## An example of stable evolution of the vortex soliton with ℓ = 1 in the isotropic system

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**Figure 5.** An example of stable evolution of the vortex soliton with ℓ = 1 in the isotropic system. Shown are the initial state, obtained by means of the imaginary-time propagation, starting from Gaussian ansatz (7) with α = 0.054 12 and γ = 0.6798, and the final result of subsequent real-time simulations. The parameters are *g* = 20, *g _{d}* = 30 and

_{1}=

_{2}= 0.1.

**Abstract**

We predict the existence of stable fundamental and vortical bright solitons in dipolar Bose–Einstein condensates with repulsive dipole–dipole interactions (DDI). The condensate is trapped in the two-dimensional plane with the help of the repulsive contact interactions whose local strength grows ~*r*^{4} from the centre to periphery, while dipoles are oriented perpendicular to the self-trapping plane. The confinement in the perpendicular direction is provided by the usual harmonic-oscillator potential. The objective is to extend the recently induced concept of the self-trapping of bright solitons and solitary vortices in the *pseudopotential*, which is induced by the repulsive local nonlinearity with the strength growing from the centre to periphery, to the case when the trapping mechanism competes with the long-range repulsive DDI. Another objective is to extend the analysis for elliptic vortices and solitons in an anisotropic nonlinear pseudopotential. Using the variational approximation and numerical simulations, we construct families of self-trapped modes with vorticities ℓ = 0 (fundamental solitons), ℓ = 1 and ℓ = 2. The fundamental solitons and vortices with ℓ = 1 exist up to respective critical values of the eccentricity of the anisotropic pseudopotential, being stable in the entire existence regions. The vortices with ℓ = 2 are stable solely in the isotropic model.