Solitary and soliton solutions of the nonlinear fractional Chen Lee Liu model with beta derivative.

The nonlinear Chen-Lee-Liu (NCLL) model is a crucial mathematical model for assessing optical fiber communication systems. It incorporates various factors, including noise, dispersion, and nonlinearity, which can influence signal quality and data transmission rates within optical fiber networks. The NCLL model can be employed to optimize the design of optical fiber systems. In this study, we investigated solitary and soliton solutions applicable to the optics of the NCLL model with a beta derivative by utilizing a new extended hyperbolic function (NEHF) and generalized exponential rational function (NGERF) methods. Using symbolic computations, the NEHF method generates closed-form solutions to the NCLL equation, which is expressed in hyperbolic, trigonometric, polynomial, and exponential form. By contrast, the NGERF method generates closed-form solutions described in hyperbolic, trigonometric, and exponential forms, offering various solution types. The model exhibits various soliton solutions, including periodic oscillating nonlinear waves, kink-wave profiles, multiple soliton profiles, singular solutions, mixed singular solutions, mixed hyperbolic solutions, periodic patterns with anti-troughs and anti-peaked crests, mixed periodic solutions, mixed complex solitary shock solutions, mixed shock singular solutions, mixed trigonometric solutions, and periodic solutions. Using symbolic computation tools, such as Mathematica 11.3 or Maple, these newly derived soliton solutions were verified by substituting them back into the corresponding system. The findings of this study demonstrate that the applied methodologies are reliable, efficient, and capable of generating optical soliton solutions for more complex wave equations in optical fiber communication systems.