Trilinearization of the Kairat-Ⅱ equation for the soliton solutions with machine learning evolution and Painlevé analysis

This research examines the integrability and soliton solution in different forms of the Kairat-Ⅱ equation. A thorough Painlevé analysis determines that the equation fulfills the criteria of compatibility conditions for integrability; thus, it is shown to exhibit soliton solutions. Applying the Hirota D-operator, we transform the Kairat-Ⅱ equation into a trilinear form. This enables the derivation of precise analytical solutions, including novel breather wave and three-wave soliton solutions by the Hirota method. These solutions reveal localized, oscillatory, and energy-exchange structures, capturing the complex nonlinear dynamics of the model. We use a machine learning analysis, the multi-layer perceptron regressor algorithm, for understanding the dynamics of these solitons, which effectively predicts the progression and dynamics of the obtained solutions. The analytical results are validated and displayed in 2D and 3D+contour plots using symbolic computation tools such as Mathematica and Python. In the end, we execute the asymptotic analysis on the gained solution and present the asymptotic behavior of the solutions. We can learn a great deal about the intricate dynamics that the Kairat-Ⅱ model governs from these visualizations.