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Communications in Theoretical Physics

Solitons in One-Dimensional Bose–Einstein Condensate with Higher-Order Interactions*

Ying Wang, Wei Wang, Shuyu Zhou

Published Nov 1, 2017• 13 citations
Publisher

Abstract

Abstract We model a one-dimensional Bose–Einstein condensate with the one-dimensional Gross–Pitaevskii equation (1D GPE) incorporating higher-order interaction effects. Based on the F-expansion method, we analytically solve the 1D GPE, identifying the typical soliton solution under certain experimental settings within the general wave-like solution set, and demonstrating the applicability of the theoretical treatment that is employed .

DOI
10.1088/0253-6102/68/5/623

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Abstract We model a one-dimensional Bose–Einstein condensate with the one-dimensional Gross–Pitaevskii equation (1D GPE) incorporating higher-order interaction effects. Based on the F-expansion method, we analytically solve the 1D GPE, identifying the typical soliton solution under certain experimental settings within the general wave-like solution set, and demonstrating the applicability of the theoretical treatment that is employed .
We model the three-dimensional (3D) trapped Bose-Einstein condensate with the Gross-Pitaevskii equation (GPE) incorporating higher-order nonlinear interaction effects. Based on the F-expansion method and self-similar approach, we analytically derive the exact soliton solutions of the 3D GPE, identifying the oscillatory mode under certain experimental settings. The oscillatory kinetic parameter values we obtained coincide very well with those reported in previous work, which is based on the variational approach, showing the applicability of the theoretical treatment presented in this work.
For study of Continuous matter waves in Bose-Einstein condensates in nonlinear and quantum atom optics, the two-dimensional Gross-Pitaevskii equation (GPE) is chosen as the reliable model for studying the dynamics of vortices in the framework of mean-field theory. In related problems in several recent studies showing that higher-order interrelationships are an indispensable component of the GPE even at the mean-field level, by numerically estimating the vortex dynamics variables. In this paper, derive the vortex soliton solutions using the variational method and investigate the effect of higher-order nonlinear corrections on the behavior of the vortex dynamics, which are shown to have an important impact on the vortex dynamics behavior.
Using the F-expansion method, we present analytical matter-wave solutions to Bose-Einstein condensates with two- and three-body interactions through the generalized onedimensional Gross-Pitaevskii equation with time-dependent coefficients, for the experimentally relevant quadratic potential strength. Various shapes of the analytical matter-wave solutions which have important applications of physical interest are studied in detail. In particular, the matter-wave solitons are obtained in the quintic Gross-Pitaevskii equation model, which may be produced by tuning the cubic nonlinearity to zero via the Feschbach resonance technique. Direct numerical simulation has been performed to show the stable region of the matter-wave soliton.
The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrödinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE.
We consider stationary matter-wave gap solitons realized in Bose–Einstein condensates loaded in one-dimensional (1D) optical lattices and investigate whether the effective 1D equation proposed in Muñoz Mateo and Delgado (2008 Phys. Rev. A 77 013617) can be a reliable alternative to the three-dimensional treatment of this kind of system in terms of the Gross–Pitaevskii equation (GPE). Our results demonstrate that, unlike the standard 1D GPE (which is not applicable in most realistic situations), the above effective model is able to correctly predict the distinctive trajectories characterizing the different gap soliton families as well as the corresponding axial wavefunctions along the entire band gaps. It can also predict the stability properties of the different gap soliton families as follows from both a linear stability analysis and a representative set of numerical computations. In particular, by numerically solving the corresponding Bogoliubov–de Gennes equations we show that the effective 1D model gives the correct spectrum of the complex eigenfrequencies responsible for the dynamical stability of the system, thus providing us with a useful tool for the physical description of stationary matter-wave gap solitons in 1D optical lattices.
The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrödinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE. In a dilute gas of N Bosons at zero temperature practically all particles are in the same single-particle state of the density matrix ρ(r,r ′;t). 1), 2) This macroscopically occupied single-particle state is called Bose-Einstein condensate (BEC) and it is well described by a complex classical field ψ(r,t) (order parameter or macroscopic