Numerical Validation of Analytical Solutions for the Kairat Evolution Equation

This study undertakes a comprehensive analytical and numerical investigation of the nonlinear Kairat model, a significant evolution equation that governs a wide range of physical phenomena, including shallow water waves, plasma physics, and optical fibers. The Kairat model effectively describes the propagation of nonlinear waves in shallow water, capturing the intricate interplay between nonlinearity and dispersion. It exhibits similarities with well-known nonlinear evolution equations such as the Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations, thereby offering insights into their common underlying dynamics. To achieve the objectives of this research, we employ the modified Khater (MKhat) and unified (UF) methodologies to derive exact solutions for the Kairat model. Furthermore, the trigonometric-quantic-B-spline (TQBS) scheme is utilized as a numerical technique to verify the accuracy of these derived solutions and validate their applicability within the domain of shallow water wave propagation. This investigation yields a collection of innovative and precise analytical solutions, elucidating the complex nonlinear behavior of the Kairat model and its effectiveness in capturing the dynamics of shallow water waves. Moreover, these analytical solutions are corroborated through numerical simulations conducted using the TQBS scheme, ensuring their reliability and practical significance in understanding and predicting shallow water wave phenomena. The significance of this endeavor lies in its contribution to a deeper understanding of the dynamics of the Kairat model and its potential applications in fields such as coastal engineering, oceanography, and related disciplines. The integration of analytical and numerical techniques offers new perspectives and methodologies for exploring nonlinear evolution equations, potentially benefiting researchers in applied mathematics, physics, and engineering. In summary, this comprehensive analytical and numerical investigation provides novel insights, precise solutions, and a robust foundation for further exploration of the physical implications and applications of the Kairat model in the context of shallow water wave propagation.