On the Korteweg-de Vries equation: an associated equation (Q1056839)

The purpose of this paper is to describe a relationship between the KdV equation \(u_ t-6uu_ x+u_{xxx}=0\) and its associated equation of the form \(z_ t+z_{xxx}-3z_ xz_{xx}/z=H(t)z.\) The authors first indicated how this connection arises and then used the result to generate explicit solutions to the KdV equation. These solutions represent solitary waves and conoidal waves. In addition, self-similar solutions in terms of Airy functions and Painlevé transcendents have also been found. The authors remarked at the outset that their approach is different from the method of inverse scattering transform and the results are not in the form of solutions to specific initial value problems, but rather in terms of solutions containing arbitrary constants.