Exploring the fourth-order Boussinesq water wave equation: soliton analysis, modulation instability, sensitivity behavior, and chaotic analysis

In this article, we reveal the novel types of exact solitons to the fourth-order nonlinear (1 + 1)-dimensional Boussinesq water wave equation. This model is obtained under the consideration of the smaller water depth and larger wavelength of the waves. The Boussinesq water wave equation is useful in understanding water wave behavior, harbor design, coastal dynamics, wave propagation in shallow seas, ocean wave models, marine environments, etc. For our aim, we used the Sardar sub-equation technique. As a result, new types of exact wave solitons involving trigonometry, hyperbolic trigonometry, and rational functions are gained. Some gained solutions are represented through 2D, 3D, contour, and density plots. In bifurcation analysis, a new planar dynamical system of the governing model is obtained by applying the Galilean transformation, and all possible phase portraits are discussed. Modulation instability is used to obtain the steady-state solutions of the concerned model. Furthermore, the chaotic behavior of the governing model is analyzed. Sensitivity analysis is utilized to determine the sensitivity behavior of the model. The achieved solutions are fruitful in distinct areas of mathematical physics and engineering fields. At the end, the technique is a useful and reliable approach to solving other important nonlinear partial differential equations. This study applies the Sardar sub-equation method to derive new analytical solutions of the fourth-order nonlinear (1 + 1)-dimensional Boussinesq water wave equation. The method demonstrates greater flexibility than traditional approaches in handling nonlinear terms. However, the results depend on specific parameter conditions, and experimental or numerical validation is left for future investigation.