A Method for Obtaining Darboux Transformations

For integrable equations which can be solved by the Inverse Scattering Transform, there exist Bäcklund transformations (BTs) [1]. These transformations were first discovered for the Sine-Gordon equation at the end of the 19th century. Usually they are treated as nonlinear superpositions, which allow one to create new solutions of a nonlinear evolution equation from a finite number of known solutions. In practice, BTs are not very straightforward to apply in the construction of multisolutions. On the other hand, the Darboux transformation (DT) is a very convenient way of constructing new solutions of nonlinear integrable equations [2]; the algorithm is purely algebraic and can be continued successively. Therefore, it is interesting to transform BTs into DTs. Many integrable equations of the form