Analytical study of the nonlinear Schrödinger equation with an arbitrary linear time-dependent potential in quasi-one-dimensional Bose-Einstein condensates

Under investigation in this paper is a nonlinear Schrödinger equation with an arbitrary linear time-dependent potential, which governs the soliton dynamics in quasi-one-dimensional Bose-Einstein condensates (quasi-1DBECs). With Painlevé analysis method performed to this model, its integrability is firstly examined. Then, the distinct treatments based on the truncated Painlevé expansion, respectively, give the bilinear form and the Painlevé-Bäcklund transformation with a family of new exact solutions. Furthermore, via the computerized symbolic computation, a direct method is employed to easily and directly derive the exact analytical dark- and bright-solitonic solutions. At last, of physical and experimental interests, these solutions are graphically discussed so as to better understand the soliton dynamics in quasi-1DBECs.     

Nonautonomous solitons of Bose-Einstein condensation in a linear potential with an arbitrary time-dependence

In the presence of a linear potential with an arbitrary time-dependence, Hirota method is developed carefully for applying into the effective mean-field model of quasi-one-dimensional Bose-Einstein condensation with repulsive interaction. We obtain the exact nonautonomous soliton solution (NSS) analytically. These solutions show that the time-dependent potential can affect the velocity of NSS. In some special cases the velocity has the character of both increase and oscillation with time. A detail analysis for the asymptotic behaviour of solutions shows that the collision of two NSSs is elastic.     

Dark soliton interaction of spinor Bose-Einstein condensates in an optical lattice

We study the magnetic soliton dynamics of spinor Bose-Einstein condensates in an optical lattice which results in an effective Hamiltonian of anisotropic pseudospin chain. An equation of nonlinear Schrödinger type is derived and exact magnetic soliton solutions are obtained analytically by means of Hirota method. Our results show that the critical external field is needed for creating the magnetic soliton in spinor Bose-Einstein condensates. The soliton size, velocity and shape frequency can be controlled in practical experiment by adjusting the magnetic field. Moreover, the elastic collision of two solitons is investigated in detail.     

Angular pseudomomentum theory for the generalized nonlinear Schrödinger equation in discrete rotational symmetry media

We develop a complete mathematical theory for the symmetrical solutions of the generalized nonlinear Schrödinger equation based on the concept of angular pseudomomentum. We consider the symmetric solitons of a generalized nonlinear Schrödinger equation with a nonlinearity depending on the modulus of the field. We provide a rigorous proof of a set of mathematical results justifying that these solitons can be classified according to the irreducible representations of a discrete group. Then we extend this theory to non-stationary solutions and study the relationship between angular momentum and pseudomomentum. We illustrate these theoretical results with numerical examples.     

Nonautonomous “rogons” in the inhomogeneous nonlinear Schrödinger equation with variable coefficients

The analytical nonautonomous rogons are reported for the inhomogeneous nonlinear Schrödinger equation with variable coefficients in terms of rational-like functions by using the similarity transformation and direct ansatz. These obtained solutions can be used to describe the possible formation mechanisms for optical, oceanic, and matter rogue wave phenomenon in optical fibres, the deep ocean, and Bose-Einstein condensates, respectively. Moreover, the snake propagation traces and the fascinating interactions of two nonautonomous rogons are generated for the chosen different parameters. The obtained nonautonomous rogons may excite the possibility of relative experiments and potential applications for the rogue wave phenomenon in the field of nonlinear science.     

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.     

Analytical study of the splitting process of a multiply-quantized vortex in a Bose–Einstein condensate and collaboration of the zero and complex modes

We study the dynamics of a trapped Bose–Einstein condensate with a multiply-quantized vortex, and investigate the roles of the fluctuations in the dynamical evolution of the system. Using the perturbation theory of the external potential, and assuming the situation of the small coupling constant of self-interaction, we analytically solve the time-dependent Gross–Pitaevskii equation. We introduce the zero mode and its adjoint mode of the Bogoliubov–de Gennes equations. Those modes are known to be essential for the completeness condition. We confirm how the complex eigenvalues induce the vortex splitting. It is shown that the physical role of the adjoint zero mode is to ensure the conservation of the total condensate number. The contribution of the adjoint mode is exponentially enhanced in synchronism with the exponential growth of the complex mode, and is essential in the vortex splitting.     

Dynamical stabilization of self-trapping of two weakly coupled Bose-Einstein condensates

We investigate the effect of an external periodic driving field on the self-trapping of two weakly coupled Bose-Einstein condensates with dissipation. It is shown that the macroscopic self-trapping can be stabilized against dissipation by a high frequency periodic driving field. The parameter ranges for stabilizing self-trapping are found analytically and confirmed numerically.     

Localization phenomena in nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities: Theory and applications to Bose-Einstein condensates

We study the properties of the ground state of nonlinear Schrödinger equations with spatially inhomogeneous interactions and show that it experiences a strong localization on the spatial region where the interactions vanish. At the same time, tunneling to regions with positive values of the interactions is strongly suppressed by the nonlinear interactions and as the number of particles is increased it saturates in the region of finite interaction values. The chemical potential has a cutoff value in these systems and thus takes values on a finite interval. The applicability of the phenomenon to Bose-Einstein condensates is discussed in detail.     

Landau Damping of Collective Modes in a Disc-Shaped Bose-Einstein Condensate

We investigate the Landau damping of collective modes in an anisotropic Bose Einstein condensate （BEC）, Based on divergence-free analytical solutions for the ground state wavefunction of the condensate and all eigenvalues and eigenfunctions for thermal excited quasiparticles, we make a detailed analytical calculation on coupling matrix elements. We evaluate the Landau damping of a quadrupole collective mode in the BEC with a disc-shaped trap and discuss its dependence on temperature and particle number of the system.     

Tunneling dynamics between two-component Bose–Einstein condensates

We present a mean-field model to study the tunneling dynamics between initially separated two-component Bose condensates in a time-dependent double-well potential. We solve the model in terms of a completely numerical procedure. In contrast to the usual Josephson effect between two coherently separated single-component condensates, we find that this system sustains a macroscopic quantum self-trapping even for sufficiently weak interatomic interactions and small initial population imbalance far below the critical value.     

Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential

High precision approximate analytic expressions of the ground state energies and wave functions for the arbitrary physical potentials are found by first casting the Schrödinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. The approach is illustrated on the examples of the Yukawa, Woods-Saxon and funnel potentials. For the latter potential, solutions describing charmonium, bottonium and topponium are analyzed. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of physical parameters. The accuracy ranging between 10⁻⁴ and 10⁻⁸ for the energies and, correspondingly, 10⁻² and 10⁻⁴ for the wave functions is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects correspondent physical systems.     

Quantum dynamics of atomic coherence in a spin-1 condensate: Mean-field versus many-body simulation

We analyse and numerically simulate the full many-body quantum dynamics of a spin-1 condensate in the single spatial mode approximation. Initially, the condensate is in a “ferromagnetic” state with all spins aligned along the y axis and the magnetic field pointing along the z axis. In the course of evolution the spinor condensate undergoes a characteristic change of symmetry, which in a real experiment could be a signature of spin-mixing many-body interactions. The results of our simulations are conveniently visualised within the picture of irreducible tensor operators.     

Discrete nonlinear Schrödinger equation with arbitrary nonlocality: Linear stability analysis and localized solution

The modulational instability of a plane wave for a discrete nonlinear Schrödinger equation with arbitrary nonlocality is analyzed. This model describes light propagation in a thin film planar waveguide arrays of nematic liquid crystals subjected to a periodic transverse modulation by a low frequency electric field. It is shown that nonlocality can both suppress and promote the growth rate and bandwidth of instability, depending on the type of a response function of a discrete medium. A solitary wave (breather-like) solution is built by the variational approximation and its stability is demonstrated.     

Schrödinger equation with delta potential in superspace

A superspace version of the Schrödinger equation with a delta potential is studied using Fourier analysis. An explicit expression for the energy of the single bound state is found as a function of the super-dimension M in case M is smaller than or equal to 1. In the case when there is one commuting and 2n anti-commuting variables also the wave function is given explicitly.     

Bogolyubov approximation for diagonal model of an interacting Bose gas

We study, using the Bogolyubov approximation, the thermodynamic behavior of a superstable Bose system whose energy operator in the second-quantized form contains a nonlinear expression in the occupation numbers operators. We prove that for all values of the chemical potential satisfying μ>λ(0)$\mu >\lambda (0)$, where λ(0)?0$\lambda (0)?0$ is the lowest energy value, the system undergoes Bose–Einstein condensation.     

Exact X-wave solutions of the hyperbolic nonlinear Schrödinger equation with a supporting potential

We find exact solutions of the two- and three-dimensional nonlinear Schrödinger equation with a supporting potential. We focus in the case where the diffraction operator is of the hyperbolic type and both the potential and the solution have the form of an X-wave. Following similar arguments, several additional families of exact solutions can also can be found irrespectively of the type of the diffraction operator (hyperbolic or elliptic) or the dimensionality of the problem. In particular we present two such examples: The one-dimensional nonlinear Schrödinger equation with a stationary and a “breathing” potential and the two-dimensional nonlinear Schrödinger with a Bessel potential.     

Hirota method for oblique solitons in two-dimensional supersonic nonlinear Schrödinger flow

In a previous work El et al. (2006) [1] exact stable oblique soliton solutions were revealed in two-dimensional nonlinear Schrödinger flow. In this work we show that single soliton solution can be expressed within the Hirota bilinear formalism. An attempt to build two-soliton solutions shows that the system is “close” to integrability provided that the angle between the solitons is small and/or we are in the hypersonic limit.     

Superfluid properties of a Bose-Einstein condensate in an optical lattice confined in a cavity

We study the effect of a one dimensional optical lattice in a cavity field with quantum properties on the superfluid dynamics of a Bose-Einstein condensate (BEC). In the cavity the influence of atomic backaction and the external driving pump become important and modify the optical potential. Due to the coupling between the condensate wavefunction and the cavity modes, the cavity light field develops a band structure. This study reveals that the pump and the cavity emerges as a new handle to control the superfluid properties of the BEC.     

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.