A unified analytical framework for chaos, sensitivity, and modulation instability with exact solutions of a newer form of Boussinesq equation

Abstract

This study explores the features and properties of a novel (3+1)-dimensional Boussinesq equation, which models the propagation of shallow water waves in higher-dimensional settings. The equation is significant for understanding nonlinear wave dynamics not only in hydrodynamics but also in fields such as plasma physics, nonlinear optics, and lattice wave theory. The Boussinesq-type equations, incorporating weakly dispersive and weakly nonlinear terms, provide a robust framework for simulating tsunami wave dynamics, particularly in modeling dispersion, wave shoaling, and inundation in coastal regions. A key contribution of this work lies in the application of the generalized exponential rational function method, which has not been previously utilized for this particular equation. Using this method, various exact analytical solutions-including solitary wave solutions-are constructed and graphically illustrated. Furthermore, the study conducts a detailed chaotic analysis, sensitivity analysis and modulation instability analysis to examine the stability characteristics of the obtained wave solutions. The results provide deeper insight into the nonlinear behavior and stability of wave propagation in the (3+1)-dimensional Boussinesq equation framework.