Gap solitons in a model of a superfluid fermion gas in optical lattices

We consider a dynamical model for a Fermi gas in the Bardeen-Cooper-Schrieffer (BCS) superfluid state, trapped in a combination of a 1D or 2D optical lattice (OL) and a tight parabolic potential, acting in the transverse direction(s). The model is based on an equation for the order parameter (wave function), which is derived from the energy density for the weakly coupled BCS superfluid. The equation includes a nonlinear self-repulsive term of power 7/3, which accounts for the Fermi pressure. Reducing the equation to the 1D or 2D form, we construct families of stable 1D and 2D gap solitons (GSs) by means of numerical simulations, which are guided by the variational approximation (VA). The GSs are, chiefly, compact objects trapped in a single cell of the OL potential. In the linear limit, the VA predicts almost exact positions of narrow Bloch bands that separate the semi-infinite and first finite gaps, as well as the first and second finite ones. Families of stable even and odd bound states of 1D GSs are constructed, too. We also demonstrate that the GS can be dragged without much distortion by an OL moving at a moderate velocity (, in physical units). The predicted GSs contain ∼10³-10⁴ and ∼10³ atoms per 1D and 2D settings, respectively.     

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.     

One-dimensional delocalizing transitions of matter waves in optical lattices

We investigate, both analytically and numerically, the conditions for the occurrence of the delocalizing transition phenomenon of one-dimensional localized modes of several nonlinear continuous periodic and discrete systems of the nonlinear Schrödinger type. We show that either non-existence of solitons in the small amplitude limit or the loss of stability along existence branches can lead to delocalizing transitions, which occur following different scenarios. Examples of delocalizing transitions of both types are provided for a class of equations which describe single component and binary mixtures of Bose-Einstein condensates trapped in linear and nonlinear optical lattices.     

Matter-wave interference in Bose-Einstein condensates: A dispersive hydrodynamic perspective

The interference pattern generated by the merging interaction of two Bose-Einstein condensates reveals the coherent, quantum wave nature of matter. An asymptotic analysis of the nonlinear Schrödinger equation in the small dispersion (semiclassical) limit, experimental results, and three-dimensional numerical simulations show that this interference pattern can be interpreted as a modulated soliton train generated by the interaction of two rarefaction waves propagating through the vacuum. The soliton train is shown to emerge from a linear, trigonometric interference pattern and is found by use of the Whitham modulation theory for nonlinear waves. This dispersive hydrodynamic perspective offers a new viewpoint on the mechanism driving matter-wave interference.     

Matter-wave bright solitons: Internal atomic recombination and external feeding

We consider matter-wave bright solitons in the presence of three-body atomic recombination, an axial periodic modulation and a feeding term, and use a variational method to derive conditions to have dynamically stabilized solitons due to compensation between the dissipation and alimentation of atoms from external sources. We critically examine how the BEC soliton is affected by the imbalance between the internal atom loss and external feeding. We pay special attention to study the influence of these terms on the soliton dynamics in optical lattice potentials that cause periodic modulation.     

Switching and Self-trapping Dynamics of Dark Solitons in a Two-Component Bose-Einstein Condensate

Two coupled dark solitons are considered in a two-component Bose-Einstein condensate, and their dynamics are investigated by the variational approach based the renormalized integrals of motion. The stationary states as physical solutions to the describing equations are obtained, and the dynamic mechanism is demonstrated by performing a coordinate of a classical particle moving in an effective potential field. The switching and selftrapping dynamics of the coupled dark vector solitons are discussed by the evolution of the atom population transferring ratio.     

Nonlinear phenomena in degenerate quantum gases

In this introductory survey, we give an overview of the main physical problems and corresponding themes of research addressed in this Special Issue. We also briefly discuss some avenues of potential interest for future research in degenerate quantum gases.     

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 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.     

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.     

Fission and Fusion of Two Bright Solitons in a Growing Bose--Einstein Condensate

We analytically study the interaction characteristics of two bright solitons in a one-dimensional growing Bose- Einstein condensate with time-dependent periodic atomic scattering length. It is shown that the interaction between two bright solitons can generate fission and fusion in the presence of both time-dependent periodic atomic scattering length and the growing case. Furthermore, we propose experimental protocols to realize these interaction phenomena by varying the scattering length via the Feshbach resonance in the future experiment.     

Dynamics of kicked matter-wave solitons in an optical lattice

We investigate effects of the application of a kick to one-dimensional matter-wave solitons in a self-attractive Bose-Einstein condensate trapped in an optical lattice. The resulting soliton’s dynamics is studied within the framework of the time-dependent nonpolynomial Schrödinger equation. The crossover from the pinning to quasi-free motion crucially depends on the size of the kick, strength of the self-attraction, and parameters of the optical lattice.     

Formal analytical solutions for the Gross-Pitaevskii equation

Considering the Gross-Pitaevskii integral equation we are able to formally obtain an analytical solution for the order parameter Φ(x) and for the chemical potential μ as a function of a unique dimensionless non-linear parameter Λ. We report solutions for different ranges of values for the repulsive and the attractive non-linear interactions in the condensate. Also, we study a bright soliton-like variational solution for the order parameter for positive and negative values of Λ. Introducing an accumulated error function we have performed a quantitative analysis with respect to other well-established methods as: the perturbation theory, the Thomas-Fermi approximation, and the numerical solution. This study gives a very useful result establishing the universal range of the Λ-values where each solution can be easily implemented. In particular, we showed that for Λ<−9, the bright soliton function reproduces the exact solution of GPE wave function.     

Matter wave soliton collisions in the quasi one-dimensional potential

We consider soliton solutions of a two-dimensional nonlinear system with the self-focusing nonlinearity and a quasi 1D confining potential, taking harmonic potential as an example. We investigate a single soliton in detail and find criterion for possible collapse. This information is then used to investigate the dynamics of the two soliton collision. In this dynamics we identify three regimes according to the relation between nonlinear interaction and the excitation energy: elastic collision, excitation and collapse regime. We show that surprisingly accurate predictions can be obtained from variational analysis.     

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.     

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.     

Complete and orthonormal eigenfunctions of excitations to a one-dimensional dark soliton

We study linear excitations to a one-dimensional dark soliton described by a defocusing nonlinear Schödinger equation. By solving an eigenvalue problem for the excitations we obtain all eigenvalues and eigenfunctions and prove rigorously that these eigenfunctions are orthonormal and form a complete set. We then use the eigenfunctions to obtain the exact form of linear excitations for any given initial condition and to investigate the transverse stability of the dark soliton. The rigorous results reported in the present work can be applied to study the dynamics of dark solitons in various nonlinear optical media and Bose-Einstein condensates.     

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.     

Localized Waves in Nonlinear Systems with Spatially Inhomogeneous Nonlinearity

A type of （2＋1）-dimensional nonlinear Schrǒdinger equation with spatially inhomogeneous nonlinearity and an external potential is studied. It is found that special external potentials and spatially nonlinearities can support nonlinear localized waves.