Study of a combined Kairat-II-X equation: Painlevé integrability, multiple kink, lump and other physical solutions

Purpose This study aims to explore a novel model that integrates the Kairat-II equation and Kairat-X equation (K-XE), denoted as the Kairat-II-X (K-II-X) equation. This model demonstrates the connections between the differential geometry of curves and the concept of equivalence. Design/methodology/approach The Painlevé analysis shows that the combined K-II-X equation retains the complete Painlevé integrability. Findings This study explores multiple soliton (solutions in the form of kink solutions with entirely new dispersion relations and phase shifts. Research limitations/implications Hirota’s bilinear technique is used to provide these novel solutions. Practical implications This study also provides a diverse range of solutions for the K-II-X equation, including kink, periodic and singular solutions. Social implications This study provides formal procedures for analyzing recently developed systems that investigate optical communications, plasma physics, oceans and seas, fluid mechanics and the differential geometry of curves, among other topics. Originality/value The study introduces a novel Painlevé integrable model that has been constructed and delivers valuable discoveries.