... , and μ have non-zero real values. In this article, we want to use the Painlevé test to show that the combined K-II-X equation (4) passes the Painlevé integrability test without any restrictions on the parameters involved. The integrability criterion of this problem will provide MS and lump wave...

... 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...