Soliton solutions to the time-fractional Kudryashov equation: Applications of the new direct mapping method

In this paper, we analyze the dynamic characteristics of the well-known Kudryashov equation with a conformable derivative in the context of pulse propagation within optical fibers. To investigate the equation, we utilize the modified direct method, a robust technique capable of deriving various soliton solutions. The study includes two-dimensional, three-dimensional, and contour plots that illustrate bell-shaped, wave, W-shaped, and mixed dark-bright soliton solutions, highlighting the significance of these novel optical soliton solutions. These forms of the wave soliton and the W-shaped soliton are constructed for the first time for the Kudryashov equation with a conformable derivative. The structure of the new soliton solutions can be used to transmit two signals simultaneously, potentially increasing transmission efficiency. A key contribution of the study is its analysis of the impact of the fractional order parameter and the time parameter on these optical solutions, illustrating how fractional calculus affects soliton behavior. This highlights the significance of the conformable nonlinear Schrödinger model and its relevance to understanding dispersion effects in optical fibers, advancing prior research in the field.