Exploration of Soliton Dynamics and Chaos in the Landau-Ginzburg-Higgs Equation Through Extended Analytical Approaches

The goal of this work is to improve our understanding of the dynamical aspects of the nonlinear Landau-Ginzburg-Higgs (LGH) equation, which offers a theoretical framework for characterizing several phenomena, including the spontaneous breakdown of symmetries and the emergence of superconducting states. The proposed model integrates the concepts of the Higgs mechanism and the Landau-Ginzburg theory when symmetry breaking appears in phase transitions in particle physics or condensed matter systems. The equation is essential for describing the Higgs field and its constituent particles, such as the Higgs boson. New extended direct algebraic approaches and modified F-expansion techniques are used to address this problem. The obtained solutions, which include kink, anti-kink, bright, dark, and periodic solitons, are essential because they shed light on the stability and nonlinear dynamics of field theories that are pertinent to cosmology and condensed matter physics. To advance the essential propagating features, a few obtained solutions are presented as 3D, contour, and 2D graphics by applying certain values to the parameters under the given constraints. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. Additionally, the chaos analysis is carried out to show the quasi-periodic and periodic chaotic patterns. The sensitivity analysis of the studied model is also looked at and presented at different initial conditions. Furthermore, we guarantee that every found solution is precise, effective, and a great addition to the literature of solitary wave theory. To calculate the soliton solutions for nonlinear models in communication engineering and operations research, the methods utilized in this work to derive inclusive and standard solutions are more accessible, effective, and quick.