<p>This study aims to extract closed-form soliton solutions and illustrate the evolving forms of solitons for a significant class of nonlinear evolution equations, focusing specifically on the (3+1)-dimensional Boussinesq equation. This equation is a mathematical model for describing complicated wave processes in fluid dynamics, oceanography, plasma physics, and earth and atmospheric sciences. Due to its multidimensional nature and strong nonlinearity, finding exact solutions to this equation poses a considerable analytical challenge. To address this, we implement two methods, namely, the Generalized Exponential Rational Function (GERF) method and the Modified Sardar Sub-equation (MSSE) method. These analytical techniques are known for their ability to generate closed-form solutions for higher-order nonlinear partial differential equations. Through these methods, we obtain various types of accurate soliton solutions, including dark and bright solitons, kink and anti-kink profiles, periodic waves, W-shaped structures, and multi-soliton solutions. The dynamic aspects of the solitonic profiles, as well as newly formed exact solutions, are graphically represented in 2D and 3D with density graphs. By tuning the free parameters present in the solution expressions, various types of wave patterns can be outlined, which highlight the rich dynamics of the system. The findings of our study enable us to gain a thorough understanding of the evolving features of higher-order nonlinear evolution equations in Earth and atmospheric sciences. They also demonstrate how modern mathematical methods, such as the GERF and MSSE methods, can effectively represent complex physical processes in various physical and engineering systems.</p>