Abstract
In the field of quantum mechanics and fluid physics, especially in the study of nonlinear geometric optics and superconductivity, the Landau-Ginzburg-Higgs (LGH) and the (2 + 1)-dimensional Novikov-Veselov (NV) equations are two significant models. In this article, we have affirmed that wave profile changes with the change of the free parameters associated with them and are mainly dominated by linear effects. The effects of nonlinearity and wave speed on the wave contours have also been analyzed. On account of this, wave solutions are computed concerning rational, hyperbolic, and trigonometric structures balancing the exponents of linear and nonlinear terms of the highest order, from which scores of typical wave profiles including kink, bell-shape soliton, lump, and periodic waves have been extracted. The wave solutions are designated through extending the typical concept of the sine-Gordon expansion (SGE) method from the lower dimensional to the higher dimensional nonlinear evolution equations. This study establishes the capability of the stated method in solving both lower and higher dimensional nonlinear evolution equations. The solutions are analyzed by sketching figures for different values of related variables, and it is observed that the attributes of these solutions are pivotal in the selection of parameters.