Dynamical analyses and dispersive soliton solutions to the nonlinear fractional model in stratified fluids

Abstract This study explores the bifurcation analysis, sensitivity analysis (SA), stability analysis, and exact solitonic wave profiles for the time-fractional Benjamin–Ono (BO) equation, which models internal waves in stratified fluids, especially where dispersive effects play a significant role. These solutions are crucial for understanding ocean engineering and mathematical physics phenomena. The BO equation simulates deep-water waves, making it essential for ocean engineering applications. We employ some diverse strategies such as the new extended direct algebraic method, generalized Arnous method, and ansatz method to extract novel dispersive wave solutions. These solutions exhibit diverse shapes, such as hyperbolic, singular periodic, exponential, rational function solutions and solitary waves including dark, singular, bright, combo, and complex solutions. Our main goal is to analyze the dynamic characteristics of the model by conducting bifurcation and SA and identify the corresponding Hamiltonian function. To ensure validity, we also conduct stability analysis using linear stability theory and outline constraint conditions. Furthermore, the bifurcation of phase portraits of ordinary differential equations corresponding to partial differential equations under investigation is also analyzed. We also demonstrate the fractional behavior of our results through visualizations (2D, 3D, contour, and density plots) by selecting suitable parametric values. Our reported results are verified using Mathematica to guarantee accuracy and validity. A detailed comparison with existing results highlights the novelty of our findings. This research contributes significantly to understand wave dynamics in nonlinear phenomena and the unique outcomes explored in this research will play a significant role in the forthcoming investigation of nonlinear problems. Moreover, the novelty of this study lies in the fact that the proposed model has not been previously explored using the aforementioned advanced methods and comprehensive dynamical analyses. This study pioneers the exploration of the fractional BO equation, yielding unique analytical results. Our techniques efficiently identify accurate solitary pulse solutions to nonlinear dynamical models with fractional parameters, making them highly successful in modeling deep-water internal waves. Our computational analytical tools are also straightforward, transparent, and reliable, reducing complexity while widening applicability. The acquired solutions are expected to have a profound impact on the study of wave propagation and related fields, offering new insights and perspectives that can inform future research and applications.