Extended new Painlevé integrable KdV-CBS equation: multiple shocks, lump, breathers, and other physical wave solutions

In this work, we construct a new evolutionary equation with multiple applications in fluids and engineering. We call it the extended (3+1)-dimensional KdV-CBS equation, an extension of the (3+1)-dimensional KdV-Calogero-Bogoyavlenskii-Schiff (KdV-CBS) equation. We apply the Painlev & eacute; integrability test to examine the compatibility conditions of this new extended model before analyzing and solving it. Subsequently, we implement the simplified Hirota's method (SHM) to analyze this model, deriving multiple soliton/shock and lump solutions, as well as breather wave solutions, based on the derived dispersion relation, with the assistance of advanced computational programs like Maple and Mathematica. Furthermore, many other techniques, such as the Tanh method and different exponential formulas, will be used to derive different physical solutions that may simulate many nonlinear phenomena that arise in fluid or plasma physics.