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

## TL;DR

This paper investigates the formation of soliton lattices in the nonlocal Gross-Pitaevskii equation, demonstrating how repulsive and attractive couplings lead to spatially periodic patterns and soliton structures, with numerical analysis of additional effects.

## Contribution

It reveals the connection between repulsive and attractive couplings in forming soliton lattices and provides an approximate soliton form using variational methods.

## Key findings

- Spatially periodic patterns emerge with dipole-dipole interactions.
- Repulsive coupling systems relate closely to attractive ones with soliton building blocks.
- Numerical studies show effects of harmonic potential and spin-orbit coupling.

## Abstract

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.