Propagation of solitary waves for hydrodynamical nonlinear complex model in a fractional derivative setting.

This study investigates soliton solutions and dynamic wave behaviors in the complex Ginzburg-Landau equation, a model that plays a central role in describing diverse physical phenomena such as superconductivity, nonlinear optical fibers, liquid crystals, second-order phase transitions, and field theory strings. To derive closed-form solutions, we employ two advanced analytical techniques: the new rational extended sinh-Gordon equation expansion method (ShGEEM) and the modified generalized exponential rational function method (mGERFM). These methods yield a wide range of solitonic structures, such as complex and singular solitons, oscillatory periodic waves, bright, dark, and multi-wave profiles. In this work, new families of exact solitary wave solutions with ShGEEM and several hyperbolic, trigonometric, and exponential solutions with mGERFM are presented. Further, the obtained solutions are checked for accuracy by substituting them back into Mathematica. For the dynamics of solutions, 2D plots, 3D surfaces, and contour graphs have been constructed for some values of parameters in the presence of the [Formula: see text]-fractional derivative to understand wave structures and their evolution. In general, the present study not only consolidates the aspects of nonlinear wave dynamics in the field of chemical and physical oceanography but also provides pathways for further research on nonlinear fractional-order models. The originality of the present study lies in the point that the complex Ginzburg-Landau equation has not been studied within the ShGEEM and mGERFM frameworks.