On the soliton dynamics, bifurcation analysis and perturbed induced chaotic behaviors of the time-fractional Boussinesq equation

Abstract The time-fractional Boussinesq equation helps in the modeling of various physical phenomena, such as shallow water waves and coastal engineering, to model tsunamis. This paper investigates the time-fractional Boussinesq equation, after introducing memory effect through the time fractional parameter β ∈ (0, 1]. Exact solutions are derived with the help of the ( G ′/ G , 1/ G )-expansion method, yielding the solitons for different cases, i.e., hyperbolic, trigonometric, and rational. The influence of the fractional order on solution profiles is examined through three-dimensional plots in terms of amplitude modulation, wave localization, and propagation dynamics. Furthermore, bifurcation analysis is performed to identify stability transitions using the formation of a Hamiltonian system. Along with that, the chaotic behavior is further examined using the three-dimensional plots, and Poincaré plots are employed to identify stability transitions and complex dynamical transitions. For validating the results, Lyapunov exponent and sensitivity analysis have been carried out.