Soliton structures and dynamical characteristics of fractional nonlinear waves in the classical Boussinesq framework.

This paper investigates the time-space fractional classical Boussinesq equation, a nonlinear partial differential equation that describes long-wave propagation in shallow water. The modified extended tanh function formalism yields bright and dark solitons, breather-type waves, and periodic waves. The dynamical behavior of these solutions is revealed with bifurcation theory and phase-plane analysis: stable and unstable wave profiles change, and there may be chaotic interactions among them. The sensitivity analysis and the linear stability analysis have guaranteed the robustness of the solutions to small perturbations. The key results indicate that the equation facilitates a deep set of stable nonlinear wave forms, which exhibit predictable dynamical actions, to promote research into nonlinear fractional wave phenomena. Shallow water hydrodynamics, plasma physics, and nonlinear lattice systems can utilize these findings. The future research could take these findings to a new level by adding stochastic effects, fractional systems of higher dimensions, and numerical simulations to make the analysis and feasible validation more thorough.