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

3

equationwhichdescribesevolutionofsmallperturbation:

iN

l

FIG.2:NumberofparticlesNvsλfordi erentstationarysolutionsatC=0.3.Numericalresultsareshownincir-cles,dashedlinesareforvariationalpredictions.Thearrowsindicatestabilitythresholdsfor

vortices.

FIG.3:NumberofparticlesNvsparameterλforvorticeswithm=1withdi erentC.Numericalresults.

IV.STABILITYANDDYNAMICS

Westartinvestigationofthestabilityofvorticeswithanalysisofsmallperturbationsappliedtothestationarysolution:

Ψ=(ψ+ )eiλt,

where| (r,t)|<<|ψ|.LinearizingtheGPE(2)invicinityofstationarysolutiononegetsthenonstationary

(t+ t)

2 t

ln

ν