Nonlocal symmetries and explicit solutions of the AKNS system

In this letter, based on the Lax pair of the Ablowitz–Kaup–Newell–Segur (AKNS) system, the nonlocal symmetry is obtained and successfully localized to a Lie point symmetry by introducing an appropriate auxiliary dependent variable. For the closed prolongation, the construction for the one-dimensional optimal system is presented in detail. Furthermore, using the obtained optimal system, we give the reductions and the explicit analytic interaction solutions between cnoidal waves and solitary waves. For some interesting solutions, the figures are given to show their properties.