Nucci based soliton structures and nonlinear dynamics of the (3+1) dimensional hyperbolic Schrödinger equation

The Hyperbolic $$(3+1)$$ -dimensional nonlinear Schrödinger equation, a basic model for multidimensional nonlinear wave propagation, is examined in this work. Exact analytical solutions covering bright and dark solitons, periodic waveforms, and M- and W-shaped soliton solutions were obtained by using Nucci’s reduction approach. The spatial-temporal features and stability of these solutions are further demonstrated by means of contour graphs, 2D graphs, and 3D surface plots. Bifurcation analysis was used to investigate the system’s dynamical changes beyond precise solutions, highlighting characteristics including multi-stability, chaotic development, and sensitivity towards initial conditions. The system’s considerable dependence on initial states and control parameters, which determine whether steady soliton propagation, oscillatory modes, or chaotic behaviors emerge, is particularly highlighted by the sensitivity analysis. The physical interpretations highlight the significance of these findings in fields such as fluid dynamics, Bose-Einstein condensates, plasma physics, and nonlinear optics, where structured multi-lobed solitons, phase defects, oscillatory states, and localized energy transfer are important. This study combines analytical soliton solutions obtained through Nucci’s reduction method with bifurcation, chaotic, multi-stability, and sensitivity analyses to investigate the dynamical behavior of the hyperbolic $$(3+1)$$ -dimensional nonlinear Schrödinger equation. These results lay down the foundation for future theoretical, computational, and experimental studies of multidimensional nonlinear wave dynamics and enhance the collection of analytical standards for higher-dimensional nonlinear systems.