Two new Painlev´e integrable KdV–Calogero-Bogoyavlenskii-Schiff (KdV-CBS) equation and new negative-order KdV-CBS equation

Abstract In this work, we develop two new (3+1)-dimensional KdV–Calogero-Bogoyavlenskii-Schiff (KdV-CBS) equation and (3+1)-dimensional negative-order KdV-CBS (nKdV-nCBS) equation. The newly developed equations pass the Painlev´e integrability test via examining the compatibility conditions for each developed model. We examine the dispersion relation and derive multiple soliton solutions for each new equation.