Darboux–Bäcklund transformation and localized excitation on the periodic wave background for the nonlinear Schrödinger equation

We construct the exact nonlinear wave solutions of the Nonlinear Schrödinger equation on the period wave background instead of on a constant background. By using Darboux–Bäcklund transformation, soliton and breather solutions on two types of cnoidal wave backgrounds are given. The density evolutions of these solutions are given under different parameters to study their wave structures and dynamical properties.