A novel hybrid analytical–PINNs approach for data-driven soliton dynamics and parameter discovery in the Kairat-II-X equationGhulam Hussain Tipu, AllahBakhsh Yazdani Cherati, Hamid Momeni and Fengping YaoEur. Phys. J. Plus, 140 9 (2025) 914DOI: https://doi.org/10.1140/epjp/s13360-025-06850-8