Spectral adventures in quantum realms: Nonlinear schrödinger dynamics, quantum vortices and time-resolved wave mechanics

In this paper, we present a comprehensive study of quantum wave phenomena using Fourier spectral numerical methods. The focus is on three interrelated topics: (1) the nonlinear Schrödinger equation (NLS) in physical systems, including optical solitons and Bose-Einstein condensates (via the Gross-Pitaevskii equation, GPE); (2) simulations of the time-dependent Schrödinger equation (TDSE) to explore quantum tunneling, wavepacket dynamics and interference; and (3) the characteristics of quantum turbulence and vortices in superfluid systems. We develop the mathematical formulations of NLS and GPE, highlighting how spectral methods efficiently capture their solutions' high-frequency content and conserved quantities. We detail the implementation of Fourier pseudo-spectral discretization combined with split-step (operator splitting) time integration, evaluating its accuracy and stability. We also discuss numerical error analysis and comparisons with alternative discretization approaches (finite differences and finite elements). The results include simulations of soliton propagation over long distances without shape distortion, quantum tunneling of wavepackets through potential barriers, and formation of vortex lattices and turbulent energy cascades in condensates. Visualizations such as soliton amplitude profiles, probability density snapshots of tunneling wave functions, and vortex lattice images are provided to illustrate these phenomena. Our findings underscore the spectral method's superior accuracy (exponential convergence for smooth solutions) and its ability to preserve physical invariants over long simulation times. We conclude that Fourier spectral techniques offer a robust and precise framework for graduate-level research and emerging applications in nonlinear and quantum wave systems.