[en] By performing numerical simulations, we discuss the collisional dynamics of stable solitary waves in the Schrödinger–Poisson equation. In the framework of a model in which part or all of dark matter is a Bose–Einstein condensate of ultralight axions, we show that these dynamics can naturally account for the relative displacement between dark and ordinary matter in the galactic cluster Abell 3827, whose recent observation is the first empirical evidence of dark matter interactions beyond gravity. The essential assumption is the existence of solitonic galactic cores in the kiloparsec scale. For this reason, we present simulations with a benchmark value of the axion mass $m_a=2\times 10^{-24}$eV, which is somewhat lower than the one preferred for cosmological structure formation if the field is all of dark matter ($m_a\approx 10^{-22}\mathrm{eV}$). We argue that future observations might bear out or falsify this coherent wave interpretation of dark matter offsets.