Nonlinear water waves (KdV) equation and Painleve technique

Abstract

The Korteweg-de Vries (KdV) equation which is the third order nonlinear PDE has been of interest since Scott Russell (1844) . In this paper we study this kind of equation by Painleve equation and through this study, we find that KdV equation satisfies Painleve property, but we could not find a solution directly, so we transformed the KdV equation to the like-KdV equation, therefore, we were able to find four exact solutions to the original KdV equation.