Two new Painleve-integrable extended Sakovich equations with (2+1) and (3+1) dimensions

Purpose The purpose of this paper is to introduce two new Painleve-integrable extended Sakovich equations with (2 + 1) and (3 + 1) dimensions. The author obtains multiple soliton solutions and multiple complex soliton solutions for these three models. Design/methodology/approach The newly developed Sakovich equations have been handled by using the Hirota's direct method. The author also uses the complex Hirota's criteria for deriving multiple complex soliton solutions. Findings The developed extended Sakovich models exhibit complete integrability in analogy with the original Sakovich equation. Originality/value This paper gives two Painleve-integrable extended equations which belong to second-order PDEs. The two developed models do not contain the dispersion term u(xxx). This paper presents an original work with newly developed integrable equations and shows useful findings.