Resonant reflection of Bose-Einstein condensate by a double delta-function barrier within the Gross-Pitaevskii equation

Resonant reflection of a Bose-Einstein condensate by a double delta-function barrier has been considered analytically using the Gross-Pitaevskii approximation for nonlinearity. The reflection coefficient has been derived taking into account a weak nonlinearity of the Schrödinger equation produced by the interaction between cold alkali atoms. Nonlinear term is given in the limit of asymptotically weak interaction and zero temperature. The one-dimensional potential is approximated by two repulsive delta-function barriers. The analytical solution was obtained for the reflection coefficient using a multiple-scale analysis in order to remove secular terms. The most interesting case corresponds to the condensate energies for which reflection is absent without nonlinear term. Thus, reflection is determined only by the nonlinearity. The reflection coefficient is derived in the first order on the nonlinearity parameter.     

Derivation of Nonlinear Schrödinger Equation

We propose some nonlinear Schrödinger equations by adding some higher order terms to the Lagrangian density of Schrödinger field, and obtain the Gross-Pitaevskii (GP) equation and the logarithmic form equation naturally. In addition, we prove the coefficient of nonlinear term is very small, i.e., the nonlinearity of Schrödinger equation is weak.     

Above-barrier reflection of cold atoms by resonant laser light within the Gross-Pitaevskii approximation

Above-barrier reflection of cold alkali atoms by resonant laser light was considered analytically within the Gross-Pitaevskii approximation. Correction for the reflection coefficient because of a weak nonlinearity of the stationary Schrödinger equation has been derived using multiscale analysis as a form of perturbation theory. The nonlinearity adds spatial harmonics to linear incident and reflecting waves. It was shown that the role of nonlinearity increases when the kinetic energy of an atom is nearly to the height of the potential barrier. Results are compared to the known numerical derivations for wave functions of the Gross-Pitaevskii equation with the step potential.     

THE NONLINEAR SCHRÖDINGER EQUATION FOR THE DELTA-COMB POTENTIAL: QUASI-CLASSICAL CHAOS AND BIFURCATIONS OF PERIODIC STATIONARY SOLUTIONS

The nonlinear Schrödinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schrödinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear stationary states of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic stationary states and a transition to spatial chaos. The relation to new features of nonlinear Bloch bands, such as looped and period doubled bands, are analyzed in detail. An analytic expression for the critical nonlinearity for the emergence of looped bands is derived. The results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks and the dynamical stability of periodic solutions in a Gaussian comb is discussed.     

Stability of gap solitons in the presence of a weak nonlocality in periodic potentials

In this work, we study the stability and internal modes of one-dimensional gap solitons employing the modified nonlinear Schrödinger equation with a sinusoidal potential together with the presence of a weak nonlocality. Using an analytical theory, it is proved that two soliton families bifurcate out from every Bloch-band edge under self-focusing or self-defocusing nonlinearity, and one of these is always unstable. Also we study the oscillatory instabilities and internal modes of the modified nonlinear Schrödinger equation.     

非均匀光纤中暗孤子传输特性研究

由于变系数非线性Schrödinger方程的增益、色散和非线性项都是变化的, 根据方程这一特点可以研究光脉冲在非均匀光纤中的传输特性. 本文利用Hirota方法, 得到非线性Schrödinger方程的解析暗孤子解. 然后根据暗孤子解对暗孤子的传输特性进行讨论, 并且分析各个物理参量对暗孤子传输的影响. 经研究发现, 通过调节光纤的损耗、色散和非线性效应都能有效的控制暗孤子的传输, 从而提高非均匀光纤中的光脉冲传输质量. 此外, 本文还得到了所求解方程的解析双暗孤子解, 最后对两个暗孤子相互作用进行了探讨. 本文得到的结论有利于研究非均匀光纤中的孤子控制技术.     

Hydrodynamics of Bose and Fermi superfluids at zero temperature: the superfluid nonlinear Schrödinger equation

We discuss the zero-temperature hydrodynamics equations of bosonic and fermionic superfluids and their connection with generalized Gross-Pitaevskii and Ginzburg-Landau equations through a single superfluid nonlinear Schrödinger equation.     

Collapse arrest in the space-fractional Schrödinger equation with an optical lattice

The soliton solution and collapse arrest are investigated in the one-dimensional space-fractional Schr?dinger equation with Kerr nonlinearity and optical lattice. The approximate analytical soliton solutions are obtained based on the variational approach, which provides reasonable accuracy. Linear-stability analysis shows that all the solitons are linearly stable. No collapses are found when the Lévy index 1 α≤ 2. For α = 1, the collapse is arrested by the lattice potential when the amplitude of perturbations is small enough. It is numerically proved that the energy criterion of collapse suppression in the two-dimensional traditional Schr?dinger equation still holds in the one-dimensional fractional Schr?dinger equation. The physical mechanism for collapse prohibition is also given.     

An exact (2 + 1)-dimensional optical soliton with spatially modulated nonlinearity and an external potential

An exact (2 + 1)-dimensional spatial optical soliton of the nonlinear Schrödinger equation with a spatially modulated nonlinearity and a special external potential is discovered in an inhomogeneous nonlinear medium, by utilizing the similarity transformation. Exact analytical solutions are constructed by the products of Whittaker functions and the bright and dark soliton solutions of the standard stationary nonlinear Schrödinger equation. Some examples of such composed solutions are given, in which these spatial solitons display different localized structures. Numerical calculation shows that the soliton is stable in propagating over long distances, thus also confirming the validity of the exact solution.     

Non-autonomous wave solutions for the Gross-Pitaevskii (GP) equation with a parabola external potential in Bose-Einstein condensates

A Gross-Pitaevskii (GP) equation with a parabola external potential is considered, and is transformed into a standard nonlinear Schrödinger (NLS) equation. By using the homogeneous balance principle and F-expansion method, we study non-autonomous wave solutions of the GP equation with a parabola external potential. In particular, based on the similarity transformation, several families of non-autonomous wave solutions of the GP equation are presented with snaking behaviors and different amplitude surfaces. These obtained bright-dark soliton solutions can give some potential applications in Bose-Einstein condensates.     

On nonlinear effects in magnetic chains

The effect of phonon unharmonism and nonlinearity in exchange integrals on soliton excitations in ferromagnetic chains in the classical and long-wave limit are studied. It has been first shown that the unharmonic effect leads to a system of coupled Boussinesque and nonlinear Schrödinger equations allowing two types of soliton solutions. The nonlinear effect on the other hand results nonlinear Schrödinger equation with saturable nonlinearity admitting stable solitons in higher dimensional models.     

New solutions in the theory of self-focusing with saturating nonlinearity

On the basis of an approximate analytic solution of a Cauchy problem for a nonlinear Schrödinger (NLS) equation describing steady-state light beams in a medium with saturating nonlinearity by the method of renormgroup (RG) symmetries, a classification of self-focusing solutions is given depending on two control parameters: the relative contributions of diffraction and nonlinearity and the saturation strength of the nonlinearity. The existence of tube-type self-focusing solutions is proved for an entering beam with Gaussian radial distribution of intensity. Numerical simulation is carried out that allows one to verify the theory developed and to determine its applicability limits.     

Periodic Solutions for a Class of Nonlinear Partial Differential Equations in Higher Dimension

We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schrödinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our result covers cases of completely resonant equations, where the bifurcation equation is infinite-dimensional, such as the nonlinear Schrödinger equation with zero mass, for which solutions which at leading order are wave packets are shown to exist.     

Stationary States for Nonlinear Schrödinger Equations with Periodic Potentials

In this paper we consider a one-dimensional non-linear Schrödinger equation with a periodic potential. In the semiclassical limit we prove the existence of stationary solutions by means of the reduction of the non-linear Schrödinger equation to a discrete non-linear Schrödinger equation. In particular, in the limit of large nonlinearity strength the stationary solutions turn out to be localized on a single lattice site of the periodic potential. A connection of these results with the Mott insulator phase for Bose–Einstein condensates in a one-dimensional periodic lattice is also discussed.     

Multi-symplectic variational integrators for nonlinear Schrdinger equations with variable coefficients

In this paper, we propose a variational integrator for nonlinear Schrdinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrdinger equations with variable coefficients, cubic nonlinear Schrdinger equations and Gross–Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space.     

Analytical solutions for the spin-1 Bose-Einstein condensate in a harmonic trap

The homotopy analysis method and Galerkin spectral method are applied to find the analytical solutions for the Gross-Pitaevskii equations, a set of nonlinear Schrödinger equation used in simulation of spin-1 Bose-Einstein condensates trapped in a harmonic potential. We investigate the one-dimensional case and get the approximate analytical solutions successfully. Comparisons between the analytical solutions and the numerical solutions have been made. The results indicate that they are in agreement well with each other when the atomic interaction is weakly. We also find a class of exact solutions for the stationary states of the spin-1 system with harmonic potential for a special case.     

Two-dimensional stability of large amplitude langmuir waves and solitons

The two-dimensional stability of nonlinear wave and soliton solutions of the exponentially nonlinear Schrödinger equation is examined. All stationary entities are unstable to two-dimensional perturbations. It is found that the saturable nonlinearity decreases growth rates in comparison with the small amplitude limit.     

Description of Pulse Propagation in Optical Strip Waveguides by the Generalized Nonlinear Schrödinger Equation

Taking into consideration the transverse confinement, the dispersion of the linear as well as of the nonlinear part of the refractive index and the nonlinear interaction of the electromagnetic field with the guiding material, the yielding equation for the envelope function of the fundamental mode propagating in a strip waveguide is the generalized nonlinear Schrödinger equation. In contrast to the nonlinear Schrödinger equation, solitons of the generalized nonlinear Schrödinger equation exist in the regimes of negative and positive group dispersion and are asymmetric.     

Symmetry-Breaking Bifurcation in the Nonlinear Schrödinger Equation with Symmetric Potentials

We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter.