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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  N th-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.

Localized Waves for the Coupled Mixed Derivative Nonlinear Schrödinger Equation in a Birefringent Optical Fiber