Bäcklund transformation for the Krichever-Novikov Equation

Abstract

(u 2 xx − r(u)) + cux, r (5) = 0 (1) appeared (up to change u = p(ũ), ˙p 2 = r(p)) in [1] for the first time in connection with study of finite-gap solutions of the Kadomtsev-Petviashvili equation. The distinctive feature of the equation (1) is that, accordingly to [2], no differential substitution exists connecting it with other KdV-type equations. This property impedes the construction of the Bäcklund transformation (BT) which in other cases can be obtained by composition of two differential substitutions. Nevertheless, we demonstrate that (1) admits BT which connects it with other equation of the same form vt = vxxx − 3