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Matter-wave solitons supported by field-induced dipole-dipole repulsion with a spatially modulated strength

We demonstrate the existence of one and two-dimensional bright solitons in the Bose-Einstein condensate with repulsive dipole-dipole interactions induced by a combination of dc and ac polarizing fields, oriented perpendicular to the plane in which the BEC is trapped, assuming that the strength of the fields grows in the radial ($r$) direction faster than $r^{3}$. Stable tightly confined 1D and 2D fundamental solitons, twisted solitons in 1D, and solitary vortices in 2D are found in a numerical form. The fundamental solitons remain robust under the action of an expulsive potential, which is induced by the interaction of the dipoles with the polarizing field. The confinement and scaling properties of the soliton families are explained analytically. The Thomas-Fermi approximation is elaborated for fundamental solitons. The mobility of the fundamental solitons is limited to the central area. Stable 1D even and odd solitons are also found in the setting with a double-well modulation function, along with a regime of Josephson oscillations.