Dynamic behavior of optical self-control soliton in a liquid crystal model

The Paraxial Model, a foundational concept in optics, provides a simplified yet effective approach to understanding the behavior of light in optical systems. This research investigates optical soliton solutions within the truncated time M-fractional Paraxial wave equation using the extended tanh method and the modified extended tanh method. Addressing the challenge of fractional order, we employ the shortened M-fractional derivative to eliminate it from the governing model, facilitating a detailed analysis. Through strategic manipulation of free parameters, soliton solutions are expressed in terms of hyperbolic and trigonometric functions, revealing a spectrum of soliton phenomena such as Lump soliton, rough wave, periodic soliton, and king wave for specific parameter values that are connected to the dispersal effect, the Kerr non-linearity effect and the diffraction effect of the governing equation. To enhance the understanding of these solutions, we utilize Matlab for graphical representations, providing a visual interpretation of the intricate dynamics. This study not only uncovers diverse soliton behaviors in the context of fractional wave equations but also offers valuable insights into their formation and complexity, bridging theoretical analysis with computational visualization. •We suggest some new form of solitons of Paraxial model through two methods.•We have explained the dispersal, the Kerr non-linearity and the diffraction effect of the Stated equation to provide the lump wave, rogue wave and periodic wave.•We study the model better by observing the change in the constants and what their impact is on the solitary waves which has not analyzed in earlier literatures.•We have studied the model in this way and presented figures showing the correctness of what we hoped to reach from the proposed model.•In addition, we presented many forms that show that the results we reached are a clear contribution to this field.