<p>In this paper, we analyze the complex coupled Higgs equation, which corresponds to a nonlinear model describing physical phenomena associated with the mass generation procedure in the context of quantum field theory. Two powerful analytical methods, namely the multivariate generalized exponential integral function method and the generalized projective Riccati equation method, have been used to develop exact travel wave solutions to the complicated coupled Higgs equation. With the application of both methods, a broad range of solution profiles has been generated, including hyperbolic, trigonometric, rational, and combined solution profiles associated with solitons. These solution profiles have been shown to have distinct physical interpretations using two-dimensional, three-dimensional, and contour plots. The research clarifies the interaction between periodic external forcing and nonlinear coupling, demonstrating the system’s transitions from periodic, through quasi-periodic, to chaotic regimes. Sensitivity analysis of initial conditions is performed, which provides priceless information on stability behavior and on parameter influence. All results provided are verified by direct substitution into the original equation, which confirms their compliance. The resulting wave and soliton patterns can be utilized as useful probes of complex physical phenomena captured by the Higgs equation, including resonance effects and nonlinear interactions between fields, as well as energy localization. Overall, the study not only improves the solution space of the coupled Higgs model but also increases the understanding of its dynamical properties, with possible applications in nonlinear science, mathematical physics, and quantum field dynamics.</p>