Yang-Baxter and reflection maps from optical solitons interactions

par Vincent Caudrelier (City University London)

jeudi 18 sept. 2014, 11:00 → 12:00 Europe/Paris

Auditorium (Annecy-le-Vieux)

The Yang-Baxter equation and its companion in the presence of a boundary, the reflection equation, are well-known fundamental equations in the theory of quantum integrable systems. They have a nice intuitive origin in the context of factorized scattering in QFT. In their purest formulation, the Yang-Baxter equation simply correspond to linear representations of the permutation group acting. We will show that this simple observation allows one to consider these equations in a very different context: that of extended nonlinear objects like solitons, governed by integrable PDEs. The vector nonlinear Schrödinger equation is a model of light propagation in optical fibers and will be taken as our main example to illustrate these ideas. By studying explicit multisoliton solutions, one can construct new (nonlinear) representations of the so-called set-theoretical Yang-Baxter. They represent the interactions of the solitons. By looking at the same problem with a fixed boundary, we will introduce the concept of set theoretical reflection equation and construct classes of solutions for it. It is the beauty of integrable systems that both quantum and classical models share such fundamental structures although the objects, both physical and mathematical, are completely different.

Slides seminar_LAPTH_2014.pdf