Ring/vortex-like extreme wave in the partially nonlocal medium with different diffraction characteristics in both directions under influence of external potential and gain/loss

In this study, we analyze a (3+1)-dimensional partially nonlocal nonlinear Schrödinger (NLS) model, which incorporates various diffraction effects, gain or loss mechanisms, and confinement within linear and parabolic potentials. By reducing this complex model to a (2+1)-dimensional framework, we uncover analytical solutions that exhibit high-dimensional extreme wave structures with Hermite-Gaussian envelopes, illustrating the model's nonautonomous characteristics. Our investigation focuses on ring-like and vortex-like extreme waves, examining how different parameters—such as radius, Hermite parameter, gain, and thickness—affect these wave structures. Specifically, we find that, for fixed thickness, Hermite, and gain parameters, the radius influences the size of the wave structures. Conversely, with a fixed radius, Hermite, and thickness parameters, the gain parameter modifies the wave properties. The introduction of the Hermite parameter p increases the number of concentric layers in the ring-like extreme waves by $p+1$. Additionally, incorporating gain and loss effects enhances the model's applicability to real-world scenarios.