On the comparative analysis for the fractional solitary wave profiles to the recently developed nonlinear system

We study the comparative exact solutions of the Kairat-II equation describing the physical behaviors of nonlinear systems. The newly introduced Kairat equation has numerous applications in the fields of plasma physics, optical communications, differential geometry engineering, oceanography and physics. Two types of fractional operators known as beta and M-truncated derivatives have been applied for creating the complex fractional Kairat-II equation. The two newly integrated methods, known as modified generalized Riccati equation mapping method and the generalized exponential rational function method approach, are under consideration to investigate the governing system. The implemented methodologies are distinguished by their efficacy, straightforwardness, and flexibility, which allow for the integration of diverse types of soliton solutions within a unified framework. In addition, visualize the solution behaviors with different parameter values, we plot the different graphs with the associated parameter values under the effect of -truncated and -fractional derivatives.