Integrable systems and squared eigenfunctions

We briefly review the Ablowitz–Kaup–Newell–Segur (AKNS) formalism for 1D$+$1D integrable systems starting with the Lax pair and continuing into integrable perturbation theory and squared eigenfunctions. We emphasize the common features of the inverse scattering transform across a wide range of known 1D$+$1D systems. We tailor the various steps to be the same as in treating higher-order systems. We briefly review both the direct and inverse scattering problems and then consider perturbations of the potentials and the scattering data. For the latter topic, we reformulate the original treatment of perturbations of the AKNS system such that it aligns with the common features of 1D$+$1D systems. We use a recent approach to derive the perturbations of the potentials due to perturbations of the scattering data in the absence of solitons. Finally, we show that recent results where the squared eigenfunctions and their adjoints were found as sums of products (not simply products) of Jost functions are determined by symmetries imposed on the potential matrix.