Reduction of KdV and Cylindrical KdV Equations to Painlevé Equation

Abstract

Similarity solutions of the KdV and cylindrical KdV equations are studied by means of Lie’s method of infinitesimal transformation groups. It is shown that the KdV equation is reduced to the Painlevé transcendental equation of the first or second kind. The similarity solutions of cylindrical KdV equation also satisfy the first or second Painlevé equation, and a soliton-like solution is obtained.