Solitons and bifurcation analysis for the fractional coupled Boussinesq Burger equation in fluids

The investigation of fluid flow in dynamic systems inevitably results in temporal fractional coupled Boussinesq–Burger equations, which are examined in this work to model the propagation of shallow water waves. These equations belong to a large class of shallow water models that account for the movement of water bodies, vertical mixing impacts, and fluid dynamics in the subsurface under conditions of different pressures. Shallow water currents have a high degree of difference between their horizontal and vertical magnitudes, making the analysis thereof difficult but highly relevant for real-world applications in hydrodynamics. In this paper, we solve the time-fractional coupled Boussinesq–Burger model, a long-wave equation, by making use of the modified extended tanh function method and the [Formula: see text] method. These methods enable the retrieval of a dense set of soliton solutions, such as kink, singular, periodic-singular, and dark-singular waves, revealing more about nonlinear wave propagation in fractional-order systems. The stability, bifurcation properties, and qualitative characteristics of these solutions are systematically investigated through phase portraits, which exhibit key dynamics and transitions between various wave states. In addition, the ensuing wave structures are represented via detailed graphical plots, such as density, 2D, and 3D plots, which facilitate visualization of intricate interactions and propagation patterns. Results of this research offer new analytical solutions and provide a mathematical framework for wave phenomena in fractional fluid models. Such results can find applications in coastal engineering, hydraulic networks, and other fields where reliable models of nonlinear shallow water waves play a critical role.