Soliton lattices in the Gross–Pitaevskii equation with nonlocal and repulsive coupling

Spatially-periodic patterns are studied in nonlocally coupled Gross–Pitaevskii equation. We show first that spatially periodic patterns appear in a model with the dipole–dipole interaction. Next, we study a model with a finite-range coupling, and show that a repulsively coupled system is closely related with an attractively coupled system and its soliton solution becomes a building block of the spatially-periodic structure. That is, the spatially-periodic structure can be interpreted as a soliton lattice. An approximate form of the soliton is given by a variational method. Furthermore, the effects of the rotating harmonic potential and spin-orbit coupling are numerically studied.