Exploring Qualitative Analysis and Interaction Dynamics in a (3+1)-Dimensional Boussinesq Equation II via Hirota Bilinear Method

Source abstract

In this work, we explore the nonlinear wave phenomena of the (3+1)-dimensional Boussinesq (II) equation, a significantly higher-dimensional model that describes dispersive wave propagation in fluid dynamics, plasma systems, and nonlinear optics. Using exact analytic and qualitative dynamic approaches, we study a wide range of solutions and stability characteristics of the model. Initially, we use the Hirota bilinear method to obtain a number of exact solutions, such as breather waves, two-wave interaction solutions, and other types of localized nonlinear waves. These solutions display remarkable physical properties, including periodic energy trapping, oscillatory modulations, and nonlinear wave interactions in higher dimensions. In addition, the (m+1G′)-expansion method is used to derive new soliton solutions, such as bright solitary waves and W-shaped solitons, which are found to be stable and undergo pulse-shaping dynamics under certain conditions. Three-dimensional, two-dimensional, and contour plots are displayed for some of the solutions to demonstrate the physical significance of the results. The visualizations reveal the presence of localized waves, wave interactions, periodical breathing, and stable soliton profiles. Furthermore, we conduct modulation instability analysis to describe the conditions under which small perturbations of continuous wave backgrounds are unstable. The dispersion relation and the instability gain spectrum are obtained, which explain the formation of breathers, soliton trains, and other coherent structures. Furthermore, a Galilean transformation converts the governing equation into a planar nonlinear dynamical system, enabling its qualitative study. The Hamiltonian structure is revealed, and the fixed points are identified as centers, saddles, and cusps through bifurcation analysis. To investigate more complex dynamics, a periodic forcing term is introduced into the system, resulting in chaos in the forced system. The chaotic behavior is confirmed via phase portraits, three-dimensional attractors, time series, Poincaré sections, return maps, fractal dimension, and positive Lyapunov exponents. We also perform a sensitivity test to show the effect of initial condition variations on the system’s long-term dynamics. The findings greatly expand the exact solution set and dynamics of the (3+1)-dimensional Boussinesq equation (II). The analytical approach presented in this paper can also be applied to other multidimensional nonlinear evolution equations of mathematical physics.