Nonclassical symmetries and Bäcklund transformations. (English) Zbl 0784.35101
Summary: 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.
MSC:
35Q53 | KdV equations (Korteweg-de Vries equations) |
58J70 | Invariance and symmetry properties for PDEs on manifolds |
58J72 | Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds |