Exploring multiple soliton solutions and lump wave solutions to two integrable (2+1)-dimensional Kairat-II-X-extended and Kairat-II-X-type equations

Purpose This paper aims to investigate two integrable (2 + 1)-dimensional Kairat-II-X-extended and Kairat-II-X-type equations. Design/methodology/approach The two equations retain the Painlevé integrability. Findings This study investigates multifaceted solution structures for each model, encompassing multiple soliton configurations, lump wave phenomena and various forms of traveling wave solutions. Research limitations/implications Hirota’s bilinear method is used to furnish these new solutions for each examined model. Practical implications This study further provides a diverse array of periodic solutions, kink solutions and singular solutions for the two integrable models. Social implications This study systematically develops algorithms tailored for analyzing newly formulated systems across diverse domains. Originality/value This study introduces a novel contribution by developing new Painlevé integrable models with distinct structures and useful explorations.