Yang-Baxter and reflection equations: unifying structures behind quantum and classical integrable systems

<strong>Seminar of the School of Physical Sciences ----------------------------------------------</strong> Title: <strong>Yang-Baxter and reflection equations: unifying structures behind quantum and classical integrable systems</strong> Speaker:<strong> Vincent Caudrelier</strong> (City University London) Dr. Caudrelier is currently the V. Ramalingaswami Chair of the Indian National Science Academy (INSA) Date: <strong>July 29, 2015</strong> <strong>Abstract: </strong>The Yang-Baxter equation (YBE) is central in the theory of quantum integrable systems. For decades, together with its companion for problems with boundaries (the quantum reflection equation), it has been studied and used in the quantum realm. But it was suggested by Drinfeld in 1990 that the general study of the so-called "set-theoretical YBE" is also important. It turns out that classical integrable field theories provide a means to construct solutions to this equation, called Yang-Baxter maps, by looking at soliton collisions. I will use the vector nonlinear Schrödinger (NLS) equation as the main example. The motivation is its high versatility and universality as it describes wave phenomena in fluid dynamics, nonlinear optics, plasma physics or quantum cold gases. In its quantum form, interactions of particles provide a solution to the quantum YBE. In its classical form, interactions of solitons provide "classical solutions of the quantum YBE". After reviewing this, we will show how the new concept of set-theoretical reflection equation naturally emerges by studying solitons in integrable boundary field theories. In the present context, factorization of interactions is the unifying principle behind integrability. This was well-known for quantum theories but was essentially unexplored classically.