Asymptotic behavior of soliton solutions of the Kairat-X model via the Hirota bilinear method: Painlevé integrability and machine learning analysis

In this article, we investigated the integrability of the nonlinear dynamical Kairat-X model through Painlevé analysis, demonstrating that the equation satisfies the Painlevé property and is therefore integrable. We applied the bilinear Hirota method to derive several exact solutions, including breather wave, novel periodic wave, periodic cross-kink wave, kink-rogue wave interaction, and one-soliton and two-soliton solutions. A machine learning multi-layer-perceptron regressor algorithm was applied to represent the behavior of the actual, and to predict, the above solutions. Furthermore, we employed an asymptotic analysis on the gain solutions to expect the demonstration of the asymptotic behavior of these analytical solutions. The soliton solutions obtained were novel and exhibited improved reliability compared to previously reported results. These findings were further validated using symbolic computation software. A comparison with the existing literature revealed that the proposed solutions were more applicable and accurate. Several of the results were visualized using two-dimensional, three-dimensional, and contour surface plots.