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SUMMARY:Luen-Chau Li: Vector soliton collisions\, Yang-Baxter maps\, and P
 oisson geometry
DTSTART:20261012T120000Z
DTEND:20261012T130000Z
DTSTAMP:20260930T214400Z
UID:indico-event-17279@indico.math.cnrs.fr
DESCRIPTION:This talk is about soliton collisions in multi-component integ
 rable 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 exampl
 e. In this case\, it is known that when two 1-solitons collide\, the map w
 hich describes the change in polarizations is a parametric Yang-Baxter map
 \, which is related to a special factorization problem on an associated ra
 tional loop group $K_\\text{rat}$. An open question in this example is whe
 ther 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 o
 f 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 polarizatio
 n 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 s
 o-called reflection maps\, which arise in soliton-boundary interactions.\n
 \nhttps://indico.math.cnrs.fr/event/17279/
URL:https://indico.math.cnrs.fr/event/17279/
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