SIMILARITY SOLUTIONS OF THE KORTEWEG-DEVRIES EQUATION

In this paper we find the general similarity solution for the Korteweg-de Vries equation in terms of the second transcendents of Painlevé. It is shown that the Bäcklund transformation for the similarity solutions reduces to an algebraic transformation on the derivatives of the starting solution. The Bäcklund transformation is expressed in the language of the second Painlevé transcendents, thus obtaining new interesting properties for these special functions. Some explicit classes of solutions of the Korteweg-de Vries equation are obtained and described. © 1979 Società Italiana di Fisica.