Painlevé integrability and analytical solutions of variable coefficients negative order KdV–Calogero–Bogoyavlenskii–Schiff equation using auto-Bäcklund transformation

The generalized form of the time-dependent variable coefficients negative order KdV–CBS equation, which represents the interactions of long wave propagations and have the numerous applications in the field of quantum and fluid mechanics, is being examined in this article. The study of Painlevé analysis effectively produced an integrable version of the considered equation. The several analytical solutions for this equation in the categories of exponential and rational functions have been reported by utilising auto-Bäcklund transformation approach. The periodic, kink-antikink, kink-soliton, and anti-kink soliton wave solutions have all been obtained using this method. To illustrate the physical applicability of the developed solutions, the results are graphically shown.