Luen-Chau Li: Vector soliton collisions, Yang-Baxter maps, and Poisson geometry

lundi 12 oct. 2026, 14:00 → 15:00 Europe/Paris

This talk is about soliton collisions in multi-component integrable soliton equations and the mathematics which grows out in its study. We will use the n-Manakov system (a.k.a. vector NLS) as our primary example. In this case, it is known that when two 1-solitons collide, the map which describes the change in polarizations is a parametric Yang-Baxter map, which is related to a special factorization problem on an associated rational loop group $K_\text{rat}$. An open question in this example is whether the change in polarization map is a symplectic map. Motivated by this simple example, we show how to construct Yang-Baxter maps on a variety of geometric objects, based on factorization problems on $K_\text{rat}$. Moreover, we also study the symplectic and Poisson geometry of these Yang-Baxter maps, which we show to be integrable maps in the sense of having natural Poisson commuting integrals. In a special case, the factorization problems we consider are associated with the N-soliton collision process in the n-Manakov system, and in this context we show that the polarization scattering map is a symplectomorphism. At the end of the talk, we will discuss how such results can be used as the starting point to understand so-called reflection maps, which arise in soliton-boundary interactions.