Darboux transformations for the reduced wave equation and its parabolic approximation

The reduced wave equation with a variable wave speed and its parabolic approximation is considered. For each equation, a Darboux transformation is constructed. For the reduced wave equation, a Darboux transformation is obtained linking solutions to Laplace's equation. For the parabolic approximation equation, a transformation is obtained linking solutions to Schrodinger's equation. It is shown that these transformations exist for the same class of variable wave speed.