Painlevé Analysis and Kink-Type Solitary Waves of the Geophysical KdV Equation Involving a Source

The geophysical KdV equation is used to explore the propagation of oceanic waves. In the present paper, the geophysical KdV equation involving a source (The source is a polynomial of degree \(n\) $n$ in the unknown function) is formally introduced. Through the Painlevé analysis, it is shown that the geophysical KdV equation with the source is not integrable. Under some necessary conditions for integrability, several kink-type solitary waves to the special cases of the governing model when \(n = 2\) $n=2$ and \(n = 4\) $n=4$ are derived using the classical Kudryashov method.