Existence of dark solitons for Kundu–Eckhaus equations

This paper focuses on dark solitons and periodic solitary wave solutions for Kundu–Eckhaus equations without and with a strong generic delay, respectively. First, in terms of the dynamic system arguments, new dark solitons and periodic solitary wave solutions for the Kundu–Eckhaus equation without delay are obtained by integrating along different orbits. Then, the limits and modulation stability of the solutions of the Kundu–Eckhaus equation are investigated. Finally, the existence of two dark solitons and two periodic solitary wave solutions for Kundu–Eckhaus equations with a strong generic delays is proven via geometric singular perturbation theory.