Dynamical Analysis of Wave Solutions for the Complex Ginzburg−Landau and (4 + 1)-Dimensional Fokas Equations With Beta Derivative

This study propounds a detailed report on the exact wave solutions of the complex fractional Ginzburg−Landau equation (CFGLE) with quadratic-cubic law nonlinearity and (4 + 1)-dimensional fractional Fokas equation (FFE). For this purpose, the aforementioned equations are transformed into nonlinear ordinary differential equations (NLODEs) within the scope of the beta fractional derivative. Then, the new version trial equation method (NVTEM) is applied effectively and featly to derive the rational, exponential, hyperbolic, and Jacobi elliptic−type function solutions. The evaluated solutions are verified by means of the Symbolic Software. Some of the achieved wave solutions are presented in 2D and 3D graphics to advance the essential propagated properties and to present new solitary-wave structures. The investigated results demonstrate that the proposed analytical method is not only a vigorous mathematical tool but also can be used for various nonlinear models to reveal the exact solutions. This analysis improves the understanding of the wave behaviors in various fields such as optics, plasma physics, fluid mechanics, and engineering.