Nonlinear dynamics and pattern formation in optical wave systems: A quantum hydrodynamic and spectral perspective

We study the two-dimensional nonlinear Schrödinger equation (2D NLSE) with cubic focusing nonlinearity, a fundamental model describing wave propagation in nonlinear optical media and related physical systems. The problem is important because it governs the formation, stability, and evolution of localized wave structures (solitons), which are essential for applications in optical communication, photonic devices, and matter-wave dynamics. To address this problem, we employ a high-resolution split-step Fourier method with Strang splitting to numerically investigate the influence of dispersion sign, nonlinearity strength, and initial amplitude on wave dynamics. The numerical experiments are designed to capture both localized and oscillatory behaviors, as well as the transition between different dynamical regimes. The results show that anomalous dispersion promotes the formation of robust, high-intensity, and spatially localized soliton structures with long-term stability, whereas normal dispersion leads to wave spreading and reduced localization. Furthermore, the inclusion of weak nonlocal and quantum-like corrections significantly enhances stability, suppresses modulational instabilities, and enables controlled pattern formation. The interplay between diffraction, nonlinearity, and dispersion is found to critically determine the emergence of cyclic oscillatory patterns and stable localized states. We conclude that the combined effects of dispersion, nonlinearity, and higher-order corrections provide effective mechanisms for controlling wave localization and stability in nonlinear media. The novelty of this work lies in the systematic numerical exploration of these competing effects within a unified framework, together with the demonstration of how weak nonlocal and quantum-like terms can be used to regulate pattern formation and stabilize nonlinear wave structures. These findings offer valuable insights for the design of advanced optical systems and for understanding complex wave phenomena in nonlinear physical settings.