  Thermodynamic Bethe ansatz equations are coupled non-linear integral
equations which appear frequently when solving integrable models. Those
associated with models with N=2 supersymmetry can be related to differential
equations, among them Painleve III and the Toda hierarchy. In the simplest such
case the massless limit of these non-linear integral equations can be solved in
terms of the Airy function. This is the only known closed-form solution of
thermodynamic Bethe ansatz equations, outside of free or classical models. This
turns out to give the spectral determinant of the Schrodinger equation in a
linear potential.
