Since the 1970s’ pioneering works of M.J. Ablowitz and H. Segur, and B.M. McCoy, C.A. Tracy and T.T. Wu the six Painleve functions have been playing the same role in nonlinear mathematical physics that the classical special functions, such as Airy functions, Bessel functions, etc., are playing in linear physics. In this context, one of the most important open problems is the study of the distribution of singularities of special solutions of the Painleve first equation.

In my seminar I will address this issue. I will show that the singularities distribution is strictly related with the (stationary) Schroedinger equation with a cubic potential (also called cubic oscillator). I will then use this relation to obtain the asymptotic distribution of the singularities through the WKB analysis of the Schroedinger equation.