Investigation of the intricate dynamics and a variety of hybrid soliton, solutions of the (3+1)-dimensional Boussinesq equation

This manuscript delves into the captivating domain of the (3+1)-dimensional Boussinesq equation, unraveling its complex dynamics and revealing mesmerizing soliton solutions. Our exploration begins with the application of the well-established traveling wave transformation, transforming the model into a set of ordinary differential equations (ODEs). Navigating through the intriguing intricacies of this system, we delve into bifurcations, chaos, and sensitivity to initial conditions, vividly brought to life through numerical simulations of phase portraits. Additionally, we present and discuss various solitary wave solutions, including singular, bright, breather-like, hybrid singular, rogue wave, and multi-singular periodic waves, vividly illustrating their existence through simulations. The (3+1)-dimensional Boussinesq equation presents rich dynamics with applications spanning fluid dynamics, climate science, optics, engineering, and more. Its soliton solutions and intricate behaviors contribute to advancements in diverse scientific disciplines.