Vector rogue wave solutions of the three-dimensional partially nonlocal three-coupled nonlinear Schrödinger model under external potentials

Rogue waves are typical localized extreme waves in nonlinear systems, whose formation and evolution are jointly influenced by various factors, including nonlinear coupling, external potentials, and gain or loss. In this work, we construct a three-component partially nonlocal nonlinear Schrödinger model incorporating linear and harmonic potentials together with gain or loss. A self-similarity transformation is designed for this system to reduce the variable-coefficient model to a constant-coefficient integrable form, and explicit solutions of one-, two-, and three-component vector rogue waves are obtained in combination with the Darboux transformation. Under periodic and exponential diffraction, the effects of external potentials and related parameters on the amplitude, spatial distribution, peak structures, and interactions of rogue waves are investigated. The results show that, in the periodic diffraction system, the Hermite parameter controls the number of rogue-wave arrays along the propagation direction, while gain and loss respectively enhance and suppress the amplitude evolution of the rogue waves. In the exponential diffraction system, the rogue-wave amplitude first increases rapidly and then gradually approaches saturation, while the system parameters enable continuous tuning of the transverse position of the rogue waves. Further parameter tuning can drive two-component vector rogue waves through a continuous interaction process involving attraction, fusion, and separation, and can induce transitions of three-component vector rogue waves among single-, double-, and triple-peak structures. These results demonstrate that the proposed three-component model and its self-similarity reduction can describe diverse evolutionary behaviors of multicomponent vector rogue waves under combined external potentials, and provide a theoretical basis for the control of extreme waves in partially nonlocal multicomponent systems.