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Explore the mathematical theory of rogue waves of infinite order through this conference talk that examines the focusing nonlinear Schrödinger equation as a universal model for wave packet amplitudes in weakly-nonlinear and strongly-dispersive systems. Discover how these novel solution families emerge in specific asymptotic regimes involving large-amplitude and near-field limits of broad solution classes, with applications spanning water waves and nonlinear optics. Learn about the anomalously slow temporal decay properties of these solutions and their connection to the third Painlevé equation. Investigate the extension of rogue wave emergence to the first several flows of the AKNS hierarchy, including scenarios with arbitrarily many simultaneous flows, and examine recent research on their space-time asymptotic behavior under general hierarchy flows. Gain insights into the exact and asymptotic properties of these special solutions and their universal emergence patterns in dispersive integrable systems.
