We consider a possibility to realize self-accelerating motion of interacting states with effective positive and negative masses in the form of pairs of solitons in two-component BEC loaded in an optical-lattice (OL) potential. A crucial role is played by the fact that gap solitons may feature a negative dynamical mass, keeping their mobility in the OL. First, the respective system of coupled Gross-Pitaevskii equations (GPE) is reduced to a system of equations for envelopes of the lattice wave functions. Two generic dynamical regimes are revealed by simulations of the reduced system, viz., shuttle oscillations of pairs of solitons with positive and negative masses, and splitting of the pair. The coaccelerating motion of the interacting solitons, which keeps constant separation between them, occurs at the boundary between the shuttle motion and splitting. The position of the coacceleration regime in the system's parameter space can be adjusted with the help of an additional gravity potential, which induces its own acceleration, that may offset the relative acceleration of the two solitons, while gravity masses of both solitons remain positive. The numerical findings are accurately reproduced by a variational approximation. Collisions between shuttling or coaccelerating soliton pairs do not alter the character of the dynamical regime. Finally, regimes of the shuttle motion, coacceleration, and splitting are corroborated by simulations of the original GPE system, with the explicitly present OL potential.
We consider a possibility to realize self-accelerating motion of interacting states with effective positive and negative masses in the form of pairs of solitons in two-component BEC loaded in an optical-lattice (OL) potential. A crucial role is played by the fact that gap solitons may feature a negative dynamical mass, keeping their mobility in the OL. First, the respective system of coupled Gross-Pitaevskii equations (GPE) is reduced to a system of equations for envelopes of the lattice wave functions. Two generic dynamical regimes are revealed by simulations of the reduced system, viz., shuttle oscillations of pairs of solitons with positive and negative masses, and splitting of the pair. The coaccelerating motion of the interacting solitons, which keeps constant separation between them, occurs at the boundary between the shuttle motion and splitting. The position of the coacceleration regime in the system's parameter space can be adjusted with the help of an additional gravity potential, which induces its own acceleration, that may offset the relative acceleration of the two solitons, while gravity masses of both solitons remain positive. The numerical findings are accurately reproduced by a variational approximation. Collisions between shuttling or coaccelerating soliton pairs do not alter the character of the dynamical regime. Finally, regimes of the shuttle motion, coacceleration, and splitting are corroborated by simulations of the original GPE system, with the explicitly present OL potential.
We study the dynamics of one-dimensional solitons in the attractive and repulsive Bose-Einstein condensates (BECs) loaded into an optical lattice (OL), which is combined with an external parabolic potential. First, we demonstrate analytically that, in the repulsive BEC, where the soliton is of the gap type, its effective mass is negative. This gives rise to a prediction for the experiment: such a soliton cannot be not held by the usual parabolic trap, but it can be captured (performing harmonic oscillations) by an anti-trapping inverted parabolic potential. We also study the motion of the soliton a in long system, concluding that, in the cases of both the positive and negative mass, it moves freely, provided that its amplitude is below a certain critical value; above it, the soliton’s velocity decreases due to the interaction with the OL. At a late stage, the damped motion becomes chaotic. We also investigate the evolution of a two-soliton pulse in the attractive model. The pulse generates a persistent breather, if its amplitude is not too large; otherwise, fusion into a single fundamental soliton takes place. Collisions between two solitons captured in the parabolic trap or anti-trap are considered too. Depending on their amplitudes and phase difference, the solitons either perform stable oscillations, colliding indefinitely many times, or merge into a single soliton. Effects reported in this work for BECs can also be formulated for optical solitons in nonlinear photonic crystals. In particular, the capture of the negative-mass soliton in the anti-trap implies that a bright optical soliton in a self-defocusing medium with a periodic structure of the refractive index may be stable in an anti-waveguide. 1 1
Abstract Basic models which give rise to one- and two-dimensional (1D and 2D) solitons, such as the Gross-Pitaevskii (GP) equations for BEC, feature the Galilean invariance, which makes it possible to generate families of moving solitons from quiescent ones. A challenging problem is to find models admitting stable self-accelerating (SA) motion of solitons. SA modes are known in linear systems in the form of Airy waves, but they are poorly localized states. This brief review presents two-component BEC models which make it possible to predict SA solitons. In one system, a pair of interacting 1D solitons with opposite signs of the effective mass is created in a binary BEC trapped in an optical-lattice potential. In that case, opposite interaction forces, acting on the solitons with positive and negative masses, produce equal accelerations, while the total momentum is conserved. The second model is based on a system of GP equations for two atomic components, which are resonantly coupled by a microwave field. The latter model produces an exact transformation to an accelerating references frame, thus predicting 1D and 2D stable SA solitons, including vortex rings.
We study the dynamics of one-dimensional solitons in attractive and repulsive Bose–Einstein condensates (BECs) loaded into an optical lattice (OL), which is combined with an external parabolic potential. First, we demonstrate analytically that, in the repulsive BEC, where the soliton is of the gap type, its effective mass is negative . This gives rise to a prediction for the experiment: such a soliton cannot be held by the usual parabolic trap, but it can be captured (performing slow harmonic oscillations, with a period that is estimated to be ∼0.01 s in realistic experimental conditions) by an anti-trapping inverted parabolic potential. We also study the motion of the soliton in a long system, concluding that, in the cases of both the positive and negative mass, it moves freely, provided that its amplitude is below a certain critical value; above it, the soliton's velocity decreases due to interaction with the OL. At a later stage, the damped motion becomes chaotic. We also investigate the evolution of a two-soliton pulse in the attractive model. The pulse generates a persistent breather, if its amplitude is not too large; otherwise, fusion into a single fundamental soliton takes place. Collisions between two solitons captured in the parabolic trap or anti-trap are considered too. Depending on their amplitudes and phase difference, the solitons either perform stable oscillations, colliding indefinitely many times, or merge into a single soliton. Effects reported in this work for BECs can also be formulated for optical solitons in nonlinear photonic crystals. In particular, the capture of the negative-mass soliton in the anti-trap implies that a bright optical soliton in a self-defocusing medium with a periodic structure of the refractive index may be stable in an anti-waveguide .
We consider effects of inter-species attraction on two-component gap solitons in the binary BEC with intra-species repulsion, trapped in the one-dimensional optical lattice. Systematic simulations of the coupled Gross-Pitaevskii equations corroborate an assumption that, because the effective mass of solitons is negative, the inter-species attraction may split the two-component soliton. Two critical values of the lattice strength are identified. Two-species gap solitons are stable in strong lattices. In an intermediate region the soliton splits into a double-humped state with separated components. Finally, in weak lattices, the splitting generates a pair of freely moving single-species solitons.
We consider effects of interspecies attraction on two-component gap solitons (GSs) in the binary BEC with intraspecies repulsion, trapped in the one-dimensional optical lattice (OL). Systematic simulations of the coupled Gross-Pitaevskii equations corroborate an assumption that, because the effective mass of GSs is negative, the interspecies attraction may split the two-component soliton. Two critical values, ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$, of the OL strength $(\ensuremath{\kappa})$ are identified. Two-species GSs with fully overlapping wave functions are stable in strong lattices $(\ensuremath{\kappa}>{\ensuremath{\kappa}}_{1})$. In an intermediate region, ${\ensuremath{\kappa}}_{1}>\ensuremath{\kappa}>{\ensuremath{\kappa}}_{2}$, the soliton splits into a double-humped state with separated components. Finally, in weak lattices $(\ensuremath{\kappa}<{\ensuremath{\kappa}}_{2})$, the splitting generates a pair of freely moving single-species GSs. We present and explain the dependence of ${\ensuremath{\kappa}}_{1}$ and ${\ensuremath{\kappa}}_{2}$ on the number of atoms (total norm), and on the relative strength of the competing interspecies attraction and intraspecies repulsion. The splitting of asymmetric solitons, with unequal norms of the two species, is briefly considered too. It is found and explained that the splitting threshold grows with the increase of the asymmetry.
The dynamical behavior of matter-wave solitons of two-component Bose-Einstein condensates (BEC) in combined linear and nonlinear optical lattices (OLs) is investigated. In particular, the dependence of the frequency of the oscillating dynamics resulting from initially slightly displaced components is investigated both analytically, by means of a variational effective potential approach for the reduced collective coordinate dynamics of the soliton, and numerically, by direct integrations of the mean field equations of the BEC mixture. We show that for small initial displacements binary solitons can be viewed as point masses connected by elastic springs of strengths related to the amplitude of the OL and to the intra- and interspecies interactions. Analytical expressions of symmetric and antisymmetric mode frequencies are derived and occurrence of beatings phenomena in the displaced dynamics is predicted. These expressions are shown to give a very good estimation of the oscillation frequencies for different values of the intraspecies interatomic scattering length, as confirmed by direct numerical integrations of the mean field Gross-Pitaevskii equations (GPE) of the mixture. The possibility to use displaced dynamics for indirect measurements of BEC mixture characteristics such as number of atoms and interatomic interactions is also suggested.