In this paper, the higher-order localized waves for the coupled mixed derivative nonlinear Schrödinger equation are investigated using generalized Darboux transformation. On the basis of seed solutions and a Lax pair, the first- and second-order localized wave solutions are derived from the Nth-order iteration formulas of generalized Darboux transformation. Then, the dynamics of the localized waves are analyzed and displayed via numerical simulation. It is found that the second-order rouge wave split into three first-order rogue waves due to the influence of the separation function. In addition, a series of novel dynamic evolution plots exhibit that rogue waves coexist with dark-bright solitons and breathers.

Higher-order localized waves for a variable-coefficient coupled Hirota equation describes the vector optical pulses in inhomogeneous optical fiber and are investigated via generalized Darboux transformation in this work. Based on its Lax pair and seed solutions, the localized wave solutions are calculated, evolution plots are constructed, and the dynamics of the obtained localized waves are analyzed through numerical simulation. It is observed that the first- and second-order localized waves interact with dark-bright solitons or breathers, and the functions α(t), β(t), and δ(t) determine the propagation shape of the localized waves. The presented results contribute to enriching the dynamics of localized waves in inhomogeneous optical fiber. Keywords: variable-coefficient coupled Hirota equation; generalized Darboux transformation; soliton; breather

In this paper, higher-order localized waves for a coupled fourth-order nonlinear Schrödinger equation are investigated via a generalized Darboux transformation. The [Formula: see text]th-order localized wave solutions of this equation are derived via Lax pair and Darboux matrix. Evolution plots are made and dynamical characteristics of the obtained higher-order localized waves are analyzed through numerical simulation. It is observed that rogue waves coexist with dark–bright solitons and breathers. The presented results also show that different values of the involved parameters have diverse effects on the higher-order localized waves.