An nth-order Darboux transformation for the one-dimensional time-dependent Schrodinger equation

Solutions for two one-dimensional Schrodinger equations are linked together using an nth-order Darboux transformation. The resulting coefficients for these Darboux transformations give rise to a system of n nonlinear partial differential equations. This system is shown to be equivalent to a matrix Burgers equation which is linearized using a generalized Hopf-Cole transformation. Solving this linear system provides the link between solutions of the two Schrodinger equations and shows how to factor an nth-order Darboux transformation into n first-order Darboux transformations.