Abstract

The inverse scattering method (ISM) is one of the major breakthroughs in Mathematical Physics and Applied Mathematics. It provides a nonlinearization of the Fourier transform to solve certain (integrable) nonlinear PDEs. But it also triggered entire new fields of research via its extension and consequences in areas areas other than analysis: geometry (classical r-matrix method, Poisson-Lie groups) and algebra (quantum Yang-Baxter equation, quantum groups, Lie bialgebras). The physical motivation of describing boundary conditions within ISM has led to, and keeps triggering, its own share of discoveries in all these areas. To date, in analysis, the most complete theory for dealing efficiently with boundary conditions in integrable PDEs is provided by the unified method of Fokas. I will explain how the question of even more realistic situations where defects might be present as well naturally leads to consider PDEs on graphs. I will show how to use elements of Fokas method to solve in principle initial-boundary value problems for PDEs on a star-graph. This is the first step of a general program towards a theory of integrable PDEs on graphs. All along, I will use the (cubic) nonlinear Schrödinger equation to illustrate the ideas.