We investigate the existence and stability of higher-order bright solitons, stripe solitons, and bright-dark solitons in a Bose-Einstein condensate with helicoidal spin-orbit coupling under a Zeeman lattice using numerical methods. The higher-order bright solitons that exist in the first-finite energy gap are stable except near the edge. The stripe solitons with parity-time symmetry and pseudospin-parity symmetry have partially overlapping norm curves; they are stable in the lower edge of the first-finite energy gap. Additionally, the bright-dark solitons discovered in the system not only exist within energy gaps but also embed within energy bands as they have periodic backgrounds. These findings offer insights into the diversity and behavior of solitons within energy bands and contribute to a deeper understanding of their distribution and dynamics.
We investigate the existence and stability of higher-order bright solitons, stripe solitons, and bright-dark solitons in a Bose-Einstein condensate with helicoidal spin-orbit coupling under a Zeeman lattice using numerical methods. The higher-order bright solitons that exist in the first-finite energy gap are stable except near the edge. The stripe solitons with parity-time symmetry and pseudospin-parity symmetry have partially overlapping norm curves; they are stable in the lower edge of the first-finite energy gap. Additionally, the bright-dark solitons discovered in the system not only exist within energy gaps but also embed within energy bands as they have periodic backgrounds. These findings offer insights into the diversity and behavior of solitons within energy bands and contribute to a deeper understanding of their distribution and dynamics.
We report a diversity of stable gap solitons in a spin-orbit-coupled Bose-Einstein condensate subject to a spatially periodic Zeeman field. It is shown that the solitons can be classified by the main physical symmetries they obey, i.e., symmetries with respect to parity (P), time (T), and internal degree of freedom, i.e., spin (C), inversions. The conventional gap and gap-stripe solitons are obtained in lattices with different parameters. It is shown that solitons of the same type but obeying different symmetries can exist in the same lattice at different spatial locations. PT and CPT symmetric solitons have antiferromagnetic structure and are characterized, respectively, by nonzero and zero total magnetizations.
Abstract We explore the existence and stability domains of stationary and moving bright solitons in spin–orbit-coupled spin-1 Bose–Einstein condensates in an external Zeeman field. Two families of bell-shape bright solitons (plane-wave (PW) phases) and two families of stripe-shape bright solitons (standing-wave (SW) phases) are obtained. We find a relation between the existence of these bright solitons and the single-particle energy spectrum, as well as the requirements of their existence for atomic interactions. The stability of four families of bright solitons is systematically analyzed using the linear stability analysis method. For two families of PW bright solitons, they are unstable only when the strength of the Zeeman field is above a critical value. The critical value is affected by the spin–orbit coupling and hardly affected by atomic interactions. When the velocities of solitons are zero (nonzero), the critical values of these two families are equal (unequal). For two families of SW bright solitons, their stability domains are identical in the absence of the Zeeman field, and they are unstable when the ferromagnetic interaction is strong. In the presence of the Zeeman field, their stability domains are complementary and complex, and one family of SW bright solitons can exist stably in the area with stronger ferromagnetic interaction. Furthermore, we find the collisions of stable bright solitons with different velocities can generate intriguing dynamics.
We report on the existence and stability of freely moving solitons in a spatially inhomogeneous Bose-Einstein condensate with helicoidal spin-orbit (SO) coupling. In spite of the periodically varying parameters, the system allows for the existence of stable propagating solitons. Such states are found in the rotating frame, where the helicoidal SO coupling is reduced to a homogeneous one. In the absence of the Zeeman splitting, the coupled Gross-Pitaevskii equations describing localized states feature many properties of the integrable systems. In particular, four-parametric families of solitons can be obtained in the exact form. Such solitons interact elastically. Zeeman splitting still allows for the existence of two families of moving solitons, but makes collisions of solitons inelastic.
We demonstrate that the model of a spatially nonuniform two-component Bose-Einstein condensates (BEC) featuring the helicoidal spin-orbit coupling (SOC), gives rise to dark-bright soliton complexes characterized by spatiotemporal periodic oscillations in each component. These solitons are formed by the superposition of dark and bright ones, and exhibit a beating state over time and a striped state across space, earning them the designation of beating stripe solitons. Our analysis demonstrates that helicoidal SOC significantly affects the formation and dynamical properties of these solitons, also serving as the primary driver for spin oscillations. Through the nonlinear superposition of the beating stripe solitons, a range of intricate scenarios of the interaction between multiple solitons emerges, including head-on collisions, bound states, and parallel states.
Abstract We investigate the properties of gap solitons in spin–orbit-coupled Bose–Einstein condensates in mixed linear–nonlinear optical lattices. The mixed linear–nonlinear optical lattices can support parity-time (PT)-symmetric soliton solutions, and these PT-symmetric solitons can stably exist in the semi-infinite gap of linear optical lattices. The PT-symmetric gap-stripe solitons are found by increasing the strength of the spin–orbit coupling. It is found that the amplitude of the nonlinear optical lattices can significantly affect the stability of these gap and gap-stripe solitons. With an increase of the amplitude of the nonlinear optical lattices, the stable domains of the gap and gap-stripe solitons are shrunk. The unstable solitons can show unique evolution characteristics and the stability analyses are also confirmed by the evolution simulations.
We analytically and numerically study the different types of solitary wave in the two-component helicoidal spin-orbit coupled Bose-Einstein condensates (BECs). Adopting the multiscale perturbation method, we derive the analytical bright and dark solitary wave solutions of the system, and the stationary and moving bright (dark) solitary waves are obtained. The effects of spin-orbit coupling, the helicoidal gauge potential, the momentum, the Zeeman splitting, and the atomic interactions on the solitary wave types are discussed, and it is found that the coupling of these physical parameters can manipulate different types of solitary waves in the system. The results indicate that the helicoidal gauge potential breaks the symmetric properties of the energy band of the system and adjusts the energy band structure, thus further effecting the solitary wave properties, i.e., stationary or moving solitary wave, bright, or dark solitary wave. Correspondingly, the analytical predictions for exciting stationary or moving bright (dark) solitary wave in parameter space are obtained. In particular, the helicoidal gauge potential changes the solitary wave types drastically for the weak spin-orbit coupling, i.e., in the absence of the helicoidal gauge potential, only dark (bright) solitary wave solutions exist in the system with repulsive (attractive) atomic interaction; however, in the presence of the helicoidal gauge potential, both dark and bright solitary waves can exist in the system regardless of whether the atomic interaction is repulsive or attractive. In addition, we investigate the stability of solitary waves and obtain the stability regions of different types of solitary waves by applying the linear stability analysis. The dynamic evolution results of the solitary waves by the direct numerical simulation not only validate the linear stability analysis but also confirm the analytical prediction of the solitary waves. Finally, the collision effects between solitary waves are also presented by the numerical simulation. It is shown that the interactions between solitary waves in the system have both elastic and inelastic collisions, which are closely related to the position of solitary wave states in the linear energy band. Our results provide a potential way to adjust the types of solitary waves in BECs with helicoidal gauge potential.