Under investigation in this paper is a (2[Formula: see text]+[Formula: see text]1)-dimensional nonlinear Schrödinger equation, which is a generalization of the standard nonlinear Schrödinger equation. By means of the modified Darboux transformation, the hierarchies of rational solutions and breather solutions are generated from the plane wave solution. Furthermore, the main characteristics of the nonlinear waves including the Akhmediev breathers, Kuznetsov–Ma solitons, and their combined structures are graphically discussed. Our results would be of much importance in enriching and explaining rogue wave phenomena in nonlinear wave fields.
Rogue wave patterns in the nonlinear Schrödinger equation