We show two kinds of inhomogeneous spin domain possessing N{e}el-like domain walls in spin-1 Bose-Einstein condensate, which are induced by the positive and negative quadratic Zeeman effect (QZE) respectively. In both cases, the spin density distribu
tion is inhomogeneous and has zeros where the magnetization vanishes. For positive and negative QZE, the spin patterns and topological structures are remarkably different. Such phenomena are due to the pointwise different axisymmetry-breaking caused by the pointwise different population exchange between the sublevels, arising uniquely from the QZE. We analyze in detail the inhomogeneous domain formation and related experimental observations for the spin-1 $^{87}$Rb and $^{23}$Na condensate.
In this paper, we consider an optimal bilinear control problem for the nonlinear Schr{o}dinger equations with singular potentials. We show well-posedness of the problem and existence of an optimal control. In addition, the first order optimality syst
em is rigorously derived. Our results generalize the ones in cite{Sp} in several aspects.
We investigate the integrability of generalized nonautonomous nonlinear Schrodinger (NLS) equations governing the dynamics of the single- and double-component Bose-Einstein condensates (BECs). The integrability conditions obtained indicate that the e
xistence of the nonautonomous soliton is due to the balance between the different competition features: the kinetic energy (dispersion) versus the harmonic external potential applied and the dispersion versus the nonlinearity. In the double-component case, it includes all possible different combinations between the dispersion and nonlinearity involving intra- and inter-interactions. This result shows that the nonautonomous soliton has the same physical origin as the canonical one, which clarifies the nature of the nonautonomous soliton. Finally, we also discuss the dynamics of two-component BEC by controlling the relevant experimental parameters.
In this paper we show a systematical method to obtain exact solutions of the nonautonomous nonlinear Schrodinger (NLS) equation. An integrable condition is first obtained by the Painlev`e analysis, which is shown to be consistent with that obtained b
y the Lax pair method. Under this condition, we present a general transformation, which can directly convert all allowed exact solutions of the standard NLS equation into the corresponding exact solutions of the nonautonomous NLS equation. The method is quite powerful since the standard NLS equation has been well studied in the past decades and its exact solutions are vast in the literature. The result provides an effective way to control the soliton dynamics. Finally, the fundamental bright and dark solitons are taken as examples to demonstrate its explicit applications.
We investigate the dynamics and modulation of ring dark soliton in 2D Bose-Einstein condensates with tunable interaction both analytically and numerically. The analytic solutions of ring dark soliton are derived by using a new transformation method.
For shallow ring dark soliton, it is stable when the ring is slightly distorted, while for large deformation of the ring, vortex pairs appear and they demonstrate novel dynamical behaviors: the vortex pairs will transform into dark lumplike solitons and revert to ring dark soliton periodically. Moreover, our results show that the dynamical evolution of the ring dark soliton can be dramatically affected by Feshbach resonance, and the lifetime of the ring dark soliton can be largely extended which offers a useful method for observing the ring dark soliton in future experiments.
In this paper 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 complete
ly integrable. Under the present model, this integrability condition is completely consistent with that proposed by Serkin, Hasegawa, and Belyaeva [V. N. Serkin et al., Phys. Rev. Lett. 98, 074102 (2007)]. Furthermore, this integrability can also be explicitly shown by a transformation, which can convert the Gross-Pitaevskii equation into the well-known standard nonlinear Schrodinger equation. By this transformation, each exact solution of the standard nonlinear Schrodinger 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 solitions and the dynamics of the Bose-Einstein condensates by using the Feshbach resonance technique.
An unstable particle in quantum mechanics can be stabilized by frequent measurements, known as the quantum Zeno effect. A soliton with dissipation behaves like an unstable particle. Similar to the quantum Zeno effect, here we show that the soliton ca
n be stabilized by modulating periodically dispersion, nonlinearity, or the external harmonic potential available in BEC. This can be obtained by analyzing a Painleve integrability condition, which results from the rigorous Painleve analysis of the generalized nonautonomous nonlinear Schrodinger equation. The result has a profound implication to the optical soliton transmission and the matter-wave soliton dynamics.
