Direct and Inverse Scattering for the Ultradiscrete KdV Equation

Explore a 54-minute lecture on direct and inverse scattering for the ultradiscrete KdV equation, presented by Ralph Willox at the Workshop on box-ball systems from integrable systems and probabilistic perspectives. Delve into a method for solving the initial value problem for the ultradiscrete KdV (udKdV) equation over real numbers, which encompasses the Takahashi-Satsuma Box & Ball system. Discover an ultradiscrete analogue of the inverse scattering transform for the continuous KdV equation. Learn about constructing bound state and non-bound state eigenfunctions to solve the direct scattering problem for udKdV with any potential over real numbers and compact support. Examine the reconstruction of the potential in the scattering problem at different time steps using an ultradiscrete analogue of a Darboux dressing transformation. Understand how to obtain data characterizing soliton content and background parts of the initial potential through successive Darboux undressing transformations. This talk is based on the paper "Darboux dressing and undressing for the ultradiscrete KdV equation" by J.J.C. Nimmo, C.R. Gilson, and R. Willox.