Bernard Deconinck, Professor and Chair of Applied Mathematics, University of Washington
Pole dynamics of solutions of integrable equations
Kruskal (1974) suggested that the dynamics of solutions of the KdV equation could be understood by examining how their pole singularities (complex x, real t) interact. I will review a biased history of the studies that followed, and I will discuss progress on three open problems in this area. The first one is the understanding of the Hamiltonian structure of the dynamical system describing the dynamics of the poles. Second is the pole dynamics of soliton solutions of the focusing NLS equation. Time permitting, I will discuss our recent "progress" on understanding the singularity dynamics emanating from initial conditions that are not compatible with the Painlevé structure of KdV solutions. This talk will involve dynamical systems, asymptotics, computation, and more.