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Acta Physica Sinica

Bright soliton solution in 1D Tonks-Girardeau gas

Hong Liu, Wei Jia-Yu, Lou Sen‐Yue, He Xian-Tu, (1)北京物资学院物流学院，北京 101149; (2)北京应用物理与计算数学研究所，北京 100088; (3)宁波大学理学院，宁波 315211; (4)中国工程物理研究院研究生部，北京 100088

Published Jan 1, 2008• 4 citations
Publisher

Abstract

Base on spherical symmetric trap treatment, we analyze how to change a generalized Gross-Pitaevskii equation to a quintic nonlinear Schrdinger equation. Analysing the nonlinear Schrdinger equation, we find a typical bright soliton solution in Tonks-Girardeau gas region. This will help experimental observations.
DOI
10.7498/aps.57.1343

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The existence of bright solitons in the model of the Tonks–Girardeau (TG) gas with dipole–dipole (DD) interactions is reported. The governing equation is taken as the quintic nonlinear Schrödinger equation (NLSE) with the nonlocal cubic term accounting for the DD attraction. In different regions of the parameter space (the dipole moment and atom number), matter-wave solitons feature flat-top or compacton-like shapes. For the flat-top states, the NLSE with the local cubic-quintic (CQ) nonlinearity is shown to be a good approximation. Specific dynamical effects are studied assuming that the strength of the DD interactions is ramped up or drops to zero. Generation of dark-soliton pairs in the gas shrinking under the action of the intensifying DD attraction is observed. Dark solitons exhibit particle-like collision behaviour. Peculiarities of dipole solitons in the TG gas are highlighted by comparison with the NLSE including the local CQ terms. Collisions between the bright solitons are also studied. In many cases, the collisions result in the merger of the solitons into a breather, due to a strong attraction between them. (Some figures in this article are in colour only in the electronic version) 1.
The existence of bright solitons in the model of the Tonks-Girardeau (TG) gas with dipole-dipole (DD) interactions is reported. The governing equation is taken as the quintic nonlinear Schrödinger equation (NLSE) with the nonlocal cubic term accounting for the DD attraction. In different regions of the parameter space (the dipole moment and atom number), matter-wave solitons feature flat-top or compacton-like shapes. For the flat-top states, the NLSE with the local cubic-quintic (CQ) nonlinearity is shown to be a good approximation. Specific dynamical effects are studied assuming that the strength of the DD interactions is ramped up or drops to zero. Generation of dark-soliton pairs in the gas shrinking under the action of the intensifying DD attraction is observed. Dark solitons exhibit the particle-like collision behavior. Peculiarities of dipole solitons in the TG gas are highlighted by comparison with the NLSE including the local CQ terms. Collisions between the solitons are studied too. In many cases, the collisions result in merger of the solitons into a breather, due to strong attraction between them.
The dynamics of linear and nonlinear excitations in a Bose gas in the Tonks-Girardeau regime with longitudinal confinement are studied within a mean-field theory of quintic nonlinearity. A reductive perturbation method is used to demonstrate that the dynamics of shallow dark solitons, in the presence of an external potential, can effectively be described by a variable-coefficient Korteweg--de Vries equation. The soliton oscillation frequency is analytically obtained to be equal to the axial trap frequency, in agreement with numerical predictions obtained by Busch et al. [J. Phys. B 36, 2553 (2003)] via the Bose-Fermi mapping. We obtain analytical expressions for the evolution of both soliton and emitted radiation (sound) profiles.
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.