Precise soliton solutions of the Landau-Ginzburg-Higgs equation via the Sardar sub-equation method

In the realm of nonlinear mathematical physics, the Landau-Ginzburg-Higgs (LGH) equation stands as a pivotal model for understanding complex physical phenomena, including superconductors, phase transitions, and particle interactions. This study applies the Sardar Sub-Equation (SSE) method to derive exact soliton solutions for the LGH equation, unveiling diverse wave structures such as kink-shaped, M-shaped, cuspon, and periodic solitons. The computational analysis, carried out using Maple software, provides a robust framework for exploring these nonlinear structures. The results demonstrate the efficiency of the SSE method in obtaining precise analytical solutions, contributing valuable insights into the dynamics of nonlinear wave propagation. These findings have significant implications across various scientific disciplines, including quantum mechanics, material science, and high-energy physics, further establishing the SSE method as an effective tool for solving nonlinear partial differential equations (PDEs). This study contributes significantly to the field of nonlinear mathematical physics by introducing the Sardar Sub-Equation (SSE) method for solving the Landau-Ginzburg-Higgs (LGH) equation. The major contributions are: