The Schrödinger-Poisson (SP) equations predict fuzzy dark matter (FDM) solitons. Given the FDM mass $\sim {10}^{-20}\mathrm{eV}/{\mathrm{c}}^2$, the FDM soliton in the Milky Way is massive $\sim {10}^7{M}_{\odot }$ but diffuse $\sim 10\mathrm{pc}$. Therefore, such FDM soliton can serve as a gravitational lens for gravitational waves (GWs) with frequency $\sim {10}^{-8}\mathrm{Hz}$. In this paper, we investigate its gravitational lensing effects by numerical simulation of the propagation of GWs through it. We find that the maximum magnification factor of GWs is very small $\sim {10}^{-4}$, but the corresponding magnification zone is huge $\sim 6\mathrm{pc}$ for FDM with mass equal to $8\times {10}^{-21}\mathrm{eV}/{\mathrm{c}}^2$. Consequently, this small magnification factor compounding over such large magnification zone results in a small antisotropy of $\sim {10}^{-4}$ over a large solid angle in the GW background. That level of anisotropy is out of the sensitivity, $<20\%$, of the pulsar timing arrays today.

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• Gravitational lenses
• Gravitational waves
• Particle dark matter
• Numerical simulations in gravitation & astrophysics