Extended (3 + 1)-dimensional Kairat-II and Kairat-X equations: Painlevé integrability, multiple soliton solutions, lump solutions, and breather wave solutions

Purpose This study aims to investigate two newly developed (3 + 1)-dimensional Kairat-II and Kairat-X equations that illustrate relations with the differential geometry of curves and equivalence aspects. Design/methodology/approach The Painlevé analysis confirms the complete integrability of both Kairat-II and Kairat-X equations. Findings This study explores multiple soliton solutions for the two examined models. Moreover, the author showed that only Kairat-X give lump solutions and breather wave solutions. Research limitations/implications The Hirota’s bilinear algorithm is used to furnish a variety of solitonic solutions with useful physical structures. Practical implications This study also furnishes a variety of numerous periodic solutions, kink solutions and singular solutions for Kairat-II equation. In addition, lump solutions and breather wave solutions were achieved from Kairat-X model. Social implications The work formally furnishes algorithms for studying newly constructed systems that examine plasma physics, optical communications, oceans and seas and the differential geometry of curves, among others. Originality/value This paper presents an original work that presents two newly developed Painlev\'{e} integrable models with insightful findings.