This research is aimed at finding exact soliton solutions of the nonlinear fractional Kairat-X equation, which describes soliton behavior in nonlinear media and has applications in quantum physics, materials science, signal processing, and telecommunications. We use a unified method that generalizes the tanh-function method to find new exact soliton solutions in trigonometric, hyperbolic, and plane wave forms. Computational simulations with fixed parameters are performed to produce two-dimensional and three-dimensional visualizations, e.g., contour and density plots, representing the physical properties of the derived solitons. The simulations result in the identification of several soliton types, namely kink wave solitons, dark solitons, bright solitons, and periodic wave solitons. Our results increase the knowledge of the solution properties of the Kairat-X equation and give a platform to interpret a variety of significant physical phenomena. The systematicity and stability of our methodology prove its usefulness as a device for solving other nonlinear partial differential equations in applied physics and mathematics, always returning different exact solutions.

The paper investigates the soliton dynamics and stability analysis of the time-fractional higher-order nonlinear Schrödinger equation. A Caputo time-fractional derivative is included in the time fractional higher-order nonlinear Schrödinger equation, along with dispersive higher-order and nonlinear terms, which allow us to describe wave propagation in arbitrarily complex nonlinear and dispersive media in greater detail. Through the use of the ${\varphi }^6$-model expansion method, a vast range of precise analytical soliton solutions is obtained, comprising nonlinear, regular, and singular periodic solitons. The effects of the fractional higher-order physical parameters on the amplitude, width, and nature of these solitons are systematically examined. Moreover, the modulational instability is studied through the use of linear stability analysis. Plots are given in order to explain the change in the form and stability behavior, and the evolution of the soliton solutions as the parameters vary.