Darboux dressing and undressing for the ultradiscrete KdV equation
arXiv:1903.07315 · doi:10.1088/1751-8121/ab45cf
Abstract
We solve the direct scattering problem for the ultradiscrete Korteweg de Vries (udKdV) equation, over for any potential with compact (finite) support, by explicitly constructing bound state and non-bound state eigenfunctions. We then show how to reconstruct the potential in the scattering problem at any time, using an ultradiscrete analogue of a Darboux transformation. This is achieved by obtaining data uniquely characterising the soliton content and the `background' from the initial potential by Darboux transformation.
41 pages, 5 figures // Full, unabridged version, including two appendices