# Darboux dressing and undressing for the ultradiscrete KdV equation

## TL;DR

This paper addresses the inverse scattering problem for the ultradiscrete KdV equation, providing explicit solutions for eigenfunctions and a Darboux transformation-based method to reconstruct potentials and analyze soliton content.

## Contribution

It introduces a novel ultradiscrete Darboux transformation approach for solving the inverse scattering problem of the udKdV equation.

## Key findings

- Explicit construction of bound and non-bound state eigenfunctions
- Method for reconstructing potentials using Darboux transformation
- Unique characterization of soliton content and background

## Abstract

We solve the direct scattering problem for the ultradiscrete Korteweg de Vries (udKdV) equation, over $\mathbb R$ 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.