11:15
The KdV equation: exponential asymptotics, complex singularities and Painlevé II
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Scott W. McCue is Professor of Applied Mathematics at Queensland University of Technology. His research spans interfacial dynamics, water waves, fluid mechanics, mathematical biology, and moving boundary problems. He is widely recognised for his contributions to modelling complex free-boundary phenomena, including thin-film rupture, Hele–Shaw flows, and biological invasion processes.
Abstract
We apply techniques of exponential asymptotics to the KdV equation to derive the small-time behaviour for dispersive waves that propagate in one direction. The results demonstrate how the amplitude, wavelength and speed of these waves depend on the strength and location of complex-plane singularities of the initial condition. Using matched asymptotic expansions, we show how the small-time dynamics of complex singularities of the time-dependent solution are dictated by a Painlevé II problem with decreasing tritronquée solutions. We relate these dynamics to the solution on the real line.
