As I've mentioned a few times, you can use eigenvalues of random matrices to model random walks of points with repulsive forces. Here's a really nice application: modelling the bus system in Cuernavaca, Mexico https://authors.library.caltech.edu/3946/1/BAIjpa06.pdf …https://twitter.com/sigfpe/status/986298301975678976 …
Those lectures look great! I remember liking the discussion in Palais's Symmetries of Solitons article: starts with FPUT on p. 355 (p.17 of the PDF), derives KdV as a ctm limit, says a little on the physics of nonlin. waves... then off to the clouds https://www.ams.org/journals/bull/1997-34-04/S0273-0979-97-00732-5/S0273-0979-97-00732-5.pdf …