Inverse scattering transform: an overview and the Toda’s chain as paradigm for discrete systems

We review the tools used in the inverse scattering transform, focusing primarily on their computational aspects. As an example, we discuss the Toda's chain, an apparently simple nonlinear discrete system, to illustrate the various steps of the process. We chose this naturally discrete nonlinear system to avoid the additional errors that can arise from discretizing a differential equation whose continuum limit represents the problem under consideration. Furthermore, the homogenized Toda's chain is equivalent to the renowned Korteweg-de Vries equation and also resembles the equally famous Fermi-Pasta-Ulam-Tsingou problem. Given that the Toda's chain serves as a prototype for nonlinear systems with known analytical solutions, it provides a valuable test case for numerical procedures. Our main goal is to outline the various steps of the inverse scattering transform, with particular attention to numerical aspects, including the reconstruction of soliton shapes.