Ring/vortex-like extreme waves for variable-coefficient axially and partially nonlocal coupled nonlinear Schrödinger equations with different diffraction characteristics in transverse plane under influence of external potential and gain/loss

The (3+1)-dimensional axially and partially nonlocal (PN) coupled nonlinear Schrödinger (NLS) model, enriched with different values of two transverse diffraction profiles and subjected to gain or loss phenomena, finds its dimensional reduction to a (2+1)-dimensional counterpart model facilitated by a converting relation. This reduction unveils intriguing insights into the excited mechanisms underlying partially nonlocal waves, culminating in the derivation of analytical solutions portraying high-dimensional extreme waves characterized by Hermite-Gaussian envelopes. Specifically, the investigation delves into the distinctive traits and evolutionary trajectories of ring-like and vortex-like extreme waves, particularly within exponential systems featuring constant or exponential chirp and gain. Notably, the radius parameter R emerges as a key determinant, governing the thickness of the first and second-order extreme waves’ ring-like and cylinder-like structures. Furthermore, the thickness parameter \(\omega \) $\omega$ influences the thickness of these structures, while the Hermite parameters p and n modulate the formation of additional layers along the z-axis represented by \(p+1\) $p+1$ and \(n+1\) $n+1$ respectively. The inclusion of gain or loss parameter permits the analysis in a more realistic environment in which the theoretical frameworks and real-world implementations are connected.