Exact Polynomial Solutions of Schrdinger Equation with Various Hyperbolic Potentials

The Schrdinger equation with hyperbolic potential 2V(x) =-V0sinhq(x/d)/cosh6(x/d)(q = 0, 1, 2, 3) is studied by transforming it into the confluent Heun equation. We obtain general symmetric and antisymmetric polynomial solutions of the Schrdinger equation in a unified form via the Functional Bethe ansatz method. Furthermore, we discuss the characteristic of wavefunction of bound state with varying potential strengths. Particularly, the number of wavefunction's nodes decreases with the increase of potential strengths, and the particle tends to the bottom of the potential well correspondingly.     

Nonautonomous bright solitons and soliton collisions in a nonlinear medium with an external potential

We present exact bright multi-soliton solutions of a generalized nonautonomous nonlinear Schrdinger equation with time-and space-dependent distributed coefficients and an external potential which describes a pulse propagating in nonlinear media when its transverse and longitudinal directions are nonuniformly distributed.Such solutions exist in certain constraint conditions on the coefficients depicting dispersion,nonlinearity,and gain(loss).Various shapes of bright solitons and interesting interactions between two solitons are observed.Physical applications of interest to the field and stability of the solitons are discussed.     

Solution of the Schrödinger equation for a particular form of Morse potential using the Laplace transform

In this paper, we have solved the Schrdinger equation for a particular kind of Morse potential and find its normalized eigenfunctions and eigenvalues, exactly. Our work is based on the Laplace transform technique which reduces the second-order differential equation to a first-order.     

Control of soliton characteristics of the condensate by an arbitrary x-dependent external potential

This paper presents a family of soliton solutions of the one-dimensional nonlinear Schrdinger equation which describes the dynamics of the dark solitons in Bose-Einstein condensates with an arbitrary x-dependent external potential.The obtained results show that the external potential has an important effect on the dark soliton dynamical characteristics of the condensates.The amplitude,width,and velocity of the output soliton are relative to the source position of the external potential.The smaller the amplitude of the soliton is,the narrower its width is,and the slower the soliton propagates.The collision of two dark solitons is nearly elastic.     

Bose-Einstein Condensate in a Linear Trap with a Dimple Potential

We study Bose-Einstein condensation in a linear trap with a dimple potential where we model dimple potentials by Dirac δ function. Attractive and repulsive dimple potentials are taken into account. This model allows simple, explicit numerical and analytical investigations of noninteracting gases. Thus, the Schrdinger equation is used instead of the Gross-Pitaevski equation. We calculate the atomic density, the chemical potential, the critical temperature and the condensate fraction. The role of the relative depth of the dimple potential with respect to the linear trap in large condensate formation at enhanced temperatures is clearly revealed. Moreover, we also present a semi-classical method for calculating various quantities such as entropy analytically. Moreover, we compare the results of this paper with the results of a previous paper in which the harmonic trap with a dimple potential in 1D is investigated.     

Soliton,Breather and Rogue Wave Solutions for the Nonlinear Schrdinger Equation Coupled to a Multiple Self-Induced Transparency System

We derive an N-fold Darboux transformation for the nonlinear Schrdinger equation coupled to a multiple selfinduced transparency system, which is applicable to optical fiber communications in the erbium-doped medium.The N-soliton, N-breather and N th-order rogue wave solutions in the compact determinant representations are derived using the Darboux transformation and limit technique. Dynamics of such solutions from the first-to second-order ones are shown.     

A theoretical and experimental study on all-normal-dispersion Yb-doped mode-locked fiber lasers

We report on a theoretical and experimental study of an all-normal-dispersion (ANDi) Yb-doped mode-locked fiber laser, in which nonlinear polarization rotation (NPR) is used to realize mode-locking without any dispersion compensation. Based on the coupled nonlinear Schrdinger (CNLS) equation, a model simulating the mode-locked process of an allnormal-dispersion ring fiber laser is developed, which shows that the achievement of stable mode-locking depends on the alignment of the polarization controller (PC) along the fast-polarization axis of the fiber, the birefringence intensity, and the net cavity dispersion. According to the theoretical analysis, stable mode-locked pulses with pulse duration 300 ps and average output power 33.9mW at repetition rate 36MHz are obtained.     

The bound state solution for the Morse potential with a localized mass profile

We investigate an analytical solution for the Schr o¨dinger equation with a position-dependent mass distribution, with the Morse potential via Laplace transformations. We considered a mass function localized around the equilibrium position.The mass distribution depends on the energy spectrum of the state and the intrinsic parameters of the Morse potential. An exact bound state solution is obtained in the presence of this mass distribution.     

An upper bound for the adiabatic approximation error

In this paper,we derive an upper bound for the adiabatic approximation error,which is the distance between the exact solution to a Schr dinger equation and the adiabatic approximation solution.As an application,we obtain an upper bound for 1 minus the fidelity of the exact solution and the adiabatic approximation solution to a Schrdinger equation.     

Effect of Time-Dependent Atomic Scattering Length on Solitons in Bose-Einstein Condensates with a Complex Potential

We consider the one-dimensional nonlinear Schrǒdinger equations that describe the dynamics of a Bose-Einstein Condensates with time-dependent scattering length in a complex potential. Our results show that as long as the integrable relation is satisfied, exact solutions of the one-dimensional nonlinear Schrǒdinger equation can be found in a general closed form, and interactions between two solitons are modulated in a complex potential We find that the changes of the scattering length and trapping potential can be effectively used to control the interaction between two bright soliton.     

A new finite difference scheme for a dissipative cubic nonlinear Schrdinger equation

This paper considers the one-dimensional dissipative cubic nonlinear Schrdinger equation with zero Dirichlet boundary conditions on a bounded domain.The equation is discretized in time by a linear implicit three-level central difference scheme,which has analogous discrete conservation laws of charge and energy.The convergence with two orders and the stability of the scheme are analysed using a priori estimates.Numerical tests show that the three-level scheme is more efficient.     

Analytic Results in the Position-Dependent Mass SchrSdinger Problem

We investigate the Schr6dinger equation for a particle with a nonuniform solitonic mass density. First, we discuss in extent the （nontrivial） position-dependent mass V（x） = 0 case whose solutions are hypergeometric functions in tanh2 x. Then, we consider an external hyperbolic-tangent potential. We show that the effective quantum mechanical problem is given by a Heun class equation and find analytically an eigenbasis for the space of solutions. We also compute the eigenstat, es for a potential of the form V （x） = Vo sinh2 z.     

Trial function method and exact solutions to the generalized nonlinear Schrdinger equation with time-dependent coefficient

In this paper,the trial function method is extended to study the generalized nonlinear Schrdinger equation with timedependent coefficients.On the basis of a generalized traveling wave transformation and a trial function,we investigate the exact envelope traveling wave solutions of the generalized nonlinear Schrdinger equation with time-dependent coefficients.Taking advantage of solutions to trial function,we successfully obtain exact solutions for the generalized nonlinear Schrdinger equation with time-dependent coefficients under constraint conditions.     

Solitons for a generalized variable-coefficient nonlinear Schrdinger equation

In this paper,we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear Schrdinger equation (NLS) with an arbitrary time-dependent linear potential which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations. Under some reasonable assumptions,one-soliton and two-soliton solutions are constructed analytically by the Hirota method. From our results,some previous one-and two-soliton solutions for some NLS-type equations can be recovered by some appropriate selection of the various parameters. Some figures are given to demonstrate some properties of the one-and the two-soliton and the discussion about the integrability property and the Hirota method is given finally.     

Detuning Effects in the One-Photon Mazer

A theoretical approach is proposed to deal with the detuning between the atom and cavity field in the one-photon mazer. Our starting point is to find a conserved quantity, which is characteristic of the atom plus field system. Resorting to this characteristic conserved quantity, we separate the Hamiltonian into two independent parts and then solve the Schr6dinger equation of quantum scattering of an incident cold atom. We investigate the detuning effects on the one-photon induced emission probability and obtain some distinct results in contrast to the previous work.     

Kuznetsov–Ma soliton and Akhmediev breather of higher-order nonlinear Schrdinger equation

In terms of Darboux transformation, we have exactly solved the higher-order nonlinear Schrdinger equation that describes the propagation of ultrashort optical pulses in optical fibers. We discuss the modulation instability(MI) process in detail and find that the higher-order term has no effect on the MI condition. Under different conditions, we obtain Kuznetsov–Ma soliton and Akhmediev breather solutions of higher-order nonlinear Schrdinger equation. The former describes the propagation of a bright pulse on a continuous wave background in the presence of higher-order effects and the soliton's peak position is shifted owing to the presence of a nonvanishing background, while the latter implies the modulation instability process that can be used in practice to produce a train of ultrashort optical soliton pulses.     

Deformed oscillator algebra for quantum superintegrable systems in two-dimensional Euclidean space and on a complex two-sphere

In this work,we study superintegrable quantum systems in two-dimensional Euclidean space and on a complex twosphere with second-order constants of motion.We show that these constants of motion satisfy the deformed oscillator algebra.Then,we easily calculate the energy eigenvalues in an algebraic way by solving of a system of two equations satisfied by its structure function.The results are in agreement to the ones obtained from the solution of the relevant Schrdinger equation.     

Twenty Parameters Families of Solutions to the NLS Equation and the Eleventh Peregrine Breather

The Peregrine breather of order eleven(P_(11) breather) solution to the focusing one-dimensional nonlinear Schrdinger equation(NLS) is explicitly constructed here. Deformations of the Peregrine breather of order 11 with 20 real parameters solutions to the NLS equation are also given: when all parameters are equal to 0 we recover the famous P_(11) breather. We obtain new families of quasi-rational solutions to the NLS equation in terms of explicit quotients of polynomials of degree 132 in x and t by a product of an exponential depending on t. We study these solutions by giving patterns of their modulus in the(x; t) plane, in function of the different parameters.     

A new approach of solving Green’s function for wave propagation in an inhomogeneous absorbing medium

A new approach is developed to solve the Green's function that satisfies the Hehmholtz equation with complex refractive index. Especially, the Green's function for the Helmholtz equation can be expressed in terms of a one-dimensional integral, which can convert a Helmholtz equation into a Schr?dinger equation with complex potential. And the Schr?dinger equation can be solved by Feynman path integral. The result is in excellent agreement with the previous work.