The nonlinear wave behavior in the tropical and mid-latitude troposphere has been simulated using the space-time fractional Landau-Ginzburg-Higgs model. These waves are the consequence of interactions between equatorial and mid-latitude waves, fluid flow in dynamic systems, weak scattering, and extended linkages. The mEDAM method has been used to obtain new and extended closed-form solitary wave solutions of the previously published nonlinear fractional partial differential equation via the beta derivative. A wave transformation converts the fractional-order equation into an ordinary differential equation. Several soliton, single, kink, double, triple, anti-kink, and other soliton types are examples of known conventional wave shapes. The answers are displayed using the latest Python code, which enhances the usage of 2D and 3D plotlines, as well as contour plotlines, to emphasise the tangible utility of the solutions. The results of the study are clear, flexible, and easier to replicate.

For the generalized integrable (2+1)-dimensional nonlinear Schrödinger system, new and creative analytical solutions were derived using a novel extended direct algebraic method incorporating conformable derivatives, which could be expressed in terms of elementary functions and yielded a variety of analytical solutions, such as single, optical periodic, and wave solitons. The analytical solutions provided key insights into the effects of conformable derivatives and temporal parameters on the behavior of optical solitons, such as their stability, propagation, and interaction. To further elucidate the dynamics of these solitons, 2D, 3D, and contour plots were created to provide a visual representation of the soliton's form, amplitude, and phase. This helps to better understand the behavior of the soliton and its potential applications in nonlinear equations. Based on the study's demonstration of the extended direct algebraic method's strength and versatility in obtaining analytical solutions for complex non linear systems, it may be a useful tool for solving a variety of nonlinear problems in science and engineering.