Dynamics of Complex Singularities of Nonlinear PDEs: Analysis and Computation JAC Weideman, Stellenbosch University, South Africa
Solutions to nonlinear evolution equations exhibit a wide range of interesting phenomena such as shocks, solitons, recurrence, and blow-up. As an aid to understanding some of these features, the solutions can be viewed as analytic functions of a complex space variable. The dynamics of poles and branch-point singularities in this complex plane can often be associated with nonlinear properties of the solution. For example, shock formation and multivaluedness in the inviscid Burgers equation can be understood as a conjugate pair of branch-point singularities that travel in the complex plane and meet on the real axis at the particular instant the shock is formed. In the first part of the talk we shall survey some of the analytical results in this area by revisiting, this time from a complex viewpoint, a few classic papers from the last century. This includes the 1950/1951 papers by Hopf and Cole (on the linearization of the Burgers equation) and the 1965 paper by Zabusky and Kruskal (on recurrence in the Korteweg-de Vries equation). In the second part of the talk we shall survey some of the numerical methods that can be used to approximate singularity dynamics in those cases where explicit solutions are unavailable.