Orateur
Description
I will discuss two families of weakly localized solutions of the focusing nonlinear Schrödinger equation, both represented by Riemann-Hilbert problems posed on large circles. In each case, the jump matrix is regular on the contour but encodes a singularity hidden inside: a Blaschke-type/logarithmic singularity in a Painlevé-V-related family, and an essential singularity for general rogue waves of infinite order. Although these solutions belong to $L^2(\mathbb{R})$ but not $L^1(\mathbb{R})$, and hence fall outside the standard inversescattering framework, their Riemann-Hilbert representations still allow for rigorous long-time asymptotic analysis. I will compare the resulting decay rates, $O(t^{-1 / 2})$ in the Painlevé-V case and the anomalously slow $O(t^{-1 / 3})$ rate for the Painlevé-III-related infinite-order rogue waves, and describe a limiting regime connecting the two families. Time permitting, I will describe ongoing work with P. Miller on another class of solutions that are weakly localized on a nonzero background. The Painlevé-V part is joint work with A. Gkogkou, G. Mazzuca, and K. McLaughlin, and the infinite-order rogue-wave part is joint work with L. Ling and P. Miller.