Darboux transformations in one independent variable have found numerous applications in various field of mathematics and physics. In this paper we show that the extension of these transformations to two dimensions can be used to decouple systems of Schrodinger equations and provide explicit representation for three classes of such systems. We show also that there is an elegant relationship between these transformations and analytic complex matrix functions.
Invariants for Darboux transformations of Arbitrary Order for $D_x D_y +aD_x + bD_y +c$
Separation of variables in the Kramers equation
A practical approach to solve coupled systems of nonlinear partial differential equations
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