Dark matter, which has no color but rather is transparent, makes up probably a quarter of the energy-matter density of our cosmos and has a lot of evidence for its existence, including gravitational lensing and cosmic microwave background fluctuations. One model for dark matter is the hypothetical quantum particle that solves the strong Charge+Parity problem in quantum chromodynamics: the axion. Axions probably clustered in the early universe, condensing into ultra-cold Bose-Einstein Condensates (BECs) that have macroscopic quantum properties. For axion clusters the mathematical description is a coupled pair of PDEs: the non-linear quantum (NLQ) equation and the Poisson equation for interactions through gravity.
We will survey BECs, axions, and the NLQ. In recent work with Partha Dey and Kesav Krishnan, we find a phase transition in the discrete NLQ. In recent work with Anthony Mirasola and Chanda Prescod-Weinstein, we calculate that the condensation of axions into a BEC is driven by gravity and should be possible within the lifetime of our cosmos.