Solitary waves pattern appear in tropical tropospheres and mid-latitudes of nonlinear Landau–Ginzburg–Higgs equation with chaotic analysis

The objective of this research is to investigate the nonlinear Landau–Ginzburg–Higgs equation, which characterizes nonlinear solitary waves exhibiting distant and feeble scattering interactions among tropical tropospheres and mid-latitudes. Additionally, the study will examine the interchange of mid-latitude Rossby waves and equatorial waves within this context. In this research article, we focus on obtaining exact traveling wave solutions for the Landau–Ginzburg–Higgs equation using a new extended direct algebraic technique. The obtained soliton solutions include various types such as combined and multiple bright-dark, periodic, bright, and multiple bright-periodic. We present these soliton solutions graphically by varying the involved parameters using the advanced software program Wolfram Mathematica. The graphical representations allow us to visualize the behavior of the wave velocity and wave number as the parameters change. Additionally, we conduct a chaotic analysis to examine the wave profiles of the newly designed dynamical framework. The results of this analysis demonstrate the reliability and efficiency of the proposed method, which can be applied to find closed-form traveling wave solitary solutions for a wide range of nonlinear evolution equations.