  In this Letter, we present a new idea of probing the distribution of dark
matter exhibiting elastic and velocity-independent self-interactions. These
interactions might be revealed in multiple measurements of strongly lensed
gravitational waves, which can be observationally explored to determine the
strength of self-scatterings. Specifically, each individual galactic-scale
strong-lensing system whose source is a coalescing compact binary emitting
gravitational waves will provide a model-independent measurement of the shear
viscosity of dark matter along the line of sight. These individual measurements
could be a probe of large-scale distribution of dark matter and its properties.
Our results indicate that with 10-1000 strongly lensed gravitational waves from
ET and DECIGO, robust constraints on the large-scale distribution of
self-interacting dark matter might be produced. More stringent limits on the
dark matter scattering cross-section per unit mass ($\sigma_{\chi}/m_{\chi}$)
relevant to galaxy and cluster scales are also expected, compared with the
conservative estimates obtained in the electromagnetic domain. Finally, we
discuss the effectiveness of our method in the context of self-interacting dark
matter particle physics.
