## Seminar (online): V. Caudrelier "Soliton interactions, Yang-Baxter and reflection maps and their Poisson properties"

**Speaker:**** ** Vincent Caudrelier (School of Mathematics, University of Leeds)

**Date and time: **** **28.02.2024, 17:00 (GMT +03:00)

**Title:** Soliton interactions, Yang-Baxter and reflection maps and their Poisson properties

**Abstract:** Using the vector nonlinear Schrödinger equation as the main example, I will briefly review how certain solutions of the set-theoretical Yang-Baxter equation, called Yang-Baxter maps, arise from the interactions of multicomponent solitons. This is best seen using the Zakharov-Shabat dressing method and refactorisation of the elementary dressing factors. In this purely classical context, it is remarkable that the Yang-Baxter equation also ensures that the total scattering map describing the collisions consistently factorises into a product of two-soliton collisions, just like in the more well-known quantum context. I will then discuss the problem of integrable boundary conditions and explain how it leads to the introduction of the set-theoretical reflection equation. Solutions to this equation, called reflection maps, arise from the reflection of a soliton on the boundary. Again, the complete analogy between this context and the more well-known quantum reflection equation introduced by Cherednik and Sklyanin holds. Finally, I will present results on the symplectic and Poisson properties of these maps. This is a natural problem to consider given the interpretation (reviewed e.g. in Faddeev-Takhtajan's book) of soliton dynamics in the scalar case as canonical transformations.

To access the online seminar please contact Anna Tolbey bekvaanna@gmail.com