<p>We utilized the enhanced Sardar sub-equation and generalized Riccati equation methods to explore precise solitary wave solutions for the coupled Higgs field equations. These equations characterize the behavior of conserved scalar nucleons interacting with a neutral scalar meson. Our findings revealed the formation of various types of solitons, including periodic, dark-bell, bright, and anti-bell solutions, which can be expressed using exponential, hyperbolic, rational and trigonometric functions. Initially, we transformed the model into a set of nonlinear ordinary differential equations through a traveling wave transformation. We utilized Maple 18 to visualize some of the solutions with both 2D and 3D plots, demonstrating their behavior by varying relevant parameters. The proposed methods do not require linearization, perturbation, or specific initial and boundary conditions. The methods discussed were straightforward to implement, effective and appropriate for addressing a wide range of nonlinear partial differential equations. To the best of the author’s knowledge, the results presented in this article are entirely original and have not been published elsewhere. These findings may open new avenues for exploring nonlinear behaviors across various physical systems, including quantum physics, data science, and optical phenomena.</p>