Abundant analytical solutions and diverse solitonic patterns for the complex Ginzburg–Landau equation

This paper explores soliton solutions and dynamic wave structures for the complex Ginzburg–Landau (CGL) equation, which characterizes the propagation of solitons in the presence of a detuning factor. To achieve this, two analytical methods the generalized exponential rational function (GERF) method and the generalized Kudryashov method-are employed to derive a diverse set of novel and closed form solutions. The derived solutions showcase a variety of solitonic phenomena including multi-wave solitons, multi-wave peakon solitons, stripe solitons, kink-wave profiles, periodic oscillating waves and wave-wave interaction profiles for the CGL equation. Symbolic computation is utilized for assistance in the derivation process. The generalized Kudryashov method introduces novel families of exact solitary waves, while the GERF method produces soliton solutions in various forms, such as hyperbolic and trigonometric functions, periodic breather-wave solitons, exponential rational functions, dark and bright solitons, complex multi-wave solutions and singular periodic oscillating wave soliton solutions across different family cases. Verification of these solutions is carried out by back-substituting them into the CGL equation using soft computing via Maple. The significance of these results lies in their contribution to understanding wave propagation and dynamics in the context of the CGL equation, particularly in the fields of physical oceanography and chemical oceanography. The computational tools Mathematica and Maple are extensively utilized for handling the intricate algebraic calculations in this research endeavor.