value. Then we can obtain a mean group velocity dispersion value of 50/125 m GI-MMF, which is -2.28710-5 fs2/nm, by calculating the sum of group velocity dispersion weight values of mode groups. The results of spectral compression with different length ratios of 50/125 m GI-MMF to 780HP SMF are also analyzed based on the generalized nonlinear Schrdinger equation and split-step Fourier algorithm. The spectral width of 2.486 nm and the compression ratio of 5.230 are calculated, when the length ratio of 50/125 m GI-MMF to 780HP SMF is 1.2. In the section of experiment, three kinds of GI-MMFs with different core diameters are used in the experiment, the influences of the core diameter and the length ratio of GI-MMF to 780HP SMF on the spectral compression are investigated. The results show that the spectral width of 2.243 nm, corresponding to the compression ratio of 5.796 is obtained, when the length ratio of 50/125 m GI-MMF to 780HP SMF is 1.2, which is consistent with the simulation result. Under the condition of the same length ratio, the use of 105/125 m GI-MMF can raise the compression ratio to 152.941, and the spectral width of output pulse is 0.085 nm. When the pulse is applied to CARS spectrum detection, the theoretical spectral resolution can be 1.386 cm-1. The experimental results show that the spectral compression way to improve spectral resolution of CARS imaging is effective. This spectral compression system is characterized by simple structure, and high and controllable compression ratio, which provides theoretical and experimental basis for the all-fiber high spectral resolution CARS excitation source research.

Application of fixed point theorem for stability analysis of a nonlinear Schrodinger with Caputo-Liouville derivative

Using the new Caputo-Liouville derivative with fractional order, we have modified the nonlinear Schrdinger equation. We have shown some useful in connection of the new derivative with fractional order. We used an iterative approach to derive an approximate solution of the modified equation. We have established the stability of the iteration scheme using the fixed point theorem. We have in addition presented in detail the uniqueness of the special solution.