Bilinear form, Bäcklund transformation to the Kairat-II-X-extended equation: <i>N</i> -soliton, anti-kink soliton, novel soliton molecule, multi-lump and travelling wave solutions

This paper aims to explore the Kairat-II-X-extended equation, which has an important part in the field of plasma physics. Making use of the Bell polynomials, the bilinear form of the considered equation is established via constructing the P-polynomials. Based on the bilinear form, we develop a bilinear Bäcklund transformation (BT), which includes four equations involving with four free parameters. Then the anti-kink soliton solution is extracted on the basis of the bilinear BT. Moreover, the N-soliton solutions are obtained by using the Hirota bilinear method and the novel soliton molecule is derived via adding the resonance constraint condition. Additionally, the multi-lump wave solutions are also extracted with the extended homoclinic test approach. Eventually, we probe the travelling wave solutions through the Kudryashov approach. In view of the obtained exact nonlinear wave solutions, we graphically depict their wave structures through the 3D, density and contour plots, and give the corresponding physical explanations.