Three-dimensional solitons and vortices in dipolar Bose-Eins(2)

Three-dimensional solitary and vortex structures in Bose-Einstein condensates are studied in the framework of Gross-Pitaevskii model including the simultaneous action of local cubic-quintic nonlinearity and nonlocal dipole-dipole interactions. Nonlocal int

2

whereg=4π¯h2a/M,aisthes-wavescatteringlength.Inthefollowingweconsiderattractivetwo-particlein-teraction(a<0)andrepulsivethree-particleinteraction(gK<0).Notethatweaccounthereonlyforthecon-servativepartofthree-particleinteraction.Thus,GPE(2)conservesthenorm(thenumberofatomsinthecon-densedstate):

2

N=|Ψ|d3r,(3)energy:

E=2

¯2h

2g|Ψ|4

1

(4)

gd|Ψ|2Θd3r,Θ=Vd(r r′)|Ψ(r′)|2d3r′,

momentumandangularmomentum.

Inthenextsection,thegeneralpropertiesofstationarysolitonsandvorticesarestudiedbyanalyticalvariationalmethodandnumerically.

III.

STATIONARY3DSOLITONSAND

VORTEXSOLITONS

StationarysolutionsoftheEq.(2)havetheformΨ(r,t)=ψ(r)exp(iλt)andobeythedimensionlessequa-tion:

λψ+ ψ+ψ|ψ| ψ|ψ|+CψΘ=0,

2 1 ,Θ=FVd(k)F|ψ|

4π

2

4

(5)(6)

denotestheFouriertransformation,V d(k)=whereF

aρ

whereρ=

|m|

ρ2

exp

2a2z

+im ,

(7)