Equivalence transformations for nonlinear evolution equations

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Abstract
A systematic approach to the study of nonlinear evolution equations based on the theory of the equivalence transformations is suggested. In this paper it is applied to the Burgers and to the Korteweg–de Vries equations. The main result is that the Hopf–Cole transformation for the Burgers equation and the Miura, Bäcklund, and Hirota transformations for the Korteweg–de Vries equation (together with the linear equations of the inverse scattering theory) are all deduced from a single general equivalence condition.