The focusing CCM equation is a completely integrable PDE that describes a continuum limit of a particle gas interacting pairwise through an inverse-square potential. This system is well-posed below the mass of the soliton, but above this threshold there are solutions that blow up in finite time.

The defocusing model arises as a modulation equation for internal waves in a deep stratified fluid. In this setting, it is a member of a family of systems known as the intermediate nonlinear Schrödinger equations.

In this talk, we will discuss some recent well-posedness results for the continuum Calogero--Moser and intermediate nonlinear Schrödinger equations. This is based on joint works with Andreia Chapouto, Justin Forlano, Rowan Killip, and Monica Visan.