Seminar| Institute of Mathematical Sciences
Time：16:00-17:00, Nov. 11, Monday
Location：Room S408, IMS
Speaker: Cheng Zhang, Shanghai University
Abstract: Roughly speaking, integrable systems are algebraic and/or differential equations that can be exactly solved. The property of exactly solvability is connected to very rich mathematical structures. In this talk, I will first give an overview of the so-called Yang-Baxter and reflection equations that are key integrable structures characterizing a wide range of quantum/classical integrable systems. They are algebraic equations arising, for instance, in the context of open-boundary problems for quantum spin chains. At the “classical” level, there are the associated classical Yang-Baxter and reflection equations. Solutions to the Yang-Baxter and reflection equations amount to the quantum R-matrix (or classical r-matrix) and K-matrix. Then, I will show some applications of the Yang-Baxter/reflection equations in the context of integrable PDEs on the half-line. Now the boundary-value problems for integrable PDE are translated into some simple algebraic constraints. Solutions to integrable PDEs on the half-line are constructed using the so-called “boundary dressing”. This construction not only overcomes the analytical difficulty of dealing with nonlinear boundary-value problems, but also gives a unified inverse scattering transform picture of the underlying problems. In the end, some connections to partial difference equations on generic planar graphs will be discussed.