AIM Seminar: Wave patterns generated by large-amplitude rogue waves and their universal character

It is known from our recent work that both fundamental rogue wave solutions (with Peter Miller and Liming Ling) and multi-pole soliton solutions (with Robert Buckingham) of the nonlinear Schrödinger (NLS) equation exhibit the same universal asymptotic behavior in the limit of large order in a shrinking region near their peak amplitude point, despite the quite different boundary conditions these solutions satisfy at infinity. This behavior is described by a special solution of again the NLS equation that also satisfies ordinary differential equations from the Painlevé-III hierarchy. We review these results and show that this profile also arises universally from arbitrary background fields. We then show how rogue waves and solitons of arbitrary orders can be placed within a common analytical framework in which the "order" becomes a continuous parameter, allowing one to tune continuously between types of solutions satisfying different boundary conditions. In this framework, solitons and rogue waves of increasing integer orders alternate as the continuous order parameter increases. We show that in a bounded region of the space-time of size proportional to the order, these solutions all appear to be the same when the order is large. However, in the unbounded complementary region one sees qualitatively different asymptotic behavior along different sequences. This is joint work with Peter Miller (U. Michigan).