Non-linear Water Waves (KdV) Equation and Painleve Property and Schwartzian Derivative

Abstract

The Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation which describes
the motion of water waves, has been of interest since John Scott Russell (1844) [1]. In
present work we study this kind of equation and through our study we found that the KdV equation
passes Painleve’s test, but we could not locate the solution directly, so we used Schwartizian
derivative technique. Therefore, we were able to find two new exact solutions to the KdV equation.
Also, we used the numerical method of Modified Zabusky-Kruskal to describe the behavior
of these solutions.