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Communications in Nonlinear Science and Numerical Simulation

Multi-fold binary Darboux transformation and mixed solitons of a three-component Gross-Pitaevskii system in the spinor Bose-Einstein condensate

Chen-Rong Zhang, Bo Tian, Qi‐Xing Qu, Yu‐Qiang Yuan, Cheng-Cheng Wei

Published Aug 11, 2021• 12 citations
Publisher
DOI
10.1016/j.cnsns.2021.105988

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We find one-, two-, and three-component solitons of the polar and ferromagnetic (FM) types in the general (nonintegrable) model of a spinor (three-component) model of the Bose-Einstein condensate, based on a system of three nonlinearly coupled Gross-Pitaevskii equations. The stability of the solitons is studied by means of direct simulations and, in a part, analytically, using linearized equations for small perturbations. Global stability of the solitons is considered by means of an energy comparison. As a result, ground-state and metastable soliton states of the FM and polar types are identified. For the special integrable version of the model, we develop the Darboux transformation (DT). As an application of the DT, analytical solutions are obtained that display full nonlinear evolution of the modulational instability of a continuous-wave state seeded by a small spatially periodic perturbation. Additionally, by dint of direct simulations, we demonstrate that solitons of both the polar and FM types, found in the integrable system, are structurally stable; i.e., they are robust under random changes of the relevant nonlinear coefficient in time.
We consider vector solitons of mixed bright-dark types in quasi-one-dimensional spinor (F=1)(F=1)(F=1) Bose-Einstein condensates. Using a multiscale expansion technique, we reduce the corresponding nonintegrable system of three coupled Gross-Pitaevskii equations (GPEs) to an integrable Yajima-Oikawa system. In this way, we obtain approximate solutions for small-amplitude vector solitons of dark-dark-bright and bright-bright-dark types, in terms of the ${m}_{F}=+1,\ensuremath{-}1,0$ spinor components, respectively. By means of numerical simulations of the full GPE system, we demonstrate that these states indeed feature soliton properties, i.e., they propagate undistorted and undergo quasielastic collisions. It is also shown that in the presence of a parabolic trap the bright component(s) is (are) guided by the dark one(s) and, as a result, the small-amplitude vector soliton as a whole performs quasiharmonic oscillations. The oscillation frequency is found as a function of the spin-dependent interaction strength for both small-amplitude and large-amplitude solitons.