Coupled Higgs Equation: Novel Solution via GSSE Method, Bifurcation and Chaotic Patterns and Series Solution via Symmetry

This article introduces the generalization of the Sardar sub-equation method, aiming to develop a more comprehensive approach. In GSSEM, we consider the more generalized assumed solution and generalized trigonometric and hyperbolic functions in the solutions. With the conventional integer order derivative model, processes characterized by long-term memory effects may fall short to capture their intricate dynamics. We consider the CHE incorporate with time-fractional derivative. The equation describes a cooperative system of scalar nucleons and neutral scalar mesons that are conserved within the system. Notably, the obtained solutions exhibit reductions to hyperbolic and trigonometric solutions in a specific scenario. Through the application of the proposed scheme, the study acquired solutions like rogue waves, damped waves, dark waves, breather and bright waves. These solutions are visually depicted through graphical representations. The advantage of the GSSE method is that it provides different kinds of solitons, such as rogue waves, damped waves, dark, bright, singular, combined dark-singular and combined dark-bright solitons. The results show that the GSSE method is very reliable, simple and can be functionalized to other nonlinear equations. The Galilean transformation has been used to conduct bifurcation analysis on the governing model and the model's behavior has been visually illustrated using phase plane portraits caused by the change in parametric values. The chaotic behavior of the governing model has been analyzed by introducing an extra perturbation term into the existing simulation methods. Furthermore, a symmetry analysis conducted to uncover additional symmetries with greater generality.