Propagation of nonlocal optical solitons in lossy media with exponential-decay response

Propagation of optical solitons in lossy nonlocal media with exponential-decay response was investigated theoretically. The analytical solutions of nonlocal solitons and breathers are obtained by variational approach which is applied to a (1 + 1)D nonlocal nonlinear Schrödinger equation. The critical power of soliton and period of breathers are also obtained in the absence of the loss. When the loss is relatively small, the average beam width of breathers has a trend to expand during propagation. The analytical results are confirmed by numerical simulation.     

Solitary wave solutions as a signature of the instability in the discrete nonlinear Schrödinger equation

The effect of instability on the propagation of solitary waves along one-dimensional discrete nonlinear Schrödinger equation with cubic nonlinearity is revisited. A self-contained quasicontinuum approximation is developed to derive closed-form expressions for small-amplitude solitary waves. The notion that the existence of nonlinear solitary waves in discrete systems is a signature of the modulation instability is used. With the help of this notion we conjecture that instability effects on moving solitons can be qualitative estimated from the analytical solutions. Results from numerical simulations are presented to support this conjecture.     

Investigation of the Dynamics of a Soliton-Like Solution to the Nonlinear Schrödinger Equation with an External Field

The properties of asymptotic soliton-like solutions to the 1-D nonstationary nonlinear Schrödinger equation with the external-field potential of a special form are studied. A comparative analysis of the asymptotic solutions and simulation results is performed to show the range of parameter values where asymptotic and numerical soliton-like solutions are in agreement, the localization being preserved.     

Spectral splitting method for nonlinear Schrödinger equations with singular potential

We consider the time-dependent one-dimensional nonlinear Schrödinger equation with pointwise singular potential. By means of spectral splitting methods we prove that the evolution operator is approximated by the Lie evolution operator, where the kernel of the Lie evolution operator is explicitly written. This result yields a numerical procedure which is much less computationally expensive than multi-grid methods previously used. Furthermore, we apply the Lie approximation in order to make some numerical experiments concerning the splitting of a soliton, interaction among solitons and blow-up phenomenon.     

Chirped soliton-like solutions for nonlinear Schrödinger equation with variable coefficients

The exact chirped bright and dark soliton-like solutions of generalized nonlinear Schrödinger equation including linear and nonlinear gain(loss) with variable coefficients describing dispersion-management or soliton control is obtained detailedly in this paper. To begin our numerical studies of the stability of the solutions, we present a periodically distributed dispersion management or soliton control system as an example. It is found that both the bright and dark soliton-like solutions are stable during propagation in the given system. The numerical results are well in accordance with those obtained by analytical methods.     

Periodic spatio-temporal oscillations governed by a globally controlled Ginzburg-Landau equation

A new type of periodic oscillations in a globally controlled subcritical cubic complex Ginzburg-Landau equation, formerly observed in numerical simulations, is explained and investigated analytically by means of a multiscale perturbation theory. Using an appropriate class of solutions of the nonlinear Schrödinger equation as a starting point, we construct a new class of asymptotic solutions of the cubic complex Ginzburg-Landau equation in the limit of large dispersion and nonlinear frequency shift.     

Soliton strings and interactions in mode-locked lasers

Soliton strings in mode-locked lasers are obtained using a variant of the nonlinear Schrödinger equation, appropriately modified to model power (intensity) and energy saturation. This equation goes beyond the well-known master equation often used to model these systems. It admits mode-locking and soliton strings in both the constant dispersion and dispersion-managed systems in the (net) anomalous and normal regimes; the master equation is contained as a limiting case. Analysis of soliton interactions show that soliton strings can form when pulses are a certain distance apart relative to their width. Anti-symmetric bi-soliton states are also obtained. Initial states mode-lock to these states under evolution. In the anomalous regime individual soliton pulses are well approximated by the solutions of the unperturbed nonlinear Schrödinger equation, while in the normal regime the pulses are much wider and strongly chirped.     

On the possibility of an analytic solution of the three-body problem

Three-body Faddeev equations in the Noyes-Fiedeldey form are rewritten as a matrix analog of a one-dimensional nonrelativistic Schrödinger equation. Unlike the method of K-harmonics, where a similar equation was obtained by expansion of a three-body Schrödinger equation wavefunction into the orthogonal set of functions of two variables (K-harmonics), the use of the Noyes-Fiedeldey form of Faddeev equations allows us to limit ourselves to the expansion in functions of one variable only. The solutions of the above mentioned matrix equation are obtained. These solutions converge uniformly within every interval of continuity of the matrix, which corresponds to the potential of that equation. Their asymptotic behavior for large interparticle distances is discussed. The solutions for the harmonic oscillator, inverse-square, and Coulomb-Kepler potentials are found. It is shown that energy levels in the last case may be calculated from a simple formula which is very similar to the corresponding formula for the two-body Coulomb-Kepler problem. This formula can be easily generalized to the case of n particles interacting with inverse distance potentials.     

Multisoliton solutions in terms of double Wronskian determinant for a generalized variable-coefficient nonlinear Schrödinger equation from plasma physics, arterial mechanics, fluid dynamics and optical communications

In this paper, the multisoliton solutions in terms of double Wronskian determinant are presented for a generalized variable-coefficient nonlinear Schrödinger equation, which appears in space and laboratory plasmas, arterial mechanics, fluid dynamics, optical communications and so on. By means of the particularly nice properties of Wronskian determinant, the solutions are testified through direct substitution into the bilinear equations. Furthermore, it can be proved that the bilinear Bäcklund transformation transforms between (N − 1)- and N-soliton solutions.     

Exact Solutions of the Schrödinger Equation with Irregular Singularity

The 1D nonrelativistic Schrödinger equation possessing an irregular singularpoint is investigated. We apply a general theorem about existence and structureof solutions of linear ordinary differential equations to the Schrödinger equationand obtain suitable ansatz functions and their asymptotic representations for alarge class of singular potentials. Using these ansatz functions, we work out allpotentials for which the irregular singularity can be removed and replaced by aregular one. We obtain exact solutions for these potentials and present sourcecode for the computer algebra system Mathematica to compute the solutions. Forall cases in which the singularity cannot be weakened, we calculate the mostgeneral potential for which the Schrödinger equation is solved by the ansatzfunctions obtained and develop a method for finding exact solutions.     

变系数非线性Schrödinger方程的孤子解及其相互作用

在光孤子通信和Bose-Einstein凝聚体动力学研究中,求解广义非线性Schrödinger方程是一个重要的研究方向.稳定的孤子模式具有潜在的应用,可为实验技术的实现提供依据.本文引进一种相似变换,将变系数非线性Schrödinger方程转化成非线性Schrödinger方程,并利用已知解深入研究变系数非线性Schrödinger方程解的单孤子解、两孤子解和连续波背景下的孤子解.同时通过选择不同的具体参数,给出它们的图像分析和相应的讨论. 关键词： 非线性Schrö dinger方程 相似变换 变系数 孤子解     

Exact spatial soliton solution for nonlinear Schrödinger equation with a type of transverse nonperiodic modulation

In this paper, we analyze (2 + 1)D nonlinear Schrödinger (NLS) equation based on a type of nonperiodic modulation of linear refractive index in the transverse direction. We obtain an exact solution in explicit form for the (2 + 1)D nonlinear Schrödinger (NLS) equation with the nonperiodic modulation. Finally, the stability of the solution is discussed numerically, and the results reveal that the solution is stable to the finite initial perturbations.     

Solving the Schrödinger equation for a charged particle in a magnetic field using the finite difference time domain method

We extend our finite difference time domain method for numerical solution of the Schrödinger equation to cases where eigenfunctions are complex-valued. Illustrative numerical results for an electron in two dimensions, subject to a confining potential V(x,y), in a constant perpendicular magnetic field demonstrate the accuracy of the method.     

Exact solutions of the Schrödinger equation for a class of hyperbolic potential well

We propose a new scheme to study the exact solutions of a class of hyperbolic potential well. We first apply different forms of function transformation and variable substitution to transform the Schrödinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same eigenstate. And then in terms of the energy spectrum equation which is obtained from the Wronskian determinant, we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential well. Such a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun function. The linearly dependent relation between two eigenfunctions is also studied.     

Induced electric dipole in a quantum ring

In this contribution, we investigate the quantum dynamics of a neutral particle confined in a quantum ring potential. We use two different field configurations for induced electric dipole in the presence of electric and magnetic fields and a general confining potential, for which we solve the Schrödinger equation and obtain the complete set of eigenfunctions and eigenvalues.     

Azimuthal modulational instability of vortices in the nonlinear Schrödinger equation

We study the azimuthal modulational instability of vortices with different topological charges, in the focusing two-dimensional nonlinear Schrödinger (NLS) equation. The method of studying the stability relies on freezing the radial direction in the Lagrangian functional of the NLS in order to form a quasi-one-dimensional azimuthal equation of motion, and then applying a stability analysis in Fourier space of the azimuthal modes. We formulate predictions of growth rates of individual modes and find that vortices are unstable below a critical azimuthal wave number. Steady-state vortex solutions are found by first using a variational approach to obtain an asymptotic analytical ansatz, and then using it as an initial condition to a numerical optimization routine. The stability analysis predictions are corroborated by direct numerical simulations of the NLS. We briefly show how to extend the method to encompass nonlocal nonlinearities that tend to stabilize such solutions.     

Emulation of the evolution of a Bose–Einstein condensate in a time-dependent harmonic trap

We emulate the ground state of a Bose–Einstein condensate in a time-dependent isotropic harmonic trap by constructing analytic simulacra of a transformed wavefunction in the regions around the origin and far from the origin. This transformed wavefunction is obtained through a pseudoconformal transformation and is a function of new spatial and temporal variables, while the simulacra are generalisations of asymptotic solutions of the nonlinear Schrödinger equation and they are matched by requiring continuity not only of the wavefunction and of its slope, but of its curvature as well. The resulting piecewise analytic simulacra coincide almost perfectly with the numerically obtained solutions of the time-dependent nonlinear Schrödinger equation and constitute an easy and accurate analytic method for describing fully the condensate ground state.     

Preliminary analysis of resonance effect by Helmholtz Schrdinger method

The Helmholtz-Schrdinger method is employed to study the electric field standing wave caused by coupling through a simple slot. There is a good agreement between the numerical results and the resonant conditions presented by the Helmholtz-Schrdinger method. Thus, it can be used in similar cases where the amplitude of the electric field is the important quantity or eigenfunctions of the Schrdinger equation are needed for complicated quantum structures with hard wall boundary conditions.     

The rotation–vibration spectrum of diatomic molecules with the Tietz–Hua rotating oscillator and approximation scheme to the centrifugal term

The Tietz–Hua (TH) potential is one of the very best analytical model potentials for the vibrational energy of diatomic molecules. By using the Nikiforov–Uvarov method and Pekeris approximation to the centrifugal term, we have obtained the solutions of the radial Schrödinger equation for the TH potential. The energy eigenvalues and corresponding eigenfunctions are calculated in closed form. Some remarks and numerical results are also presented for some diatomic molecules.     

Exact and approximate solutions of Schrödinger’s equation for a class of trigonometric potentials

The asymptotic iteration method is used to find exact and approximate solutions of Schrödinger’s equation for a number of one-dimensional trigonometric potentials (sine-squared, double-cosine, tangent-squared, and complex cotangent). Analytic and approximate solutions are obtained by first using a coordinate transformation to reduce the Schrödinger equation to a second-order differential equation with an appropriate form. The asymptotic iteration method is also employed indirectly to obtain the terms in perturbation expansions, both for the energies and for the corresponding eigenfunctions.