Résumé : In this talk, I discuss the energy-critical half-wave maps equation (HWM). It has been known for quite some time that (HWM) is completely integrable with a Lax pair structure. However, the question about global-in-time existence of solutions has been completely open so far — even for smooth and sufficiently small initial data. I will present very recent results that prove global well-posedness for rational initial data (with no size restriction) along with a general soliton resolution result in the large-time limit. The proofs strongly exploit the Lax structure of (HWM) in combination with an explicit flow formula. This is joint work with Patrick Gérard (Paris-Saclay).

Keywords : half-wave maps equation; Calogero-Moser system; soliton resolution