On the uniqueness of Bäcklund transformations for KdV-type equations

Abstract

The restricted Backlund transformations for the KdV and MKdV equation are uniquely determined up to two real parameters. One of them can be incorporated into the solutions of the equations and is, therefore, inessential. The only possible generalization of the Backlund transformations for the KdV and MKdV equations which can be obtained is through enlarging the number of independent variables of the functions P, Q, i.e. through extending the dependence of the Backlund transformations so that they include zxy, zyy and possibly higher derivatives. The derivation of these more general Backlund transformation by the Clairin's method seems to be very difficult. Theorem 1 was found to be a powerful tool for the proof of the uniqueness of the restricted Backlund transformations for the KdV and MKdV equations. Actually, it was used only for very special cases of its range of validity, and it seems that it may be useful for the investigation of the class of the KdV-type equations which possess a restricted Backlund transformation. We hope to publish the results of that investigation in the near future. The origin of these investigations on Backlund transformations stems from a seminar held with Stanly Steinberg and Kurt Bernardo Wolf. The author is grateful to both of them for his introduction to the problem.