Investigation of lump, breather and multi solitonic wave solutions to fractional nonlinear dynamical model with stability analysis

In the current research, the new extended direct algebraic method (NEDAM) and the symbolic computational method, along with different test functions, the Hirota bilinear method, are capitalized to secure soliton and lump solutions to the $(2+1)$-dimensional fractional telecommunication system. Consequently, we derive soliton solutions with sophisticated structures, such as mixed trigonometric, rational, hyperbolic, unique, periodic, dark-bright, bright-dark, and hyperbolic. We also developed a lump-type solution that includes rogue waves and breathers for curiosity’s intellect. These features are important for controlling extreme occurrences in optical communications. Additionally, we investigate modulation instability (MI) in the context of nonlinear optical fibres. Understanding MI is essential for developing systems that may either capitalize on its positive features or mitigate its adverse effects. Also, a comprehensive sensitivity analysis of the observed model is carried out to evaluate the influence of different factors. 3D surfaces and 2D visuals, contours, and density plots of the outcomes are represented with the help of a computer application. Our findings demonstrate the potential of using soliton theory and advanced nonlinear analysis methods to enhance the performance of telecommunication systems.