On the Darboux transformation of second order ordinary differential operator (Q1116004)

The Darboux transformation of the second order differential operator \(L(u)=\partial^ 2-u(x)\) in the complex domain is investigated. Define the differential polynomials \(Z_ n(u)\) by the recursion relation \(Z_ n(u)=\partial^{-1}(2^{-1}u'+u\partial -4^{-1}\partial^ 3)Z_{n- 1}(u),\) \(n\in {\mathfrak N}\) with \(Z_ 0(u)=2\). Put \(C_{\pm}(\zeta)=\pm \partial +2(\partial /\partial x)\log (\xi_ 1y_ 1(x)+\xi_ 2y_ 2(x)),\) where \(y_ 1(x)\) and \(y_ 2(x)\) are the fundamental systems of solutions of \(L(u)y=0\) and \(\zeta =[\xi_ 1:\xi_ 2]\in {\mathfrak P}_ 1\). The identity \(C_+(\zeta)Z_ n(u^*(x;\zeta))=C_-(\zeta)Z_ n(u(x)),\) \(n=0,1,2,..\). and its application to the transformation theory of the higher order KdV equation are discussed, where \[ u^*(x;\zeta)=u(x)-2(\partial /\partial x)^ 2\log (\xi_ 1y_ 1(x)+\xi_ 2y_ 2(x)) \] is the coefficient of the Darboux transformation of L(u). Moreover the Darboux transformation in the elliptic case is also mentioned.