Abstract
PurposeThis 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/approachThe Painlevé analysis confirms the complete integrability of both Kairat-II and Kairat-X equations.FindingsThis 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/implicationsThe Hirota’s bilinear algorithm is used to furnish a variety of solitonic solutions with useful physical structures.Practical implicationsThis 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 implicationsThe 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/valueThis paper presents an original work that presents two newly developed Painlev\\'{e} integrable models with insightful findings.
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