Higher-order Darboux transformations with foreign auxiliary equations and equivalence with generalized Darboux transformations

We show that a recently developed modified Darboux transformation that uses foreign auxiliary equations, can be unified with the Darboux transformation for generalized Schrodinger equations. As a consequence of this unification, we obtain explicit Darboux transformations with foreign auxiliary equations of arbitrary order. (C) 2012 Elsevier Ltd. All rights reserved.