The Cole-hopf and Miura Transformations Revisited

An elementary yet remarkable similarity between the Cole-Hopf transformation relating the Burgers and heat equation and Miura's transformation connecting the KdV and mKdV equations is studied in detail. 1. Introduction Our aim in this note is to display the close similarity between the well-known Cole{Hopf transformation relating the Burgers and the heat equation, and the celebrated Miura transform connecting the Korteweg{de Vries (KdV) and the modiied KdV (mKdV) equation. In doing so we will introduce an additional twist in the Cole{Hopf transformation (cf. (1.28), (1.29)), which to the best of our knowledge, appears to be new. Moreover, we will reveal the history of this transformation and uncover several instances of its rediscovery (including those by Cole and Hopf). We start with a brief introductory account on the KdV and mKdV equations. The KdV equation 42] was derived as an equation modeling the behavior of shallow water waves moving in one direction by Korteweg and his student de Vries in 1895 1. The landmark discovery of the inverse scattering method by Gardner, Green, Kruskal, and Miura in 1967 19] (cf. also 20]) brought the KdV equation to the forefront of mathematical physics, and started the phenomenal development involving multiple disciplines of science as well as several branches of mathematics. The KdV equation (in a setting convenient for our purpose) reads