Modulation instability, localized wave solutions of the modified Gerdjikov–Ivanov equation with anomalous dispersion

Abstract Solving new integrable systems and exploring their physical applications have been a hot topic. This paper gives the modulation instability in the continuous wave background of the new modified Gerdjikov–Ivanov equation with anomalous dispersion. Based on the extend Lax pair, the localized wave solutions of the new model are obtained via generalized Darboux transformation method, various types solutions including breather, rogue waves, interaction solutions are presented and their dynamic properties are analyzed. The results obtained have certain application value for nonlinear optics and long-distance transmission.