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The defocusing nonlinear Schr\"odinger (NLS) equation has no the modulationalinstability, and was not found to possess the rogue wave (RW) phenomenon up tonow. In this paper, we firstly investigate some novel nonlinear wave structuresin the defocusing NLS equation with real-valued time-dependent andtime-independent potentials such that the stable new RWs and W-shaped solitonsare found, respectively. Moreover, the interactions of two or three RWs areexplored such that the RWs with higher amplitudes are generated in thedefocusing NLS equation with real-valued time-dependent potentials. Finally, westudy the defocusing NLS equation with complex PT -symmetric potentials suchthat some RWs and W-shaped solitons are also found. These novel results will beuseful to design the related physical experiments to generate the RW phenomenaand W-shaped solitons in the case of defocusing nonlinear interactions, and toapply them in the related fields of nonlinear or even linear sciences.

Rogue wave formation and interactions in the defocusing nonlinear Schr\"odinger equation with external potentials