Analytical Dynamics of Complex Soliton Structures Characterized by Hyperbolic, Trigonometric, and Jacobi Elliptic Solutions in the Coupled Higgs System

This work is devoted to the analytical investigation of the 1+1-dimensional coupled Higgs system, which arises in a spectrum of physical contexts including plasma physics, nonlinear optics, field theory, and hydrodynamic wave propagation. Understanding the exact wave behaviors supported by this system is crucial for revealing its nonlinear dynamical behavior and potential physical applications. To this end, two efficient analytical techniques, namely, the Jacobi elliptic function expansion (JEFE) method and the f/μf+g-expansion method, are utilized to construct new exact solutions of the coupled Higgs equations. A wide family of soliton solutions is derived, encompassing bright, periodic, dark, singular, mixed hyperbolic, mixed trigonometric, and Jacobi elliptic function–based wave structures. Several known solutions reported in earlier studies are recovered as special limiting cases, while many of the obtained solutions are shown to be new and have not been reported previously for the coupled Higgs system. To demonstrate the qualitative features of the derived solutions, extensive graphical representations in the form of three-dimensional surface plots, two-dimensional profiles, and contour plots are provided, emphasizing the influence of key parameters on wave localization, propagation, and stability. The results illustrate that the proposed analytical frameworks significantly extend the solution space of the coupled Higgs system and offer deeper insight into its nonlinear wave dynamics. The findings not only enrich the existing literature on Higgs-type models but also provide useful mathematical tools for further investigations of nonlinear evolution equations arising in applied physics.