Exact Nonlinear Wave Solutions and Interaction Dynamics of the Integrable Kairat-II-X Equation via Improved Riccati Neural Networks

This article studies the nonlinear wave dynamics of the recently introduced integrable combined Kairat-II-X (K-II-X) equation, which combines dynamical features of the Kairat-II and Kairat-X models. The considered model possesses relevance in nonlinear wave propagation, geometric curve dynamics, and localized optical pulse evolution, thereby providing a mathematical framework for describing curvature-driven nonlinear phenomena in higher-dimensional systems. To obtain exact analytical solutions, a symbolic neural analytical framework based on the improved Riccati neural networks (IRNNs) method is employed. The proposed framework integrates trial functions within multilayer neural network structures, where each neuron in the first hidden layer is constructed through solutions of the improved Riccati equation. The symbolic outputs obtained from the neural network computations are subsequently employed as trial functions for the integrable combined K-II-X equation. Using this framework, several classes of exact wave solutions are derived in the form of hyperbolic, trigonometric, rational, including localized solitary waves and interaction-type structures. In particular, the symbolic neural representation produces both single- and multisoliton wave profiles exhibiting nonlinear localization and interaction behavior. Furthermore, representative wave structures are illustrated through two-dimensional, three-dimensional, contour, and density visualizations to examine the qualitative influence of governing parameters on wave amplitude, localization, propagation behavior, and interaction patterns. The reported results demonstrate the capability of the IRNNs framework to generate diverse nonlinear wave structures in integrable higher-dimensional systems and provide a useful analytical reference for future investigations in nonlinear science and applied mathematical physics.