Lie symmetry analysis, conservation laws and soliton solutions by complete discrimination system for polynomial approach of Landau Ginzburg Higgs equation along with its stability analysis

In this article, we study Landau Ginzburg Higgs equation (LGHE) for Lie symmetries (LS), conservation laws (CLs) and soliton solutions via complete discrimination system (CDS) to polynomial method (CDSPM). We also discuss the qualitative analysis for our governing model with the help of stability analysis (SA). LGHE has huge applications in radially inhomogeneous plasma with constant phase relation of ion-cyclotron waves. The equation was introduced by Lev Devidovich-Landau and Vitaly-Lazarevich Ginzburg and it shows unidirectional wave propagation and superconductivity in nonlinear media. Our governing model also discuss the nonlinear waves that comprises of long-range connections and weak scattering in the mid-latitude troposphere (MLT) and tropical phenomenon, where MLT shows the interactions between mid-latitude and equatorial Rossby waves. The concept of LS gives a systematic way to obtain solutions and helps to understand the underlying structure of differential equations. They are linked with Lie algebras and Lie groups which describe continuous symmetries. CLs of differential equations (DEs), possessing both conserved densities and fluxes, are critical in distinct real-life scenarios. These laws govern the evolution and behaviour of physical quantities over space and time. SA is a mathematical technique used to discuss the behaviour of solutions to DEs over time. Phase plane analysis shows plotting solutions in a phase space. This representation assists to analyse the stability of equilibrium points, identify limit cycles, and understand the overall behaviour of the system.