On connection formulas for the second Painleve transcendent. Proof of the Miles conjecture, and a quantization rule

The method of isomonodromy deformations is used to prove connection formulas for the second Painleve transcendent, which is exponentially decreasing on one side of a turning point and has a Kuzmak–Luke–Whitham decomposition on the other. The phase advance turns out to be equal to $\pi/2$ ($\operatorname{mod}\pi$). These connection formulas lead to the determination of the asymptotics of the eigenvalues for the Sturm–Liouville equation with a cubic nonlinearity.