The Formation of Invariant Exact Optical Soliton Solutions of Landau-Ginzburg-Higgs Equation via Khater Analytical Approach

This work aims to enhance our comprehension of the dynamical features of the nonlinear Landau-Ginzburg-Higgs evolution equation, which provides a theoretical framework for identifying various phenomena, such as the formation of superconducting states and the spontaneous breakdown of symmetries. When symmetry breaking is involved in phase transitions in particle physics or condensed matter systems, the Landau-Ginzburg-Higgs model combines the ideas of the Landau-Ginzburg theory and the Higgs mechanism. The equation plays a crucial role in characterizing the Higgs field and its related particles, including Higgs boson. In a standard model of the particle physics, Higgs mechanism explains precisely how mass is acquired. The Lie invariance requirements are taken into account by the symmetry generators. The method produces a 3-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3-$$\end{document}dimensional Lie algebra of the Landau-Ginzburg-Higgs model with translational symmetry (dilation or scaling) and translations in the space and the time associated with the mass and energy conservation. It is shown to be the optimal sub-algebraic system after similarity reductions are also performed. The next wave transformation method reduces the governing system to ordinary differential equations and yields a large number of exact travelling wave solutions. The Khater approach is used to solve an ordinary differential equation and investigate the closed-form analytical travelling wave solutions for the considered diffusive system. The obtained results include a singular, mixed singular, periodic, mixed trigonometric, complex combo, trigonometric, mixed hyperbolic, plane, and combined bright-dark soliton solution. The results of the sensitivity analysis demonstrate how vulnerable the suggested equation is to various initial conditions. The findings are visually displayed in contour, three-dimensional, and two-dimensional forms to emphasize the features of pulse propagation.