Matter-wave solitons in the presence of collisional inhomogeneities: Perturbation theory and the impact of derivative terms

We study the dynamics of bright and dark matter-wave solitons in the presence of a spatially varying nonlinearity. When the spatial variation does not involve zero crossings, a transformation is used to bring the problem to a standard nonlinear Schrödinger form, but with two additional terms: an effective potential one and a non-potential term. We illustrate how to apply perturbation theory of dark and bright solitons to the transformed equations. We develop the general case, but primarily focus on the non-standard special case whereby the potential term vanishes, for an inverse square spatial dependence of the nonlinearity. In both cases of repulsive and attractive interactions, appropriate versions of the soliton perturbation theory are shown to accurately describe the soliton dynamics.     

Modulating the amplitude of dark soliton by scattering-length management in Bose-Einstein condensates

We present a family of soliton solutions of the quasi-one-dimensional Bose-Einstein condensates with time-dependent scattering length, by developing multiple-scale method combined with truncated Painlevé expansion. Then, by numerical calculating the solutions, it is shown that there exhibit two types of dark solitons—black soliton (the zero minimum amplitude at its center) and gray soliton (the minimum density does not drop to zero) in a repulsive condensate. Furthermore, we propose experimental protocols to realize the exchange between black and gray solitons by varying the scattering length via the Feshbach resonance in currently experimental conditions.     

Controlling Chaos Probability of a Bose-Einstein Condensate in a Weak Optical Superlattice

The spatial chaos probability of a Bose-Einstein condensate perturbed by a weak optical superlattice is studied. It is demonstrated that the spatial. chaotic solution appears with a certain probability in a given parameter region under a random boundary condition. The effects of the lattice depths and wave vectors on the chaos probability are illustrated, and different regions associated with different chaos probabilities are found. This suggests a feasible scheme for suppressing and strengthening chaos by adjusting the optical superlattice experimentaJly.     

Suppression of Chaos in a Bose-Einstein Condensate Loaded into a Moving Optical Superlattice Potential

We have shown that the application of modulating the secondary lattice is an efficient route to suppressing the generation of chaotic traveling waves of a Bose-Einstein Condensate with attractive interatomic interaction loaded into a moving optical superlattiee consisting of two lattices. With the Melnikov method, we obtain the optimal value of the relative phase between the two lattice harmonics for the control of chaos. We also find that the regularization route as the potential depth of the secondary lattice is varied and fairly rich, including the period-doubling bifurcations.     

Stabilities of one-dimensional stationary states of Bose-Einstein condensates

We explore the dynamical stabilities of a quasi-one-dimensional (1D) Bose-Einstein condensate (BEC) consisting of fixed N atoms with time-independent external potential. For the stationary states with zero flow density the general solution of the perturbed time evolution equation is constructed, and the stability criterions concerning the initial conditions and system parameters are established. Taking the lattice potential case as an example, the stability and instability regions on the parameter space are found. The results suggest a method for selecting experimental parameters and adjusting initial conditions to suppress the instabilities.     

Dynamics of Bright Solitons in Bose-Einstein Condensates with Complicated Potential

We present analytical solutions of the one-dimensional nonlinear Schrodinger equations of Bose-Einstein condensates in an expulsive parabolic background with a complex potential and gravitational field, by performing the Darboux transformation from a trivial seed solution. It is shown that under a safe range of parameter, the shape of bright soliton can be controlled well by adjusting the experimental parameter of the ratio of axial oscillation to radial oscillation and feeding condensates from a thermal cloud. The gravitational field can change the contrail of the bright soliton trains without changing their peak and width.     

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.     

Collapse of triaxial bright solitons in atomic Bose-Einstein condensates

We study triaxial bright solitons made of attractive Bose-condensed atoms characterized by the absence of confinement in the longitudinal axial direction but trapped by an anisotropic harmonic potential in the transverse plane. By numerically solving the three-dimensional Gross-Pitaevskii equation we investigate the effect of the transverse trap anisotropy on the critical interaction strength above which there is the collapse of the condensate. The comparison with previous predictions [A. Gammal, L. Tomio, T. Frederico, Phys. Rev. A 66 (2002) 043619] shows significant differences for large anisotropies.     

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.     

Ratchet-induced matter-wave transport and soliton collisions in Bose-Einstein condensates

We study the dynamics of bright matter-wave solitons in a Bose-Einstein condensate with negative scattering length under the influence of a time-periodic ratchet potential. The potential is formed by a one-dimensional bichromatic optical lattice which flashes on and off so that the time average of its amplitude vanishes. Due to the broken space and time-reversal symmetries of the potential, the soliton is transported with a nonzero average velocity. By employing the non-dissipative mean-field model for the matter waves, we study the dependence of the transport velocity on the initial state of the soliton and show how the properties of the individual localized states affect the outcome of their collisions. A useful insight into the transport properties is provided by Hamiltonian theory for the mean field, which treats the extended matter-wave excitation as an effective classical particle.     

Matter-wave vortices and solitons in anisotropic optical lattices

Using numerical methods, we construct families of vortical, quadrupole, and fundamental solitons in a two-dimensional (2D) nonlinear-Schrödinger/Gross-Pitaevskii equation which models Bose-Einstein condensates (BECs) or photonic crystals. The equation includes the attractive or repulsive cubic nonlinearity and an anisotropic periodic potential. Two types of anisotropy are considered, accounted for by the difference in the strengths of the 1D sublattices, or by a difference in their periods. The limit case of the quasi-1D optical lattice (OL), when one sublattice is missing, is included too. By means of systematic simulations, we identify stability limits for two species of vortex solitons and quadrupoles, of the rhombus and square types. In the attraction model, rhombic vortices and quadrupoles remain stable up to the limit case of the quasi-1D lattice. In the same model, finite stability limits are found for vortices and quadrupoles of the square type, in terms of the anisotropy parameter. In the repulsion model, rhombic vortices and quadrupoles are stable in large parts of the first finite bandgap (FBG). Another species of partly stable anisotropic states is found in the second FBG, subfundamental dipoles, each squeezed into a single cell of the OL. Square-shaped quadrupoles are completely unstable in the repulsion model, while vortices of the same type are stable only in weakly anisotropic OL potentials.     

Integrability of the Gross-Pitaevskii equation with Feshbach resonance management

In this Letter we study the integrability of a class of Gross-Pitaevskii equations managed by Feshbach resonance in an expulsive parabolic external potential. By using WTC test, we find a condition under which the Gross-Pitaevskii equation is completely integrable. Under the present model, this integrability condition is completely consistent with that proposed by Serkin, Hasegawa, and Belyaeva [V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98 (2007) 074102]. Furthermore, this integrability can also be explicitly shown by a transformation, which can convert the Gross-Pitaevskii equation into the well-known standard nonlinear Schrödinger equation. By this transformation, each exact solution of the standard nonlinear Schrödinger equation can be converted into that of the Gross-Pitaevskii equation, which builds a systematical connection between the canonical solitons and the so-called nonautonomous ones. The finding of this transformation has a significant contribution to understanding the essential properties of the nonautonomous solitons and the dynamics of the Bose-Einstein condensates by using the Feshbach resonance technique.     

Solutions of Two Kinds of Non-Isospectral Generalized Nonlinear Schrodinger Equation Related to Bose-Einstein Condensates

Two non-isospectral generalized nonlinear Schrodinger （ONLS） equations, which are two important models of nonlinear excitations of matter waves in Bose-Einstein condensates, are studied. Two novel transformations are constructed such that these two GNLS equations are transformed to the well-known nonlinear Schr6dinger （NLS） equation, which is an isospectral equation. Therefore, once one solution of the NLS equation is provided, we can immediately obtain one solution for two ONLS equations by these transformations. Thus it is unnecessary to solve these two non-isospectral GNLS equations directly. Soliton solutions and periodic solutions are obtained for them by two transformations from the corresponding solutions of the NLS equation, which are generated by Darboux transformation.     

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.     

Bright matter wave solitons and their collision in Bose–Einstein condensates

We obtain the bright matter wave solitons in Bose–Einstein condensates from a trivial input solution by solving the time dependent Gross–Pitaevskii (GP) equation with quadratic potential and exponentially varying scattering length. We observe that the matter wave density of bright soliton increases with time by virtue of the exponentially increasing scattering length. We also understand that the matter wave densities of bright soliton trains remain finite despite the exchange of atoms during interaction and they travel along different trajectories (diverge) after interaction. We also observe that their amplitudes continue to fluctuate with time. For exponentially decaying scattering lengths, instability sets in the condensates. However, the scattering length can be suitably manipulated without causing the explosion or the collapse of the condensates.     

Controlled Manipulation with a Bose--Einstein Condensates N-Soliton Train under the Influence of Harmonic and Tilted Periodic Potentials

A model of the perturbed complex Toda chain （PCTC） to describe the dynamics of a Bose-Einstein condensate （BEC） N-soliton train trapped in an applied combined external potential consisting of both a weak harmonic and tilted periodic component is first developed. Using the developed theory, the BEC N-soliton train dynamics is shown to be well approximated by 4N coupled nonlinear differential equations, which describe the fundamental interactions in the system arising from the interplay of amplitude, velocity, centre-of-mass position, and phase. The simplified analytic theory allows for an efficient and convenient method for characterizing the BEC N-soliton train behaviour. It further gives the critical values of the strength of the potential for which one or more localized states can be extracted from a soliton train and demonstrates that the BEC N-soliton train can move selectively from one lattice site to another by simply manipulating the strength of the potential.     

Integrability of a general Gross-Pitaevskii equation and exact solitonic solutions of a Bose-Einstein condensate in a periodic potential

We consider a general form of the Gross-Pitaevskii equation with time- and space-dependent effective mass, external potential and strength of interatomic interaction. Using the inverse-scattering method, we derive the integrability condition of this equation within a general scheme that can be used to find exact solutions of a wide range of linear and nonlinear partial differential equations. We use this condition to derive exact solitonic solutions of the one-dimensional time-dependent Gross-Pitaevskii equation corresponding to a Bose-Einstein condensate trapped by a periodic potential. Both attractive and repulsive interatomic interactions are considered. The values of the parameters of the potential can be chosen such that the periodic potential becomes almost identical to that of an optical lattice.     

Bose-Einstein solitons in time-dependent linear potential

In this paper, Bose-Einstein soliton solutions of the nonlinear Schrödinger equation with time-dependent linear potential are considered. Based on the F-expansion method, we present a number of Jacobian elliptic function solutions. Particular cases of these solutions, where the elliptic function modulus equals 1 and 0, are various localized solutions and trigonometric functions, respectively. Specially, for V_ext = ZF(T) = Z[mg + Hcos (ω₁T)], we discussed the Bose-Einstein condensate trapped in the coupling external field with considering the effect of gravity; for F(T) = constant, it describes the wave (Langmuir or electromagnetic) in a linearly inhomogeneous plasma with cubic nonlinearly.     

A supersymmetric Sawada-Kotera equation

A new supersymmetric equation is proposed for the Sawada-Kotera equation. The integrability of this equation is shown by the existence of Lax representation and infinite conserved quantities and a recursion operator.