Nonclassical symmetries and Bäcklund transformations

In this paper we show how to obtain well-known Bäcklund transformations of several equations by applying the nonclassical symmetries method. In particular, we derive the Bäcklund transformation between the modified Korteweg-de Vries and the Korteweg-de Vries equations, the Bäcklund transformation between the Calogero and the third-order linear equations, the auto-Bäcklund transformations of the potential Korteweg-de Vries, the potential modified Korteweg-de Vries equation, and the Burgers equations. As a consequence, the n-solitons solution of the Korteweg-de Vries equation can be obtained by the nonclassical symmetries method.