Contact structure

Short stories (continued)

How to make a salad dressing, Fibonacci the marathon runner, the beauty of structure (see image). Just three of the many short films we have made over the last few weeks. The films aim to make mathematics both entertaining but also challenging, encouraging people to think and maybe make connections and realise the part maths might be quietly playing in their lives. But, course, viewers can just scroll on (after sometimes posting a comment...).
Optimal control of the non-linear Fokker-Planck equation
 Hambly, B Jettkant, P Stochastic Processes and their Applications
Wed, 17 Sep 2025
11:15
L3

The KdV equation: exponential asymptotics, complex singularities and Painlevé II

Scott W. McCue
(School of Mathematical Sciences Queensland University of Technology Brisbane)

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Further Information

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.

 

 

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